This is the latest of my web pages. It  was started on  the 4th of July , 2010. My intention is to discuss in an informal manner relevant topics in technology from my own mathematical perspective. The discussions should be understandable for most Internet readers.  I will start each section with the date and a question. My answer will follow by clicking on the underlined word HERE. Should you be interested in more advanced mathematical topics then you might want to visit my other web page at  http://www2.mae.ufl.edu/~uhk/MATHFUNC.htm . There you will find lots of information on ordinary, partial, and integral equations and variational methods plus some of our latest thoughts on prime numbers, arctan approximations, Diophantine equations, integer spirals, formulas for Pi, etc listed in chronological order.

  my e-mail contact is:   kurzweg@ufl.edu    and  home-page at:   https://mae.ufl.edu/~uhk/HOMEPAGE.html


 Let's begin-



July 4, 2010-Are hydrocarbons of inorganic(abiogenic) or organic origin?
 The recent BP oil spill in the Gulf of Mexico and the unabated leakage of oil and methane (about 60,000 barrels a day) from the broken well for a period of now more than 70 days has got me thinking about the origin of this oil located at least 13,000 ft below the bottom of the Gulf. The standard view held by most oil geologists is that it has a biogenic origin and was produced by decaying vegetation under high temperature and pressure. If this is so then the world is indeed running out of oil and we may already have reached a peak in oil recovery. However, there is a small minority (Russian School especially Mendeleev of periodic table fame and Kurdryavtsev, plus some individuals in the United States including Gold and Kenney) who cite evidence for an abiogenic origin involving the chemical conversion of  methane arising from deep within the earth mantle being converted to larger chain hydrocarbons by the high pressures and temperatures existing at greater depth. If the latter group is correct, then it would appear there is an unlimited supply of hydrocarbon oils and gases available, provided one drills deep enough into pockets capable of holding such products, especially methane. After some thinking on the subject, I find myself siding with the minority in believing that most hydrocarbons are indeed of abionic origin. Go HERE  to see my points supporting an abiogenic origin.

 July 8, 2010-How much oil can be expected to leak from a blowout involving an oil well of cross section A and depth L?

 When the leak of oil from BP well in the Gulf of Mexico was first revealed they claimed the spill was controllable with less than 1000 barrels per day spilling into the Gulf. It soon became clear , especially after the underwater films of the leak became available , that things were much worse than the initially announced. The estimated number crept up week by week to the present estimate of 60,000 barrels (2.5 million gallons) per day gushing into the water 5000ft below the sea surface. To give you a rough estimate of how much leakage will actually occur during an oil well  blowout of the type we have been experiencing with BPs Deep Horizon well, we look at things from a fluid mechanics viewpoint.  Go HERE for the details.                                                                                                                       

July 23, 2010-Why is human vision confined the very narrow wavelength electromagnetic range of 4000A<λ<7000A and its hearing confined to a frequency range of 20Hz<f<20,000Hz?

 Of the five human senses of vision, hearing, taste, smell, and touch it is the first two which are perhaps the most important for survival. Our homo sapiens ancestors and those species preceding clearly required acute seeing and hearing capabilities to function in the environment in which they existed. The question that arises is how were these senses developed in the process of evolution. Go HERE for our thoughts on the matter. 


July 29, 2010-What is the rate of energy consumption in the United States and the World and what are the energy sources available now and in the future?

The world consumption of energy is estimated to be about   474 exajoules=474x1018J of which 88% comes from fossil fuels.

This implies an energy consumption rate of-


The sources for this energy are oil, coal, gas, hydro, nuclear, and renewables in descending order. Go HERE for further discussions.

July 30, 2010-Was Malthus right?

A little over two-hundred years ago the Anglican clergyman Thomas Malthus (1766-1834) published his famous book �An Essay on the Principles of Population�. In it he pointed out that (1)human population will increase with an increase of subsistence resources,(2) that such an increase will inevitable produce a population over shoot, and (3) that this will lead to a decrease in  population levels until it falls below the subsistence level. After that the process repeats itself. Go HERE for details of why I think he was right.

August 25, 2010- What are the energy densities of various energy conversion processes and how do their magnitudes suggest a direction the world will have to take to meet its ever increasing energy demands?

Most of us back in our elementary physics courses in high school and undergraduate college learned about the various different forms of energy ranging from kinetic energy  Ek=(1/2)mv2,  through chemical energy Ec, to nuclear where En=mc2. We want here to look at the concept of energy content per volume and use it to point out some reasonable directions the world might take to meet its future energy needs. Go HERE for the discussion.

September 23, 2010- How can one determine in a non-destructive manner if a gold bar is made of pure 24K gold?

Recent interested in gold and silver due to a potential demise of paper currencies worldwide has brought to the forefront the question of how pure are those gold bars being  held by ETFs , international banks, and at Ft.Knox . Rumors have been spreading ( possibly initiated by short sellers) concerning some of the commonly traded 400oz gold bars being just cladded tungsten. How can one tell? There are several ways, some simple others more elaborate and expensive. Go HERE to see our thoughts on this topic.


September 25, 2010-What is the origin and purpose of the ancient Trident Geoglyph found along the Bay of Paracas in Peru

Several decades ago while travelling in southern Peru I came across the well known rock carving in the shape of a 600 ft long trident geoglyph located on a sloping hill off of the Bay of Paracas. We discuss HERE our thoughts on what its purpose might be.

October 12, 2010 -What is the Goldilocks zone for planets orbiting about a central star?

In the last decade or so astronomers have found the presence of several hundred planets orbiting about stars using optical occultation techniques. Most of these planets lie in regions about the central star incapable of supporting life, however, some probably do. The presence of such planets lends strong support for the nebular hypothesis of planetary formation first proposed by Kant and Laplace. In this theory a distributed disc of matter collapses by gravitational collapse leaving a central star plus a rotating disc of matter which eventually condenses to planets and moons all lying in essentially the same plane(the ecliptic). By the conservation of angular momentum most of the resultant planets will be spinning with a rotation axis nearly perpendicular to the ecliptic. This rotation allows a nearly uniform distribution in temperature at a given latitude of the planet and should make the probability of extraterrestrial life quite likely provided the planet is not too close or too far away form the central star. The zone where life is possible has been termed in the literature as the Goldilocks Zone. The term is taken from the Grimm�s brothers fairy tale �Goldilocks and the Three Bears�.Go HERE to see our discussions on these Goldilocks Zones about stars in our galaxy.

October 16, 2010-What is the approximate speed versus height of a free-fall parachutist jumping from a high altitude balloon?

About 50 years ago air force colonel Joseph Kittinger (alumni of UF) carried out a free fall parachute jump from a balloon at an altitude of 102,800 ft. He reached a maximum speed of about 600mph(268m/s) before deploying his chute at 18,000ft and landing safely. This record jump is to be finally challenged in the next few months by Felix Baumgartner who, with sponsorship of Red Bull, will attempt to jump from a height of H=120,000ft(36.6km) and hopes to reach the speed of sound before chute deployment.We want here to examine the speed as a function of height a parachutist will experience during a jump from heights above 100,000ft. Go HERE for the details.

November 1, 2010- How did early stone carvers construct massive cyclopean walls using boulders of irregular shape and how where they able to produce extremely tight fits between neigboring stones without the use of mortar?

One finds throughout the world in Greece, Ireland, Stonehenge and Avebury, Incan Peru, and Easter Islands stone structures built with  huge non-rectangular cross-section boulders with very little modification of the original stone surfaces. Of particular interest are the cyclopean structures found near Cuzco, Peru. There huge irregular shaped boulders were cut in such a manner that they fit together like pieces of a jigsaw puzzle. We discuss HERE how this was most likely accomplished.

December 17, 2010-What is the origin of the Decimal Numbering System?

With only a few exceptions, the base ten number system is the dominant one used throughout the world for commerce and measurements. Why is that? Clearly it stems from the fact that early man used his ten fingers to count objects and that such finger counting was soon replaced by symbols such as sticks and by marks recorded on surfaces. The Roman number system of R={I, II, III, IV, V, VI, VII, VIII, IX, X}clearly hints at such stick figures as does the earlier Chinese method of representing numbers by bamboo sticks. We want HERE to discuss the probable origin of the Decimal Numbering System and indicate how the basic mathematical operations of addition, subtraction, multiplication, and division are accomplished. A few other number systems are also discussed.

December 22, 2010-Why are the most prosperous and technologically advanced and creative countries around the world located within two relatively narrow temperature bands ranging from  yearly lows no lower than -10degC and highs no higher than +30degC ?

If one looks at the most technologically advanced countries around the world it is clear that these lie within two temperature bands which I call the creativity(or JUST RIGHT) bands. These zones have produced and are producing the majority of the literature, art, science, and technology  and have been the home of such creative individuals as Plato, Confucius, Michelangelo, daVinci, Shakespeare, Newton, Rembrandt, Bach, Tolstoy, Beethoven,Voltaire, Edison, Picasso, and Einstein.The northern hemisphere band includes western Europe, the western part of Russia , Iran, northern India, China, Korea, Japan, the southern part of Canada, and the United States. The southern hemisphere band includes the southern and eastern parts of Australia, New-Zealand, Chile, the northern parts of Argentina, the southern part of Brazil, and South Africa. We discuss HERE why it is that within these bands most of the advancements in the arts, literature, and science have been and are being made and why most regions outside of these bands remain underdeveloped exibiting very little in the form of advanced human intellectual activity or creativity.

January 22, 2011- What is the big deal with Binary and how does one manipulate things mathematically in this Number System?

Although most of us were brought up in school counting things in a base ten (decimal) number system it is clear that other number systems and especially the Binary System are becoming more and more important. Most computer manipulations, data storage, and the transmission of writing and music are now pretty much handled in binary because of the conveneince of a base requiring just two digits instead of the ten associated with decimal. We dicuss HERE some of the properties of the Binary Number System, look at several problems which are well handled in binary, and show how one can mechanically perform basic mathematical operations using a binary abacus.

March 1, 2011-What is relative time and how is it measured?

Time in an absolute sense does not exist as made clear by numerous philosophers(Kant, Heidegger) and by modern technical analysis including relativity. It can however be measured in a relative sense by comparing the interval beteen two events to that of a known time interval such as the period of one earth orbit about the sun. We discuss HERE how relative time measurements were first developed using astronomical observations and later employing atomic clocks.  We use a modified Deborah number to measure relative time. An example of a digital combination of date and time is also presented.

March 27,2011- What are the Economics of Solar Energy Conversion using Photovoltaic Methods?

The recent events in Japan will probably set-back the use of nuclear energy for electricity generation by a decade or more. In the meanwhile there appears to be sufficient fossil fuel supplies(especially coal and natural gas)available to muddle through into the near future despite of the clear rise in costs of these available energy sources and their possible contribution to global warming.  If the nuclear option cannot be revived because of political and economic reasons and also the recovery of fossil fuels becomes economically prohibitive, then one needs to go to renewable energy sources such as wind, biofuels, and solar. Of these renewable energy sources, solar energy via photovoltaic energy conversion appears to be most promising. This energy source is plentiful throughout much of the world but has the disadvantage that one is dealing with a very diffuse energy source requiring multiple square kilometers just to supply a large city with its energy needs.We want HERE to re-look at the economics of solar energy using photovoltaic conversion in order to see if it would make sense as a viable and economic energy source.

May 5, 2011- What is the BMI and how is it Measured?

The ever increasing problem of  obesity in this country has increased  interest in having a quick way to estimate the fat content of the human body. Clearly too much fat is unhealthy and indicates a calorie food intake above that required for the individuals energy output. An excellent approximate measure of fat content is the Body Mass Index(BMI) first proposed by the Belgium polymath A. Quetelet in which one looks at the ratio of a persons weight W ( kg)divided by the square of the heigh H( meters). Population statistics show that the normal range lies between 20 and 25, with values above 25 considered overweight up to 30 and obese above 30. Today nearly 70% of the United States population has a  BMI above 25. Go HERE to see our discussion on this index and how one can construct a simple circular sliderule to quickly determine ones BMI.

MAY 17, 2011- What is Symmetry and where is it Encountered?

Most recognize symmetry in an object , be it in the human face, in architecture, or  mathematics , but often find it difficult to pinpoint exactly what makes the object so. We discuss HERE some of the characteristics of line, plane, and rotational symmetry by analyzing several different examples. Bilateral symmetry is demonstrated by looking at mirror images of the face of Charlize Theron and of the Taj Mahal. Next hexagons, circles, ellipsoids,  cubes , the Laplace equation , and the five petal Rhodonea are examined for their symmetry.

July 1, 2011-Do Stock Indexes World-Wide Correlate?

We have been aware for a long time that stock averages throughout the world have a tendency to correlate . This has been especially true in recent years probably do to the increasing speeds of electronic communications and the use of computer arbitrage. We present HERE a graph of the excellent correlation observed between the S&P500, the German DAX Index, the DOW World Index, the Hang Sang Index, and the ILF ETF over a ten year period. Also a discussion of why such correlations should exist is presented.

July 12, 2011-What is RSA Cryptography and its Connection to Large Prime Numbers?

One of the best and most used cryptography techniques is the RSA approach which uses both public and private keys. It involves a public key which relies on the use of two large prime numbers p and q. The product N=pq  is essentially impossible to factor with even the fastest electronic computers in any finite amount of time. Go HERE to see our explanation of the RSA method in simple terms, view an unbreakable public key which I constructed in just a few minutes, and find an example of sending a short encoded message from party B to party A.

July 26, 2011-How does an Energy Balance relate to Weight Control?

It is well known that a large fraction of the American public are overweight according to BMI measurements. This fact has become more prevalent in recent years and has been attributed to overeating accompanied by insufficient energy expenditure due to sedentary working conditions. Mathematically one can say that DW/dt=I-E=B, where W is a person's weight, I is  the food intake, E the energy output and B the energy balance factor. Go HERE for a discussion of this problem and how it can most effectively be solved. Decrease in calorie intake and increase in  physical activity are the well known factors making for weight loss ( B<0). Weight gain occurs when B>0.

August 1, 2011-What is meant by Exponential Growth and Decay and what are the Properties of an Exponential Function?

Although one often hears and reads about exponential growth in the literature, it is not always clear that the authors know precisely what they are talking about. In discussing true exponential behaviour, it is first necessary to understand what is meant by an exponential function, how it arises and what its properties are. We discuss HERE some of these. Both population growth and radioactive decay are described and we show how one can quickly estimate the value of an exponential function via a quotient of two polynomials.

August 4, 2011-Could the US experience Hyperinflation?

Actions by the Federal Reserve in recent years has essentially tripled the monetary base to about 2.5trillion dollars by simply printing money without any hard assets backing up these dollars. This is a recipe for hyperinflation as witnessed in Weimar during the 1920s and  more recently in Zimbabwe. The reason we have not yet seen any major spike in inflation is due to the fact that banks are reluctant to loan out this new money as they have a risk free way to make money directly of the government via government bonds. Once house prices reverse their downtrend and the velocity of this extra money increases, large increases in inflation can be expected. Go HERE for our discussion of some of the histrorical aspects of hyperinflation and the possibility that it could happen in this country.

August 27, 2011-What are Self-Similar Patterns and how are they Generated?

Recent increased interest in fractals  and how they may be used to explain many natural phenomena has got me to thinking more about how intricate 2D patterns can be generated from some very simple laws of replication. In particular, we look  HERE at self-similar patterns which are generated from elementary geometrical figures such as squares, triangles and hexagons. The problem is treated as one of  biological  generation.

September 1, 2011-Can one create an easy to understand 3D Fractal based on a simple Reproduction Law?

Still thinking about fractals from our August 27th note, we recently posed ourselves the question is there an elementary three dimensional fractal whose basic initiator structure is simple and one where this structure is replicated forever through an infinite number of generations? We answer this question in the affirmative  using a simple cube to generate smaller and smaller self-similar cubes. Go HERE for the details. We have not seen this type of 3d fractal before although some other more complicated 3D fractals have received attention earlier. It is hoped that this type of discussion might lead to a future increased emphasis on 3D fractal structures with the anticipation that they might prove helpful in describing the geometry of certain viruses. We also show how a simply cubic fractal  model can be constructed from wood with the right tools and a lot of patience.

October  3, 2011-What is a Pelekinon and how is it used to measure Date and Time?

Before the advent of mechanical and electronic clocks, people depended on sundials to determine the time of day and used elaborate structures (such as Stonehenge) to measure the time of year when the solstices occur. The ancient Greeks used one of the simplest sundials called the Pelekinon. It was capable of telling both the hour of the day and the day of the year. The Pelekinon consists esentially of a vertical pole(the gnomon) and a flat plane on which they empirically observed and marked down the locus of the shadow produced by the pole throughout the year. We want here to discusss the mathematics behind the Pelekinon and show how one can use spherical trigonometry involving the astronomical triangle to accurately predict the movement of the shadow. Go HERE for the discussion.

October 23, 2011-What is the impact speed of a bullet fired straight up upon its return to earth?

Recent vidios showing Libyan freedom fighters firing their guns randomly into the air during victory celebrations, has brought the question to my attention of whether or not such acts pose dangers to individuals nearby. We analyse the problem of the returning bullets by first calculating what  height H a bullet , fired from a standard Kalashnikov AK-47, will reach and then determine the impact speed Vi with which it impacts the ground on its return. Go HERE for the details. We will show that the returning bullets do not pose a  lethal threat. In addition, we put to rest an  urban legend that dropping a penny off of the Empire State Building can kill someone walking on the sidewalk below.

December 12, 2011-What
is Geometric Art what are the Basic Elements used in its Creation?

It is possible to create appealing abstract art work based on the superposition of  simple geometrical shapes. I term this Geometric Art and give examples arising from religion, philosopy, and other movements. In addition I show how this type of art , involving straight lines, circles, squares  and other simple 2D geometric figures, can  readily be created by computer graphics or via handicraft approaches involving, for example, wood work. Go HERE for the details.
February 1, 2012-How can an Abacus be used to Increase one's Ability to quickly Add and Multiply Numbers?

The standard method of adding, subtrtacting, multiplying and dividing numbers by the standard techniques taught in schools is very often not the quickest way to obtain an answer. Rather using an Abacus which requires a decimal concept of numbers can produce much faster results. We discuss HERE how this is accomplished by looking at how elementary mathematical operations are handled with an Abacus. Several different shortcuts involving multiplication based on recognizable powers of intergers and expresing numbers as differences of simpler numbers are discussed.

February 3, 2012-What are Logarithms and how may they be Used to Multiply and Divide Numbers?

Prior to the advent of hand calculators and electronic computers, the way to handle multiplication and divisions arising in complex mathematical expressions was to first convert the numbers contained therein to logarithms , perform the logarithmic operations, and then invert to get the final answer. We discuss HERE what is meant by a logarithm, how one manipulates them, and how slide rules can be used to quickly calculate approximate answers .

February 26, 2012-How does one express Large and Small Numbers?

Although most readers will be familiar with the exponetial notation for number such as 1000 being equvalent to 103 and 0.000001 equal to 10-6, they are less likely to be familiar with the designations involving Greek and Latin prefixes for both very large and very small numbers. Such designations are becoming more and more commenly used and one needs to become familiar with them not just for understanding the scientific literature. Go HERE for a discussion on this topic. As examples we show that a light year equals 9.46 Petameters, the worlds energy consumtion in 2008 was 474 Exajoules, and the Big Bang occured 13 Gigayears ago.  A circular slide rule capable of performing multiplications and divisions  over a wide 48 order of magnitude range is also described.

March 20, 2012-How does Facial Recognition by Eletronic Computer work and what steps might be taken to improve the Procedure?

One of the more important and growing tasks in digital recognition is how to quickly identify a face from a collection of millions of other faces electronially and do so with one hundred percent accuracy. So far such approaches have been only partially successful and will fail when simple changes in hairstyle, aging, or facial orientation are introduced. This seems strange in view of the fact our brain allows facial identification in split seconds by an as yet incompletely understood process. We suggest here that improved computer recognition in shorter times might be accomplished by comparing a given face with the norm of a human face and then identifying things by only the differences. Go HERE for our thoughts on such a process.

May 17, 2012-How does the Concept Life and Evolution not violate the Law of Entropy?

The concept of life and its accompanyting evolution seems to be in contradiction to the Law of Entropy which requires that all systems move in time from an ordered state (low Entropy) to one of disorder (high Entropy). After first discussing the very slow rate of change produced by evolution over thousands of generations , we show how life and accompanying evolution is indeed possible without a violation of the Entropy Law by  considering the entire environment . Go HERE for a discussion of the details.

May 27, 2012-What is the Difference between a Mean and a Median?

When reading articles in the popular press concerning population trends, wealth distribution , and weather records one quite often encounters the terms mean and median
without the articles explaining the difference . This can lead to confusion on part of the readers and in addition is sometimes used to hide certain facts such as the true unemployment rate and the great wealth disparity existing between diffenert segments of the US population. We discuss HERE these two concepts in greater detail and show how highly skewed input data can lead to large differences between the median and mean of a data set. The AGM method for integral evaluation and the Pareto Curve used by economists is also discussed.

June 17, 2012-What are Pareto Curves and how are they be used to discuss Wealth Distribution?

Over a hundred years ago the Italian engineer and economist Vilfredo Pareto noticed that some 20% of any population typically controls about 80% of the wealth. His statistical analysis led to the power law curve y=xn, where y is the fraction of the wealth owned by fraction x of the population. The constant n is a measure of the inequality of the wealth distribution with n=1 corresponding to an egalitarian society where wealth is evenly distributed to n>>1 for increasing income inequality. According to Pareto, the n=8 curve appeares closest to equilibrium conditions. At the present time the value for US wealth distribution lies between n=16 and n=32 and hence is out of wack with equilibrium condition and calls for adjustments. Go HERE for further discussions.

July 4, 2012-What is the Relation between Dip-Stick Readings and the Volume of Gasoline left in a Storage Tank?

Several years ago one of our undergraduate students who was working part-time at a local gas station, asked me to explain to him the relation between dip-stick readings and the volume of gasoline remaing in an underground storage tank. In looking at the problem it became clear that one should expect a non-linear relation between fluid level and fluid volume. The relation depends very much on the type of tank cross-section one is dealing with. Go HERE to see some detailed calculations for both cylindrical cross-section and spherical cross-section tanks.

July 17, 2012-How were the Pyramids at Giza constructed using only 2500 BC Technology?

One of the most impressive sights in the world  are the three pyramids on the Giza Plateau just west of  Cairo, Egypt. People have speculated for years how a civilization having not yet invented the wheel nor familiar with iron tools and other simple machines such as  pulleys could possibly have been able to construct such massive monuments. We show HERE how this was indeed possible and how primitive technologies including the use of ropes, inclined planes, sleds, and oil lubrication were used in conjunction with  a masive labour force to construct these pyramids. We also show how a limited knowledge of elementary astronomy allowed the builders to allign these pyramids precisely with a north-south axis.

August 14, 2012-What are the characteristics of an ICBM's Trajectory in its Flight from Launch to Impact ?

We examine the laws of conservation of energy and of momentum to discuss the properties of an ICBM trajectory. Using a minimum of mathematics , we obtain explicit formulas for the height H reached by the missile and the time to impact . An important parameter entering the analysis is the non-dimensional parameter alpha=2gR/Vo2, where R is the earth radius, Vo the launch speed and g the acceleration of gravity. Go HERE for details of the discussion including the effect of launch angle beta.

August 19, 2012-What shaped Tiles can be used to cover a Flat Surface without leaving Gaps?

An interesting problem in 2D geometry is what type of tiles can be used to cover a flat surface (such as a floor) without leaving empty spaces between the tiles. Square and rectangular tiles are obvious examples which can do this. But as we will discuss HERE there are an infinity number of other configurations which can produce a contiguous array. We show how one can use a square and rhomboidal base pattern to generate some intricate tile patterns which meet this no gap condition. Also regular polygons are used to generate two tile configurations in the shape of kites and arrows.

September 26, 2012-What is the Sagitta and how can it be used to calculate certain Dimensions associated with Circles and Spheres?

When dealing with geometrical figures such as circles and spheres one is often interested in the maximum distance from the figure edge to a chord running beteween two points A and B also located on the edge. This quantity is known as the Sagitta (from the Latin word sagittarius for arrow) and has important applications in architecture and line of sight radar among other areas. Go HERE for details of our discussion an this topic. Among other subjects, we briefly look at the maximum depth a straight line tunnel of length 2L dug between two points on earth will have.

September 27, 2012-What are the Characteristics of a Straight-Line Tunnel dug through the Earth between Two Points on its Surface?

Most of you who have taken an introductory physics course  during your life know that if a mass is dropped into an imaginary shaft drilled through the earth center that the mass will undergo  simple harmonmic motion converting potential energy to kinetic energy and visa versa and will make the round trip in about 85 minutes. The same continues to hold true for a mass moving in an off-center shaft . Such a shaft may be looked at as a tunnel connecting two points A and B on the earth's surface. We discuss HERE some of the properties of such a tunnel and look in detail at the case of a 204 mile long  tunnel between Washinghton DC and New York City and passing some 1.3 miles under Philadeplphia.Theoretically a mass moving through such a tunnel would be cost free assuming all friction can be eliminated.

November 26, 2012-How does one distinguish between a Composite and a Prime Number when the Number under consideration becomes large?

Most of you where taught in middle or high school how to distinguish between a prime number which is divisible only by itself and one and a composite number which has multiple additional factors. What is often not  clear is how does one know when a number is prime or composite when the number N becomes very large. We show you HERE how one can quickly distinguish between these two groups by using our recently established fact that all primes above N=3 are of the form Q=6n+1 or Q=6n-1. Also we introduce a new parameter termed the Number Fraction which can rapidly distinguish between the two groups of numbers  using a simple computer routine. Also the Sieve of Eratosthenes is discussed and a table for the first few Q primes is presented.

November 28, 2012-How much will the Sea Level rise if all the ice in the Polar Caps were to melt?

Although the validity of the theory of global warming is still in some dispute, there is little question that ocean levels have varied from much below the present level during the ice ages and  above the present  during the times of the dinosauers. We look HERE  at what the maximum sea level rise would be if all ice in the Arctic and Antarctic regions  were to melt. The answer turns out to be 99 meters. This would put most coastal regions and cities under water. However, for this to happen it would take thousands of years giving man sufficient time to evacuate and rebuild. We also briefly discuss beneficial effects of global warming including increased agricultural production and recovery of fresh water from icebergs.

December 14, 2012-What is a Fractal Square and how is it Constructed?

We examine a new type of fractal consisting of the superposition of progressively smaller squares whose locations are dictated by very simple rules. Termed a Fractal Square it is an extention of the well-known Koch Snowflake but unlike it has finite area and either infinite or finite perimeter depending on the magnitude of the fraction factor f between neighboring generations. Some interesting graphics are obtained including an intriguing figure which I name the Black Snowflake. In addition, we show how such fractal squares can be constructed by simple genetic algorithms. Go HERE for the details.

December 28, 2012- What are the four basic Temperature Scales and how does one convert between them?

We look at the four commonly used temperature scales, their history and callibration points. Starting with the Fahrenheit Scale based on the freezing of a brine mixture(0 deg F) and the boiling of water(212 deg F).  This is followed by the Celsius Scale  using freezing water as 0 deg C and boiling water as 100 deg C. Finally the Kelvin and Rankine Scales are discussed and how their zero points correspond to absolute zero. It is shown how the conversion formulas F=(9/5)C+32=R-459.67 and K=273.15+C are derived. Go HERE for the details.


January 4, 2013-What is a Complex Number and what are some of its uses?

We present an elementary discussion of complex numbers and how they arise and how they may be used to find certain trignometric formulas. After demonstrating how the use of such numbers arose in connection with trying to understaning the meaning of numbers such as sqrt(-2), we show how one takes roots of the complex number z=a+ib and how to use it to evaluate certain integrals. Go HERE for the discussion.

January 11, 2013-How did the Wheel develop?

One of man 's earliest and most significant inventions is the wheel. It developed over thousands of years starting with simple wheels constructed from wooden planks to the latest 541 ft diameter Singapore Flyer Ferris wheel using tension spokes. Go HERE for the details of the historical development. Among the discussion is a treatment of rolling friction of wheels and how bearings are used to minimize frictional resistance at a wheel hub.

February 16, 2013- What is the Impact Speed of an Asteroid hitting the Earth?

In the last few days the news has been saturated with reports of a near miss of the earth by a football field sized asteroid  and also the direct impact of a much smaller asteroid near Chelyabinsk, Russia which caused sonic boom damage. We examine the trajectories of such asteroids and in particular calculate the impact speed an asteroid will have when originating from the asteroid belt. Go HERE for the details.

February 25, 2013-What is Mental Arithmetic and how does it work?

One often encounters situations where the ability to make quick  mental  math calculations without the aid of pocket calculators or pen and paper come in handy. For example , calculating the amount of an 18% tip on a dinner at a restaurant or the miles per gallon one's car  has achieved during a recent trip. We discuss HERE how such simple mathematical calculations involving just addition, subtraction, multiplication, and division are performed mentally. Also we show how quick approximations to certain mathematical questions can be obtained without outside help or before someone pulls out an iphone to calculate or google things.

April 3, 2013-What are the Negative Effects of Quantitative Easing to Infinty?

Those believing in Keynesian Economivcs claim that when a country is in recession or depression the way to get out of it is to devalue the currency by running the printing presses. Ben Bernanke and the Federal Reserve have been doing this since 2008 at an ever increasing rate with little to show for it except a dilution of the currencty by over 300%, the stoking of a stock market mania , and an artificial increase in bank earnings. Their policies have not helped unemployment and have had the direct effect of depriving  retirees of a reasonable return on their life savings by running an essentially zero interest rate policy. It is time for the Fed to  change course and to do so before international currency wars erupt and the dollar looses the little remaining value it has left. The best solution would be to immediately clean house at the Fedreal Reserve and in particular remove Chairman Bernanke before further damage is done.  Go HERE for the details of the discussion.

April 8, 2013-How do the Seasons come about and what is the procedure for locating the North and South Celestial Poles?

If you ask individuals how is the earth's tilt axis is related to seasonal changes and how one locates the north and south poles on the Celestial Sphere, most will not be able to answer. We give HERE a brief answer to these questions including how to determine the  local latitude by measuing the altitude of the celestial poles above the horizon. A time lapse photo showing the circular paths stars take in the night sky about the south celestial pole is also presented.

April 14, 2013-What are the Platonic Solids and how may they be constructed by Electronic Computer using a Guiding Sphere?

There are five basic Platonic Convex Polyhedra known as  the Tetrahedron, the Hexahedron, the Octahedron, the Dodecahedron, and the Icosahedron. They were discovered by the ancient Greeks and form the starting point for most most discussions on 3D geometry. We show how these solids can be constructed via computer by using a guiding sphere on which all vertices of these polyhedra are located. Go HERE for the details. We also show how one can use cardboard cut-outs and wood polygons to construct these polyhedra .

June 6, 2013-What is the Volume and Surface Area of Standard Pyramids?

We use simple geometry and elementary calculus to determine the volume and surface areas of pyramids consisting of a regular polygon base and a vertex placed at height H above the centroid of the base. The results show that all of these pyramids have a volume equal to the product of the base area times the height divided by three. The properties of the square base Great Pyramid of Cheops at Giza are also discussed including the volume of such pyramids during construction. Go HERE for the details.

June 19, 2013-Does the recently revealed Program of Massive Surveilance by NSA constitute a violation of the 4th Amenment?

It has now been several weeks since the Guardian Newspaper's revalation of massive warantless  spying by the National Security Agency(NSA) on all Americans . A heavy defense of these actions has been mounted by the mass media and the Executive, Legislative and Judicial Branches of the US Government. The essence of their arguments being that such surveilance is neseccary to protect against acts of terrorism. We are expected to believe the recent testimony by General Keith Alexander that this program has indeed protected us from over fifty attacks although he could not tell us what they were because its 'secret'. There seems to be only a minority including myself who recognize that there has been a violation of the 4th Amenment to the US Constitution and the right to privacy. Go HERE to see our discussion on this matter as I wrote about it several weeks ago.

June 24, 2013-Does it make Economic Sense to build new Ocean-Connecting Canal across Nicaragua ?

The Nicaraguan government has recently agreed to  the building of a canal across southern Nicaragua by a Hong Kong based company. We discuss HERE the ecomomics of such a proposal and compare it with the cost of modernizing the Panama Canal. The difficulties with the construction of  a trans-Nicaraguan canal, which would be about twice the length of the Panama Canal, would be the need to cut through higher terrain than in Panama and also be subject to potential earthquake damage, not to mention border disputes with Coasta Rica. We conclude that a more economic approach for all involved would be to not  undertake the construction of such a canal but rather increase the size of existing locks of the Panama Canal in order to accomidate larger ships. The financial support  for such an undertaking could come from a world wide consortium of companies, not unlike the AirBus consortium, if Panama can be made to agree.

July 20, 2013-At what height H above the equator must a satellite be placed in order to be Geosynchronous?

A geosynchronous satellite is one whose angular velocity  matches that of the earth's rotation rate of
ω=2π /(365.25x24x3600) r/s. Using Newton's second law one finds that such a match occurs when the satellite height H above the equator is H=R{-1+[g/ωR]1/3}, where g is the acceleration of gravity and R the earth radius. The height turns out to be about 22 thousand miles. Go HERE for the details of the discussion,  including finding the orbital period for satellites in a circular orbit at any height and the expected delay in two-way conversations between any two points on earth using geosynchronous satellites.

July 24, 2013-What are the Conic Sections and how are they derived mathematically?

One of the more important concepts students encounter in analytic geometry are the Conic Sections. These are 2D curves formed by the intersection of a plane intersecting a cone at different angles. They may be circles(e=0), ellipses(e<1), parabolas(e=1), or hyperbolas(e>1) depending upon the eccentricity e.  We show HERE how to use some simple mathematics to derive these curves. Applications for each of the figures are also briefly discussed.

August 21, 2013-What is Number Theory and what are some of its most important Results?

Number Theory is that branch of mathematics which deals with the relation between the positive integers 1,2,3,4,5..   . It has become of more practical importance in recent years with the advent of high speed electronic computers and the use of digital encryption. Basically the integers break up into two groups- the prime numbers such as 3,5,7,11,13.. and the composite numbers 4,6,8,9,10,.. We state some of the known properties for these groups and introduce some new concepts such as the number fraction f(n) and the plotting of Q primes along a hexagonal spiral. Go HERE for the details.

August 25, 2013-What is the Volume of an Irregular Hexahedron such as those seen in cut Cantaloupe Pi

Have you ever noticed that the cut melon pieces one has for breakfast are often in the form of irregular six-sided polyhedra ? It is  natural to ask what is their volume. Clearly the problem is a lot more involved than just calculating the volume of a rectangular solid and will require some vector analysis to obtain the answer. We carry out such an analysis HERE. It is shown that the irregular hexahedron can be broken up into six irregular pyramids having quadrangle bases and a common vertex placed at a point within the hexahedron. The volume of each pyramid is found to be Vol=(1/6){|V2�(V1xV3)|+|V4�(V1xV3)|}, where V1,V2,V3, and V4 are vectors defining the side lengths of a pyramid.

September 14, 2013-What is the Struve Geodetic Arc?

Prior to the advent of earth satellites , laser reflectors and radar, the standard way to determine the precise distance between two points  on earth was by means of triangulation based on a known base line  and the ability of theodolites to accurately measure the angles from the ends of the base line to a third point.  We discuss HERE how such measurements are carried out  and pay special attention to one of the longest triangle chains ever divised and now referred to as the Struve Geodetic Arc. A simulation of the triagulation procedure using just three oblique triangles is discussed in detail. Also we indicate how the elevations of mountains such as Mt. Everest or Mt.McKinley were originally obtained.

September 16, 2013-What is the Area of An Irregular S-Sided Polygon?

We examine the area of irregular polygons with S sides by evaluationg the areas of T=S-2 sub-triangles constructed by L=S-3 straight lines. Several different examples are considered and a brief discussion of how land areas are determined knowing only the length between neighboring vertices and the angles the connecting lines between vertices make relative to a north-south line. It is also shown how these area calculations can be greatly simpilfied by making use of symmetry. Go HERE for the details.

October 3, 2013-What is the Future of the US National Debt?

Perhaps the most critical economic problem  facing this country at the present time is a run-away national debt which has increased by a factor of over fifty  in the last half century and at the present time stands at 17.065 trillion dollars and accelerating. We make a linear extrapolation of this debt into the near future and show that in about seven years the US Treasury will be paying well over a trillion dollars a year just to service this debt. Clearly this is unsustainable and will lead to a default of the dollar. Printing of money by Ben Bernanke will work only as long as the lenders don't recognize the futility of his actions and recognize that their dollar holdings are being depreciated on a daily basis. A calculation based on the assumption that the US monetary system will collapse when the debt reaches 130% of GDP, suggests this point will be reached shortly before 2020. Go  HERE for the details of the discussion.

October 28, 2013-How can one use Generalizations to establish Mathematical Principles?

In the history of mathematics and especially in number theory investigators have established many mathematical principles based on the generalization of observations based on special cases. In many instances such generalizartion have proven to be valid for all integers and have led to the establishment of certain universal principles while in other cases the generalizations have failed. We want here to discuss some of the more important results obtained by generalization  in number theory and then add a few thoughts on our  own generalizations involving prime numbers. Go HERE for the discussions.

November 13, 2013-What is the Future of Fission Generated Nuclear Power after Chernobyl and Fukushima?

The recent nuclear reactor accident at Fukushima and earlier ones at Three-Mile Island and Chernobyl have called into question the future use of nuclear fission as a viable energy source. Since many of the pro and con arguments appearing in the literature and on the internet are based on an incomplete understanding of the facts, we want HERE to help clarify things by discussing the basic process involved in nuclear power generation, the radioactive species produced in the process, their half-lives, detection methods , and the basic units of radiation measurement. After presenting these in some detail, we conclude with a short summary of our own views on the future of nuclear power and its necessary co-existence with fossil fuel energy generation.

November 28, 2013 (Thanksgiving)-How does one use Vector Operations to determine Angles between Straight Lines in Space?

It is known that any straight line connecting points [x1,y1,z1] and [x2,y2,z2] in space can be represented  by the vector V=i(x2-x1)+j(y2-y1)+k(z2-z1). Furthermore the scalar product of any two vectors normalized to unit length equals the cosine of the angle between them. We use this fact to treat a variety of problems requiring a precise determination of the angle between lines. Among the problems discussed are those of  Eratosthenes for determing the circumference of the earth, finding the shortest distance between a parabola and a straight line, and locating the center of mass of a regular tetrahedron. Go HERE for the details.ast

December 6, 2013-How does one determine the Volume of an N sided Polyhedron?

A polyhedron is any 3D solid which has four or more flat faces. The cube, icosaheron and dodecahedron are examples of regular polyhedra. We show HERE how to calculate the volume of any polyghedron with a number of sub-tetrahedra whose exact volume are given by one sixths of the scalar triple product of the three vectors representing its three lateral edges. Several different examples are examined including the volume of a  triangle base truncated pyramid.


anuary 1, 2014-What is the Relationship
between Integer Spirals and Spider Webs?

In earlier articless we have studied the properties  of certain spirals and radial lines intersecting them. Such curves can be used to distinguish composite from prime numbers. It is our purpose here to show the similarities of these curved structures to spider webs by approximating an actual spider web by an Integer Spiral pattern. Go HERE for the details. Also a brief discussion is given of how such a mathematical approximation may lead to a new type of electrostatic dust cleaner.

January 23, 2014-What is the Mathematics behind the popular Numbers Game Sudoku?

As many of you are aware, Sudoku is a very popular and addictive number puzzle in which one is asked to find all integers in  a square array given a certain number of starting values. It is governed by several simple rules and is related to Euler's Latin Square.We show that the solution procedure is simply an extensive manipulation of elements in an n x n square matrix and involves finding the compliment to known numbers found in the row, column and sub-matrix which contains the element ai,j. After the possible solutions for an element have been found, one eliminates most of these by simple rules to  yield just one final answer. The sum of the n elements in any row, column or submatrix will always be equals to S(n)=n(n+1)/2.
Go HERE for the details of our discussion.

February 1, 2014-How did the Ancient Egyptians measure Slopes during Pyramid Construction?

It is well known that about 4000 years ago the ancient Egyptians started building tombs for their kings in the form of huge square base pyramids with precise equal slopes for the slanting four sides. How was this accomplished with the very primative tools at their disposal ? The answer for slope determination was use of the Egyptian Level and Square which was essentially an A-frame structure where the two sides intersect at a right angle and a blumb-bob is suspended from this intersection. We describe how this instrument was used to measure slopes(and hence angles) of surfaces relative to the horizontal. We also show how their use of length measurements based on cubits, palms and digits allowed for the definition of a slope in terms of the Seked. Go HERE for details of the discussion and also to view a modern version of the Egyptian Level-Square which we just constructed in our workshop.

February 20, 2014-What are Magic Squares and how are they Constructed?

A magic square is an array of integers having n rows and n columns. The square obeys the law that the integer elements in each row, column, and diagonal add up to S=n(n2+1)/2 and that no integer appears more than once in the n x n array. We discuss how these squares are constructed using both a Sudoku approach and one involving adding together smaller magic squares to form larger ones. We discuss in detail n=3, 4 and 6 magic squares. Go HERE for details of the discussions.

March 1, 2014-What is the relation between Grade and Angle of an Inclined Surface?

The standard way of measuring the incline of a surface is to define its grade which represents the ratio G of the vertical height 'b' of the incline  to the corresponding horizontal distance 'a'. It is essentially a tangent measure which converts to the incline angle in radians by the simple formula α=arctan(G). We graph this function and its approximation plus present a table relating the grade to the incline angle. Also the grades of the PA turnpike, railway beds, and pyramid surfaces are discussed. Go HERE for the details.

March 9, 2014-What is a Temperature Well and how can its properties be used to build a better Heat Exchanger?

We examine the heat flow into a 1D Temperature Well and show that the early phases produce extremely large heat transfers. This information is used to suggest a way to quickly remove heat from a solid in a way analogoues to the quenching of a hot iron slab by dunking it into a batch of cold water. A design involving periodic injections of fluid into the well is then used to suggest a possible design for an improved a fluid-fluid or fluid solid heat exchanger. Go HERE for the details.

May 7, 2014-Is there an alternate way to generate and easily store large Prime Numbers?

We discuss a new way to generate and then store in compact form large prime numbers. The technique depends on using the first m digits of combinations of several different irrational numbers including Pi ,sqrt(2), ln(2) and exp(1). Calling the combination of a group of these mathatical constants taken out to m places A and then eliminating the decimal point by multiplying  by 10m, produces a large number N whose digits are essentially random.The N mod(6) value is noted and a small constant 'a' is added to N to make it confirnm to the fact that all primes above three must have the form 6n+1 or 6n-1. Next carrying out a search of the sequence N+a+6k over a small range for integer k will produce  primes for certain values of k. We show how to generate a number of large primes and demonstrate how they may can be stored and also how they are connected with public key cryptograhpy. Go HERE for the details.

May 25, 2014-What are the Characteristics of Financial Bubbles?

We look at the characteristics of finacial bubbles starting with the Dutch Tulip Mania in Holland during the 1630s, through the South-Sea Bubble of 1720, to the 1929 Market Crash and the latest Bernanke-Yellen Bubble. Typically one finds that these bubbles are characterized by a steep acceleration in price followed a  rapid decline which catches most investors and speculators by surprise and can lead to financial ruin. Such finacial bubbles can occur  for metals, stocks, other commodities, art, and home prices. Typically they last just a few years, but new and different bubbles will always re-emerge. Gold at $1800 per oz in 2011 and the dot.com peak of technollogy stocks in 2000 are clear examples of earlier financial bubbles. Go HERE for details of our discussion.

May 27, 2014-What is the Area of any N sided Irregular Polygon?

A polygon is a closed 2D figure consisting of N vertexes connected to each other by straight line boundaries. If the lines all have equal lengths then one has a regular polygon such as a standard hexagon or octagon. Irregular polygons have border lines of different lengths. It is shown that any polygon can have its area expressed as the sum of the areas of sub-triangles which in turn can be evaluated quickly using a vector product of two of its sides. We discuss in some detail the area of a hexagon, a pentagram, and a four sided irregular quadrangle. Go HERE for the details

JUNE 1, 2014-What is Thermal Pumping?

About 30 years ago I found a new way to transfer heat from a hot to a cold reservoir by oscillating a fluid in a bundle of open ended capillaries connecting the reservoirs and have termed the process Thermal Pumping. It works by the coupling a large periodic transverse conduction heat flow with a time-averaged axial convection process confined mainly to the Stokes boundary layers formed along the conduit walls. Go HERE to understand the details of this heat transfer process.

JUNE 11, 2014-What are Sequences and how are they Generated?

A sequence is an array of numbers  whose elements f[n] are generated by  formulas of the form f[n]=F(n), where F is a specified function of the integer n. Simple examples are the sequence {1, -1, 1, -1, 1, -1, 1,...} generated by F=(-1)n and the sequence {1, 4, 9, 16, 25,...} generated by F=n2. We look HERE at more complicated sequences including ones where the elements are complex numbers. One of the more interseting new sequences examined has F(n)=sum((n+k)!/(n-k)!, k=0..n-1).

June 22, 2014-How does one generate Sudoku Squares?

Sudoku is a math game in which one is given a few numbers in a square array and asked to find the remaining ones in order to generate a complete Sudoku Square. These squares have the property that all rows and columns in the n x n array contain each of the n numbers only once. The n sub-matrices of the square also contain each of the n symbols only once. It is our purpose here to look at the reverse  (and much easier) problem of first generating some generic Sudoku Squares, examine their properties, and then use them to construct Sudoku Puzzles. Go HERE for the details of the discussion.

July 6, 2014-How does one determine Latitude and Date using only Astronomical Observations?

Suppose one was stranded on a desert island without the benefit of a GPS receiver, a nautical Almanac, and any form of electronic receiver or transmitter. How would such an individual (say Robinson Crusoe) find his latitude, the current date, and hour of the day? The answer is via astronomical observations. We discuss HERE how this would be accomplished using only the most primitive observation methods consisting of a pair of vertical posts oriented in a north-south direction as defined by Polaris.

July 9, 2014-How does one use the Big Dipper in Ursa Major to determine Time?

We show how a pointer defined by a line going through the North Star Polaris and the two front stars of the Big Dipper yield an accurate 24 hour clock. The pointer moves counter-clockwise at the rate of one degree evry four minutes(15 degrees per hour). This astronomical clock is very precise in measuring time increments since one does not need an accurate local time to carry out such measurements.Go HERE for the details of the discussion.

July 16, 2014-What are Sun-Shadow Trajectories and how are they calculated?

The shawdow projected on a horizontal surface by a vertical pole depends on the local latitude(LAT), hour angle(HA), and the sun's declination(DEC). We examine the time-dependent trajectory of the length of this shadow at local noon(HA=0) throughout the year. Shadow trajectories for Gainesville, FL, Washington, DC and Moscow, Russia are presented in detail. The larger the latitude of the observation point the longer the shadow line becomes at Winter Solstice. It is also shown how the shadow length between two different places in the Norethern Hemisphere along the same longitude can be used to calculate the distance between them. Go HERE for the details.  

August 2, 2014-What are Reuleaux Triangles and how may they be used to make Square Holes?

A Reuleaux Triangle is constructed from an equilateral triangle of side-length s by drawing circular arcs of  this radius from each of the three vertexes. When such a triangle has its centroid follow a closed path about a fixed rotation axis, the triangle can sweep out a near perfect square. This allows one to use such triangles to drill holes of nearly square cross-section. We show HERE how such drills function by looking at both the mathematics behind Reuleaux Triangles and by demonstrating their rotational behavior  via a wooden model.

August 12, 2014-How does one determine the Curvature of a Function f(x) and its corresponding Evolute?

In introductory calculus one learns about the curvature of 2D functions y=f(x). The treatment is many times incomplete and students often have trouble understanding the 3/2 power term arising in the discussions. We help clarify this problem here by deriving in detail the radius of curvature and the evolute of any continuous function f(x). Among other curves discussed in detail are the parabola and also the Witch of Agnesi. Go HERE for details of the discussion.

August 30, 2014-What i s the Gaussian and how is it Derived?

We look at a modified Pascal triangle to derive the Normal Gaussian Distribution for a continuous function. After taking the appropriate limit, on finds the function F(z)={1/sqrt(Pi)} exp(-z2).  Its properties are discussed, some integrals involving F(z) are presented, and finally its relation to the probability density distribution P(w) are looked at. Values under the P(w) curve at one, two , and three standard deviations are also discussed. Go HERE for the details.

September 1, 2014-What is an Asymmetric Pascal Triangle and how can it be Used to Predict Gambling Odds?

We examine a modification of the Pascal Triangle where each element is defined by the binomial expression C[n,m,a,b]=n!/{m!(n-m)!}{anb(n-m). Asymmetry is induced into the triangle whenever 'a' does not equal 'b'. Gambling involving Coin Flipping and Roulette are discussed in some detail including the probability of winning or loosing and the payouts for wins. Go HERE for details of the discussion.

September 4, 2014-What was Stonehenge's Purpose?

The Neolithic monument Stonehenge near Salisbury England was built about 4500 years ago. It's reason for being have fluctuated over the years from that of a primitive astronomical observatory to a place of worship and burials. We look here at the basic equations for determining the azimuth at sunrise and sunset during the solstices, to show that the orientation of the monument strongly supports its role as an instrument for measuring the time of the Summer and Winter Solstice, but nothing more elaborate than that. We find. among other things, that the hour angle (HA) along sun's path during the winter solstice is very close to one radian (3hrs 49min) after local noon. Go HERE for the discussion.

September 25, 2014-What are the Properties of  Lenses?

A lens is a transparent structure having curved surfaces. It has the ability to refract light rays to a focal point and thus finds multiple applications in optical instruments. We show here how a light ray passes throuh various lenses, discuss their multiple uses, and describe how magnification occurs when several different lenses are allined along an optical axis. In addition we show how Fresnel lenses are constructed and make some suggestions concerning their use in solar energy studies. Go HERE for the details.

December 18, 2014-What are the Dimensions of the Great Pyramid of Cheops?

We re-look at the dimensions of the Great Pyramid at Giza and show that there is no need for hidden mathematical constants to account for its shape. All that is required to build these is the capability to accurately measure length(cubit) and angle(seked) already known to the pyramid builders in 2500 BC. Furthermore we show how  knowledge of a 3-4-5 right triangle is sufficient to build a pyramid very close in shape to that of the Great Pyramid. Go HERE for details of the discussion.

 January 28, 2015- How much does a Football's Pressure drop with Decrease in Air Temperature?

A major controversy among football fans at the moment concerns what has been termed Deflategate. It involves the question of whether  the underinflated footballs used by the New England Patriots in their recent trouncing of the Indianapolis Colts was due to temperature changes or deliberate deflation  of  several of the footballs by the Patriot staff. We look here at the problem from a technical viewpoint to see if the measured 2lb/in2 deflation in some of the gameballs could have been caused by temperature changes alone as claimed by Patriots' head coach Bill Belichick. Go HERE for the discussions concerning this problem and some suggestions for future ball handling protocol.

January 31, 2015-How does one Construct 2D Figures using only Straight Edge and Compass?

It is known that one can construct an infinite number of two-dimensional figures using just a straight edge and compass. Classic examples of such figures include n sided regular polygons and squares and their vaiations including the Rhombus. We want here to show how such constructions are carried out. Among other figures we show how a pentagon can be generated and how a combination of straight lines and circles  can lead to some interesting computer art. Go HERE for the details.

February 7, 2015-What is the Gaussian, its Properties and better known Applications?

We examine the Gaussian  y(x)=a exp-{(x-b)2/c} including the special case of the Probability Density Function. The Gaussian is shown to  have a bell shape with its maximum at x=b and a standard deviation of sigma=sqrt(c/2). It is shown how one measures IQ in terms of the number of standard deviations from the peak of the Gaussian. Several identities involving y(x) are  derived. Go HERE for the details.

February 9, 2015-What is an Integer Spiral and what are some of its more important Properties?

We examine the complex function F[z,n,m]=n exp(i n Pi/m) for integer n and m. It produces a series of points in the complex plane which represent all positive integers n=1,2,3,..  when m is set to an integer. Among other results, we detail the historical development of integer spirals starting with (1+i)n. In  particular, we concentrate on F[z,n,4] and F[z,n,3] . For the first time it is shown that F[z,n,3] represents all positive integers with all primes above n=3 falling along just two straight lines instead of being scrambled over the entire z plane as they are in a standard Ulam spiral. Go HERE for the details.

February 23, 2015-What are some of the Areas which can be created by the Intersection of Circles and Straight Lines?

We look at some of the  areas which can be created by combining circles and straight lines. In particular, we emphasize different types of crescents. Also Heron's Formula for triangles is derived and used to find the area of an equilateral triangle incribed in a circle. In addition it is shown how to find the central triangular scalloped triangle formed by three equal radius circles just touching each other. Go HERE for the details.

April 15, 2015-How can one Construct an Infinite Number of 2D Curves using only the Length and Orientation of its Straight Line Edge Components ?

We examine the concatenation of line increments L with specified orientation theta to form both open ended  and closed curves in two dimensions. After discussing how a piece of land areas can be defined by the length of  its sides measured in paces and the compass direction these sides take, we proceed to construct  curves such as the rectangular pulse, regular polygons, and spirals. Go HERE for the details.

April 20, 2015-What are the basic Mathematical Constants and how are they Derived?

We look at the basic six mathematical constants, show how they are derived, and how they can be evaluated to any desired degree of accuracy using some new approaches. We give their values to 100 digit accuracy using various approaches including iteration, continued fractions, and integrals involving Legendre polynomials. Go HERE for details of the discussion.

May 8, 2015-What are some of the Shortcuts one can Employ to speed up Mental Arithmetic Calculations?

We look at the four basic mathematical manipulations of addition, subtraction, multiplication, and subtraction and show how these operations can be performed quickly and accurately in one's mind without use of pencil and paper or electronic calculator. Also we indicate how to take powers and roots of integers plus calculate percentages of numbers. Go HERE for details of the discussions.

May 13, 2015-How does one find the Focal Points of Parabolic and Elliptic Reflectors?

We use the vector form of the  reflection law to  find the focal point of a parabolic reflector and the two focal points of an ellipsoid. Also we discuss how such focusing capabilities are used in reflector telescopes such as the Hubble and in kidney stone pulverizers. Our own work on centrifugally spun large diameter parabolic mirrors is also discussed. Go HERE for the details.

May 17, 2015-What are some additional properties of 2D curves generated by Genetic  like Algorithms?

We construct some two dimensional curves formed by the concatenation of straight line segments connected to each other by specified angles. Figures like a Swiss cross, a multiple pulse function, a magnified and upside down pentagon, and a hexagonal spriral are generated. Closed curves are formed when the sum of the connection angles after m elements add up to 2 pi. Go HERE for the details.

May 20, 2015-How can a Train's Speed and other factors cause Derailment in a Curve?

The recent train derailment near Philadelphia which caused eight deaths and injuries to many others was a clear case of a train's excessive speed entering a sharp curve. We discuss HERE what the forces are involved in such a derailment and what speeds can be considered safe for a given radius curve.  The use of track embankment at a curve is also discussed. The main non-dimensional quantities encountered in the problem  are V2/gR  and w/2H. Here V is the train's speed, g the acceleration of gravity, R the curve radius, w the axle width, and H the distance from the track to the center of mass.

August 1, 2015-How can one use a Square to generate intricate Fractal Patterns?

We show how one can take a simple square to construct different fractal patterns. Among the more interesting are the Black Snowflake and the Dancing Clown. The fractals are generated by a generation growth process . We show how the nth generation of squares must be smaller than a certain amount to prevent overlap in the (n+1) generation. Go to MORE-FRACTAL-SQUARES.pdf for further details.

August 21, 2015-How does one generate complicated 2D curves via a concatination of straight line elements?

We re-examine the problem of generating two dimensional curves by genetic codes which define only the straight line increments  and their orientaion. In addition to regular polygons, we generate a rectangular pulse train, a staircase function and several versions of a Swiss cross. Also more complicated figures such as a square spiral and a crosslet are discussed. Go HERE for the details.

October 1, 2015-How does one determine the Area of an N sided Polygon?

We look some more at determining the area of both regular and irregular N sided polygons. It is shown that partitioning such polygons into T=N-2 oblique triangles and then finding the areas of such triangles, allows one to uniquely determine the area of any polygon. Several different examples are considered including an irregular quadrangle and also N sided regular polygons. Go HERE for the details.

October 15, 2015-What is a Cruise Missile, its development History, and future Directions?

A cruise missile is any unmanned aerodynamic vehicle capable of delivering a warhead over long distances flying at low altitude but  at high speed. We briefly discuss its history starting with the V-1 and take the discussion through the latest and most sophisticated supersonic BrahMos misile. Also,  the US Tomahawk sea-launched missile is discussed and its needed improvements pointed out. Go HERE for the details.

December 1, 2015- How can one transmit a large Semi-Prime N and its prime components p and q in secret?

We define any semi-prime N=pq via two new parameters M=(p+q)/(2sqrt(N)) and K=(q-p)/(2N) making it possible to transmit N, p, and q in secret. When M and K are used as a public key in cryptography only a receiver familiar with these parameters will be able to recover the prime p involved  and use it to send encrypted messages back to the sender without a third party being able to figure out the message being sent. Go HERE for details of the discussion.

December 12, 2015-How can one find Super-Composites?

It is well known that all positive integers are either primes or composites.  Those numbers N which can be divided only by themselves and one are primes such as 5, 11, and 41 while numbers having three or more divisors are composites such as 6,12,60. We can distinguish any number by looking at its unique number fraction f(N)={sigma(N)-N-1}/N, where sigma(N) is the divisor function of number theory. Primes have f(N)=0 while composites will typically have values of f(N) near 0.5. A small fraction of the composites will yield values of f(N) in excess of 1,2. We call these numbers super-composites. It is our purpose here to determine some aditional properties of f(N) and in in particular find some formulas capable of generating super-composites.Go HERE for the details.

December 20, 2015-What is the Black Snowflake and Cellular Automata generated by it?

A cellular automaton is a structure defined by n generations beginning with a single  closed area as the first generation followed by later generations obeying a certain specified growth program. A good example of such a structure is the Black Snowflake which uses a square of side-length s as its starting point with later generations consisting of ever smaller squares with side-length sf , where 0<f<1. An infinite number of different intricate patterns can result. We develop several additional cellular automata using the Black Snowflake as the template. Go HERE for the  discussion.    



January 15, 2016-What are the Seven Rules for Successful Stock Market Investing?

We discuss the basic investment rules I follow for successful investment in equity markets. These rules have evolved over a fifty year period ever since my first stock purchases as a teenager in the mid fiftees. The basic idea is to be long stocks only in bull markets and short stocks only during bear markets. The type of market one is in is determined by looking at  the longer term price trend  of the market averages  when compared to a running lag curve. Investment risks are lowered by not going on margin and dealing with ETFs as opposed to individual stocks. Go HERE to learn more about the details of my investment observations.

February 1, 1016-What are the Properties of Oblique Triangles?

We derive the well known laws for triangles including a determination of their areas by both geometric and vector means. Also the formulas for the radii and centers of circles
circumscribing and inscribing a triangle are derived. Next the possibility of tiling with oblique triagular tiles is demonstrated . Also a tile flooring involving the simultaneous use of  equilateral and isosceles triangles are shown. Finally we discuss how oblique triangles are important in the functioning of  trusses and tripods. Go HERE for the details.

April 17, 2016-How do Probabilities enter Games of Chance?

We examine the probabilities associated with games of chance. Starting with the most elementary of such games, namely, coin flipping, it is shown that to get n heads or tails in a row has the very low probability of (1/2)n. Although the probability of any number t through 6 coming up is always p=1/6 when one casts a single die, conditions change dramatically when dealing with two or more dice being thrown simultaneously. For a two dice game such as craps the probability of making snake eyes(1+1) or boxcars(6+6) is only p=1/36. We also look at the probability of a favorable outcome in both American and European Roulette .To only play red-black or even-odd is probably an optimum strategy considering the house advantage. A brief discussion of short term stock movements indicate that this is  essentually a random process and has no predictive value.. Go HERE for further details.

May 31,2016-How does one determine the Properties of an Equilateral Triangle using only Intersecting Interior Lines?

We show how, using simple geometry, algebra, and some analytical methods, to solve a mathematical puzzle recently posted in the Wall Street Journal. The problem involves finding the value of the constant d knowing that the line lengths of lines emanating from the three vertexes of an equilateral triangle have lengths d, 2d, and 3d/2 when all meet at a fixed interior point P(x,y). Simple geometry is used to solve part of the problem while more advanced methods based on Heron's triangle and analytic forms for circles are also employed. Go HERE for details of the discussion.

June 6, 2016-How do the Pascal Triangle and the Gaussian relate to Games off Chance?

We discuss further how games of chance can be described as face value S versus possible combinations T and how such relationships lead to the probability of an event occurring. Multiple coin flipping, marble withdrawals from a jar, and rolling of dice are all analyzed in detail.It is shown how multiple coin tosses lead naturally to the Pascal Triangle and that the combinations involving multiple die games produce a Gaussian distribution. The probabilities involved with Russian Roulette are also discussed. Go HERE for the details.

June 21, 2016-How can one quickly construct segments of  Irrational Numbers to find their neighboring Primes?

We show how one can take the products of certain irrational numbers ak taken to their pkth power to construct  an irrational number L of infinite length. By choosing a segment of length m from this number, one obtains a large finite length number M which can be conveniently represented by a short number code.  With appropriate manipulations one can then find the neighboring primes to M. Calling these p1,p2,..., one can generate semi-primes such as N=(pa )(pb). Such numbers can be conveniently used as public keys in RSA cryptography. Go HERE for the details.

July 9, 2016-What is the Orbit Period of two Gravitational Masses rotating about their common Mass Center?

We examine the period of a binary mass system rotating about its center of mass. Special cases considered are binary stars, the Earth moving about the Sun, and the behavior of earth satellites. By balancing the gravitational attractive force with the centrifugal forces about the barypoint one is able to relate orbit period to distance between the two masses and the total value of the masses. The discussions are confined to only circular orbits which are simpler to discuss than the more general case of elliptic trajectories. Go HERE for the details.

July 28, 2016-What are Power Towers and the Tetration of Complex Numbers?

 We consider the complex numbers N=a+ib and ask if Z=z[infinity]  exists when undergoing the iteration
z[n+1}=(a+ib)z[n] , subject to z[0]=1. Such iterations are equivalent to power towers such as (1/e)^(1/e)^(1/e)^ etc. For such iterates to converge at z[infinity] it is necessary  that z[n+1]=z[n] for large n. Once this is achieved  one has the explicit form for Z expressed in terms of the Lambert Function result  Z=W(ln(1/(a+ib)))/ln(1/(a+ib)). The value of Z for sqrt(2) is 2 , for 1/e is W[1]=0.56714329, and for (1+i) is 0.6410+i0.5236. Go HERE for the details.

August 16, 2016-What is the increment ratio b/a for a  Chord of a Circle inscribing an Equilateral Triangle?

In this weekend's Wall Street Journal they posed in their Puzzle Page the  problem of finding how the chord to a circle breaks up into two small increments a+a and one larger one b when an equilateral triangle is inscribed in the circle.. The question asked was to find the ratio of b/a. The ratio turns out to equal the Golden Ratio phi=[1+sqrt(5]/2. Go HERE for the details of the solution. We also extend the discussion to ratios obtained for other inscribed regular polygons including for a standard square.

August 25, 2016-What are the Properties of a Cam?

We examine the properties of cams as they arise in connection with the opening  and closing of intake and exhaust valves in internal combustion engines. Starting with an examination of the simplest cam, consisting of a circular disc rotated at constant angular velocity about an off-center rotation axis, we proceed on to elliptical type cams capable of creating conditions where a valve driven by a follower is capable of remaining shut during a good part of the rotation cycle. Also we show how the shape of cams with desired on and off capabilities can be constructed graphically starting with a chosen elliptical shape as a first approximation. Go HERE for the details.

September 4, 2016-What is the Volume of  a glass Aquarium having twelve identical Rhombic Faces ?

We examine the latest math problem posted in this week's Wall Street Journal Puzzle Page. The problem is to find the fluid volume inside an aquarium constructed from twelve identical rhombic faces. The edges of the individual faces each have length 5sqrt(3) and the length ratio of the two diagonals of a face go as sqrt(2). The problem is readily solved by just a little geometry and some spatial visualization. Also we work out a related problem of the filling of a spherical tank, This latter analysis is related to the low tech dip-stick measurements used by garage owners to measure the remaining gasoline in an underground storage tank. Go HERE for the details.

September 14, 2016- What is the Length of the shortest side of a Right Triangle given the Hypotenuse and Median Lengths ?

We examine the latest math problem posed in this week's Wall Street Journal Puzzle Page. The problem is to find the length of the shortest leg 'a' of a right triangle given a hypotenuse length of L=10 and a median length of sqrt(ab), where b is the second leg of the triangle. We use two approaches to solve the problem. The first is an intuitive approach using symmetry and simple geometry. The second makes full use of the basic trigonometric formulas for two oblique sub-triangles. Both yield the same answer but the first can be considered the more elegant. Go HERE for the details.

October 1, 2016- How many Diagonals fit into a regular N-sided Polygon? We look at a generalization of this week's Wall Street Puzzle Page question dealing with the number of unique diagonals D one can draw inside a regular n-sided polygon.The answer turns out to be D=(n/2)[n-3]. A decagon with ten sides will have 35 unique diagonals. The lowest number of  diagonals occur for the square where D=2. Go HERE for the details of the discussion.

October 12, 2016-How many unique Diagonals and Sub-Areas can one create inside an N sided Polygon?
This is an extention of an article we wrote on this web page a few days ago. We are interested in finding the number of unique diagonals D one can draw inside an N sided polygon and then use this information to determine the number of sub-areas A created. As expected the number of diagonals and number of sub-areas increases rapidly with increasing N, It is shown that N is a quadratic function of  diagonal number D given by D=N(N-3)/2. Although no direct functional relation between N and A was found, we show how to find A by a counting process involving the use of colorization. Also, some interesting designs are created using the sub-areas A. Go HERE for the details.

October 27, 2016-What is the shortest Path between N Vertexes of a regular Polygon and its Central Point?
We examine the shortest path connecting multiple points in the x-y plane. The particular problem considered deals with finding the optimum trajectory between n vertexes of a regular polygon and its center using an approach where the area in the polygon is broken up into smaller sub-triangles into some of which we position Steiner Forks which are known to minimize the path between the three vertexes of any triangle. Squares, hexagons, and dodecagons are considered. Go HERE for the details.

November 26, 2016-What are the Angles and Areas created by the Intersection of a Regular Polygon with multiple Circles?
We solve the latest puzzle posed in the mathematical  recreation page of the Nov.19, 2016 issue of the Wall Street Journal and then consider several variations thereof including the spacing left by a bundle of seven circles tightly bound by a regular hexagon. Among other problems considered  are Heron's Formula for any oblique triangle whose sides are tangent to a single radius R circle placed within the triangle. Also we show how the Golden Ratio follows naturally from the radius ratio of two different circles intersecting a pentagon. Go HERE for the details.

December 5, 2016-What are the Characteristics of Two-Stroke Internal Combustion Engines?
We discuss the properties of two-cycle ( also known two-stroke ) engines. These devices consist of a single cylinder inside of which  the upward movement of a piston causes a fuel-air mixture to be compressed. At the maximum compression point a spark plug ignites the mixture and a downward power stoke ensues. At the end of the power stroke the burnt gases are expelled by one port while simultaneously a new atomized fuel mixture is injected through a second port.  These piston movements are repeated and produce, with aid of a flywheel, a continuous rotation movement of a connected crank shaft. Heat removal is achieved via air-cooled fins. All higher powered four-cycle engines require more intricate liquid cooling. Two-cycle engines are used mainly for lower power needs such as for motor-cycles, outboard motors, lawn mowers and chain saws.  Go HERE for further details of the discussion.

December 19, 2016-How does one Express Numbers in Different Base Systems?
We show how one can express a number N in different base b systems.The main emphasis is on decimal(b=10), binary(b=2) and hexadecimal(b=16)forms. After showing how one adds, subtracts, multiplies and divides in such systems, we give the formula for quickly converting a number between any two different bases. Finally we also look at the historical used bases  of b=20 of the Mayans and the base b=60 of the Sumarians. Go HERE for the details of our discussion.

December 29, 2016-What is the Relation between a Hexagonal Spiral and the Prime-Numbers?

We show that all primes five or greater have the form 6n1. This fact makes it possible to locate all but the primes 2 and 3 along two radial lines 6n+1 and 6n+5 which cross the vertexes of a hexagonal spiral. This geometrical picture of all positive integers allows one to quickly distinguish between prime and composite numbers. The combination of hexagonal spiral and radial line is reminiscent of a spider web and is the reason we often refer to this pattern as such. Any odd number of the form 6n+3 and all even numbers must always be composite numbers no matter how large N becomes. Go HERE for details of the discussion.


January 20, 2017-What are the Properties of Squares?

We examine the properties of squares, give their mathematical representation in polar coordinates, their use in deriving the Pythagorean Theorem, use in construction of more intricate patterns, and the construction of spirals. In addition we consider the stability of towers built with cubes and determine the area ratios when cutting any square with a straight line. Finally we show a couple paintings by famed modern artist Jopseph Albers on homage to the square. Go HERE for the details.

February 6, 2017-What are the Laws of Geometrical Optics?

We use the Fermat Principle to derive the basic laws of reflection and refraction for a ray of light striking the interface between different index of refraction media. After deriving the reflection law we demonstrate how it may be used to explain the properties of corner reflectors. Next the Snell's(alias Reflection) Law is derived and used to show how , via ray tracing, one is able to calculate ray deflection as it propagates through a glass wedge. Finally we conclude by deriving the first Fresnel formula for the reflection coefficient at an interface. Go HERE for the discussions.

March 18, 2017-What is the Relation between Archimedes and Van Ceulen and the Constant Pi?

Undoubtebly, the most famous mathematical constant is the number Pi =3.145159265.. It measures the length ratio  between a circle's circumference and its diameter. Archimedes was the first some 2200 years ago to introduce a rigorous mathematical method to calculate its value  By inscribing and circumscribing a 96 sided polygon by circles, he was able to show that  Pi has a value somewhere between 3+10/71 and 3+1/7. Although the value of 3+1/7 is good to only two decimal places it was taught to students in middle school for the next 2000 years. Many mathematicians, before the invention of calculus, attempted to increase the digit accuracy by using larger n sided polygons. The most heroic of these attempts was by the Dutch mathematician Ludolph Van Ceulen who in the 16 hundreds was able to find Pi accurate to 35 places using the Archimedes Method. The calculations took him essentially his entire life and led to Pi being named the Ludolph number in Germany until the end of the 19th century. Today of course this method for finding Pi has become obsolete and replaced by artan formulas, AGM methods, and simple iteration. We show here a new method of Pi calculation which is related to the Archimedes  method but differs from the former by using an integral involving even Legendre polynomials instead of polygons. The technique converges much faster than the Archimedes approach and does so with a minimum of mathematical effort. Go HERE for the details.

April 11, 2017-What are Difference Equations?

A difference equation is an expression where the value F[n+1] at n+1 is directly related to the function F[n] and satisfies an initial condition F[0]=const. Equations of this type arise in a variety of different areas including economics,  radioactive decay, compound interest, and iteration formulas for various different mathematical constants such as π,π the golden ratio  φφ. and the exponential e. We discuss here various specific solutions of such equations. In the process we show , among other things, how they may be used to sum the  pth powers of the first n integers.Also we discuss special forms for which F[n1] yields integer values. Go HERE for the details.

April 17, 2017-How does one factor Large Semi-Primes? 

We show how to factor large semi-primes N into its prime components p and q  using an approach involving the right choice of k in the integer evaluation of two specific radicals R and S. The new method used generates starting values k1[alpha) depending on the fraction alpha=p/sqrt(N)=sqrt(N)/q.   For N mod(6)=1 primes a lower  bound on the value of k1[alpha]m is sqrt(N)/18. Go HERE fore the details of the discussions.  

April 27, 2017-How can alpha(N) and beta(N) functions be used to locate Primes, Twin Primes, and Rich Numbers? 

We introduce two new functions alpha(N)={sigma(N)-N-1}/N and beta(N)={tau(N)-2}/N0.25   , where sigma(N) and tau(N) are standard point functions found in number theory. These new functions have the interesting property that they have zero value whenever N is a prime number. The functions exhibit local maxima for N referred to rich numbers. Special rich numbers N are shown to produce twin (or double) primes at N1. Go HERE for details of the discussions.

April 30, 2017-What is the Solution to the April 29th math Puzzle in the Wall Street Journal?   

The mathematics problem posed in the April 29th
issue of the WSJ  is to find the length of the fourth segment of the hypotenuse of a right triangle  . The triangle has a superimposed  circle centered along the diagonal and just tangent to the two opposite sides of the right triangle. We solve the problem in generic form. Go HERE for the details. The fourth segment is found to have length 135 when then circle radius is 120 and the smallest segment along the diagonal is 16. 

MAY 2, 2017-What is the Shape of a Hanging Rope under its own Weight and when Weights are Hung from it?

We examine two classical statics problems, namely, the shape a rope assumes when suspended between two fixed end points and the shape when the rope becomes weightless but weights are suspended at regular  intervals along the rope. The first problem leads to the catenary governed by the differential equation y"=sqrt[1+(y')2] . The second problem leads to straight line segments between suspended weights. Examples of catenaries include the St. Louis Arch  and the attached weight solution includes the mechanic of tightrope walking. Go HERE for the details.

MAY 6, 2017-What are the Properties of  Cannonball Stacks?

We develop the formulas  for stacked cannonballs giving not only the number of balls per layer but also the number for the entire stack. Close-packed configurations for both triangular and square bases are considered.
In the analysis one comes up with triangular numbers (1, 3, 6, 10,....) and square numbers (1, 4, 9, 16,...) . The angles associated with square base stacks are shown to be remarkably similar to those existing for the Great Pyramid in Giza, Egypt. Go HERE for the details.  

MAY 29, 2017-What are my latest thoughts on factoring large Semi-Primes ?

Over the last few years I have been very interested in finding a way to quickly factor large semi-primes . If this could be accomplished in a simple manner then conventional public key cryptography would become obsolete since an adversary could then break any code involving public keys. Although elaborate procedures for factoring large semi-primes have been proposed in the literature these methods still require an inordinate amount of computer time using even the fastest supercomputers. We summarize HERE a simple way to quickly factor larger semi-primes N=pq using only the simplest of mathematical techniques. After deriving the formulas we use for achieving the factorization we also show how they may be applied to quickly solve four specific semi-primes.

JUNE 2, 2017-What is the shortest Path connecting a Source with n Sinks?   

We examine the problem of finding the shortest distance which  connects a point source with n sinks using only straight lines? This problem was first looked at by Jacob Steiner (1796-1863). He showed that connecting three points leads to a characteristic  y shaped path with two of the points located at the upper end of the two pronged y and the remaining point at the bottom of the y. For obvious reasons we term this structure a Steiner Fork. It is shown in this article  how one can combine several Steiner Forks to create optimum paths between one source and n sinks. Go HERE for the details. Note that this is not the same as the equally complicated traveling salesman problem but the two do share some similarities.

June 10, 2017-How can one use a Dip-Stick to measure the Volume of Beer in a Barrel? 

We determine the volume of beer in a standard wooden barrel and then show how a dip-stick inserted at the bung hole can be used to find this volume. This is a classic problem dating back to J. Kepler in the sixteen hundreds . It is shown via simple calculus that dip-stick measurements are quite precise if the sticks are appropriately graduated. The relation between fluid level and fluid volume is shown to be non-linear. Go HERE for the details.

June 14, 2017-How does one determine longer term Price Trends for Commodities?  

One of the more intriguing problems in economics is how to distinguish longer term price trends compared to short term price fluctuations. One way to accomplish this is to first look at a longer term price history and use this to filter out the higher frequency components. Once this has been done one can then draw a type of long term moving average referred by us as a lag curve, When the lag curve lies below the smoothed price one has a buy signal and when the lag curve lies above the smoothed price one has a sell condition. As we will show HERE this action will lead more often then not to profitable transactions both on the long and short sides of a market. We demonstrate the existence of long term price trends by examples from the housing market, behavior of GE stock, and the price of  West Texas Crude Oil.  

June 27, 2017-What are the Properties of the Exponential Function Exp(x)? 

One of the best known functions in mathematics is f(x)=ex=exp(x). We show in this article how it arises naturally when taking the derivative of the function f(x)=ax . In addition, the properties of exp(x) and the related hyperbolic functions are determined. Also we introduce a new technique based on an integral containing even Legendre polynomials times cosh(x/2) to find the approximate value of e=exp(1) to sixty place accuracy. Go HERE for the details. 

June 29, 2017-What is the Ulam Spiral and how can it be simplified by a Morphing Procedure?

The Ulam Spiral is a spiral array of integers which exhibit a distinct and non-periodic distribution of primes. We show how this rather intricate distribution of primes within the spiral essentially offers no new information on primes other than they are odd numbers for p=3 or greater. This fact is established by a simple transformation where the spacing between integer n and n+1 equals exactly n. The resultant morphed pattern places all primes(with the exception of two), along two left leaning diagonals. An even better distribution of primes  occurs using our own hexagonal integer spiral. Go HERE for the details.

July 18, 2017-What is a quick way to determine the numerical value of the basic six mathematical
 constants to 100 place accuracy?

We show how one can quickly evaluate the irrational numbers sqrt(2), phi, exp(1), Pi, γγ, and ln(2) by a new iteration approach and via integrals involving the Legendre polynomials. Unlike series expansions, the methods used  converge much faster. For example the root of two follows from the iteration S[n+1]=(A+BS[n])/(C+dS[n], where the values of the constants A,B,C, and D depend on the number of terms used in a continued fraction expansion. Integrals involving Legendre polynomials are easily evaluated using the basic computer codes for rem and quo in MAPLE. Go HERE for the details.

July 26, 2017-What is the Witch of Agnesi Curve?

This is a 2D mathematical curve which in its simplest form reads y(x)=1/(1+x2). It is named after Maria Agnesi (1718-1799) a polymath from Milan and professor at the University of Bologna. Fluent in six languages and conversant in philosophy and theology, she was the first person to study the curve in greater detail although Fermat had already looked at it a century earlier. The term Witch became attached to the curve do to an incorrect translation of the Latin-Italian word averisera into English. After deriving the parametric form of the curve from a geometrical viewpoint we proceed in determining its various properties. We also look at several related curves. Go HERE for the details.

August 23, 2017-What is a Solar Eclipse and some of  it's Properties?

An eclipse of the Sun by the Moon occurs whenever the Moon finds itself between the Sun and Earth along a straight line lying in the ecliptic plane. It is a relatively rare occurrence with yesterday's eclipse being the first one I have seen here in Florida since 1970.It is our purpose in this short article to explain some of the characteristics of a solar eclipse using diagrams and some simple trigonometric relations. Among other things we show why the path of totality in yesterday's eclipse was along a 23.5 deg tilted diagonal and had a width of about 50 km. Also the angles subtended by the Sun and the Moon as seen from Earth are determined. Go HERE for the details.

September 1, 2017-Why do fixed Time-Increments become Subjectively Shorter as one Ages?

It is well known that individuals judge fixed time increments, such as the length of a year, as becoming shorter as they age. We explain here why this is so and in the process come up the tentative law that -"Subjective measurements of time intervals decrease as one ages as the reciprocal of the logarithm of the observer's age". We motivate this as yet unproven law via various examples such as why one's eighteen year old freshman students seem to be getting younger every year and why the judged time by a four year old to his next birthday seems to be about three times as long as that judged by an advanced age senior citizen for the same time interval. Go HERE for the discussion.

September 16, 2017-What are the Properties of n sided regular Pyramids?

We discuss the properties of regular pyramids including their volume and surface area. Cones as pyramids with a regular polygon base with an infinite number of sides are also examined. The volume of all such pyramids equals one third the product of their base area multiplied by the height. A graphic of a cone in a cylinder is also presented. Go HERE for the discussion.

September 21, 2017-What is a Void Fraction and its values for bundled Regular Polygons?

We examine the voids produced when bundling polygons in either a square centered or close packed arrangement. Starting with the limiting case of circles we then proceed to polygons with either 4x2n or 6x2n sides. Both cases produce periodic arrangements of voids. The size of these voids have importance in certain practical applications such as the cooling superconductors and providing conduits in oscillatory heat transfer. Go HERE for the details.

October 1, 2017-What is the Largest Rectangle which can be placed into an Oblique Triangle?

We solve a math puzzle posed in this week's Wall Street Journal (Sept.30th). The question is to find the largest rectangle which just fits into an oblique triangle. First solving the problem in its most general form we show that no matter what the side-length of the triangle are, the triangle area is exactly twice the optimized rectangular area. For the special case of a triangle with sides 9-19-17 we find the triangle area to be 36 so that the optimized rectangle area becomes 18. Go HERE for the details.

October 8, 2017-What is ASCII and how may it be used to convert Numbers and Letters to Binary Form?

We show how ASCII (American Standard Code for Information Interchange) converts both numbers and letters to binary form suitable for electronic communication. The idea behind the method is to first construct a 3x26 table in which the first row contains all integers  1 through 26, the second row gives the binary form for these numbers and the third row contains the 26 letters A through Z.One uses an eight bit code for each number and letter. The first few bits on the left read 0011 for numbers and 010 for capital letters. The terms on the right of a group are the bits  in binary form of the particular letter or number in question. Thus, for example, the letter K reads 01001011 in binary. In addition to numbers and letters the 8 bit code offers forms for punctuation marks, brackets and spacings plus mathematical symbols and lower case letters. Go HERE for details of the discussion.

October 12, 2017-How can one use Irrational Numbers to code Messages?

We generate large random numbers using the product of irrationals. Such products N act as very long passwords for encrypted messages.  The Ns can be expressed in compact form for easy transmission. An example of  one such number and its compact  construction form is  N=2LPSG=544147299248829936628591544616211725396215111691. Only the sender and his friendly receiver have the same list of Ns in their repertoire but no one else does. This means they can easily decipher the product M(N) but an adversary can't since he doesn't know the N being used and is highly
unlikely recover it by a trail and error approach when N is fifty or more digits long. Go HERE for the details.

October 16, 2017-How does one quickly distinguish between Prime and Composite Numbers?

We derive a new prime number function defined as F(N)=1/[Nf(N2)] to distinguish between prime and composite numbers. Here f(N2) is the number fraction. Whenever F(N)=1 we have a prime number and when F(N)<1 it is composite. We obtain graphs for F(N) versus N and show how one can quickly calculate the number of primes present in any given range of N. Comparison is also made with the prime density function of Legendre and Gauss. Go HERE for the details of the discussion.

October 24, 2017-What are Twin-Primes and how are they Generated?

We re-look at twin primes such as [5,7], [11,13], and [59,61] by use of a hexagonal integer spiral. These primes , when greater than three, have the property that their average value equals a multiple of six. That is" The condition for the existence of a twin prime is that both 6n1 are primes and that N=6n has N mod(6)=0". We also use the number-fraction f(N) to generate some interesting patterns were the sequence  of increasing N goes as prime-supercomposite-prime. We have named this the PSP pattern. Go HERE for the details of the discussion.

November 1, 2017-How do Nodes, Connectors, and Sub-Areas relate to each other in Graph Theory?

It is well known that any polyhedron has its vertexes V related to its edges E and its faces F by the Euler Formula V+F-E=2. There is an exact 2D analogue to this formula which states the number of nodes plus inside areas minus the sum of connecting lines equal unity. It is our purpose here to explore this second possibility further from the viewpoint of graph theory. We start with a collection of N points (nodes) lying in the x-y plane and then connect them to each other by straight lines(collectors C) producing a number of sub-areas (A). By looking at several different examples , we arrive at the formula N+A-C=1. This continues to hold  for any distribution of points lying in a plane. Go HERE for the details of the discussion.

December 21, 2017- What is the relation between Temperature and Time for Cooling a Cup Of Coffee?

We examine the classic problem of the cooling of a cup of coffee both from an experimental and analytic point of view to show that the process follows essentially a Newton Cooling Law. The cases of plain water, black coffee, and coffee mixed with cream are considered. They all follow the same law where the non-dimensional temperature goes as θθ(t)=exp(-kt) with k an experimentally determined constant. It is found that one need not add the cream to hot coffee when first served in order for the mixture to stay warm as long as possible. Go HERE for the discussion.

December 25, 2017-What is the Area of a Spherical Cap drawn on a Sphere by a Compass?


We answer a question appearing in last Sunday's Wall Street Journal puzzle page. The question is to see how the area of a spherical cap created by a compass tracing out a circle of radius L on the surface of a sphere varies with sphere radius.We approach the problem from both an elementary viewpoint and then also via more rigorous mathematics. In both case one finds the spherical cap area generated is always equal to A=ππL2 regardless of the sphere size. Go HERE for the details of the discussion.

January 5, 2018-How does one determine Price Trends for any Commodity including Stocks?

It is well known that any commodity will have periods of long uptrends and long downtrends during its existence.If these trends can accurately identified early after the onset of a new trend large profits can be made.The secret for positive financial returns is to first identify the trend one is in and then stay with the trend until there is a reversal. Whether one acts on the long or short side of a market is immaterial as long as the price stays above a lag curve during bull markets and below a lag curve for bear markets. Go HERE to see the details of our discussion.

January 30, 2018-How does one convert Infinite Series to Infinite Products?

It is known that  any polynomial f(x) may be expressed as a product function involving all roots of such a polynomial. This fact continuous to hold for certain periodic functions with an infinite number of distinct zeros  so that an infinite series can be expressed as its equivalent form as an infinite product..We show in this article how this may be accomplished by looking at the specific functions sin(x)/x, cos(x). cos(x)2, and tan(x)/x. Often such equalities can produce very interesting results such as Leonard Euler's famous identity  Pi2/6=1+1/4+1/9+1/16+... . Go HERE for details of the discussion.

February 10, 2018-How can one use Sticks to construct any Polygon?

We show how one can construct any N sided polygon by the concatenation of straight sticks of specified length L connected to their neighbor by a specified angle θθ. The conditions that the resultant figure close on itself are that the sum of the external angles add up to
2ππ and that the sum of the x and y components of the stick arrangement vanish. The oblique triangle and non-symmetric quadrangle are discussed in detail with specific examples given. Also the interior area for any regular polygon is derived. Go HERE for the details.

March 1, 2018-How can one evaluate any Trigonometric Function using only the Values of Tan(a) in 0<a<45deg?

We use a new approach involving higher order Legendre Polynomials to obtain twenty digit accurate approximations for the tangent function tan(a) over the limited range  of  0<a<45deg. It is then shown that this information may be used to determine the values of any of the other trigonometric functions at any 'a' including points outside the region. Also the location of poles of the tangent function are well estimated by setting the polynomial in the denominator of its approximation to zero. Go HERE for details of the discussion.

March 6, 2018-When is the Time to Buy and the Time to Sell a Stock or other Commodity?

Applying the aphorisms (1) For every season there is a time to sow and a time to reap  and (2) Know when to hold them and know when to fold 'em to stock and commodity investments, we explore the financial adage Buy Cheap and Sell Dear. That is, we discuss what procedure one should use to be on the right side of market moves using both long term price charts and especially designed lag curves. Analysing the S&P500 Stock Index, we indicate the points where one should buy and be long and the points in time where one should be short. A long term following of the outlined procedure has produced excellent returns for me over what is now a time span of over sixty years. Go HERE for the details.

March 11, 2018-What are the basic Principles of Calculus?

Calculus is that field of mathematics to which students are exposed right after analytic geometry and trigonometry. It deals with continuous functions defined over given intervals. The two parts of the subject are known as Differential Calculus dealing with derivatives of functions and Integral Calculus involving the determination of areas under specified curves. We present in this article a very much abbreviated version of calculus which is intended to be used as a handy supplement to existing calculus books often exceeding five-hundred pages in length or so. We call the present description Calculus in a Nutshell. Once you master it you will br well on the way to understanding of the entire subject. Go HERE for the details of the discussion.

March 22, 2018-What was the famous grave marker placed at the now lost grave of Archimedes of Syracuse?

It is well known that the famous ancient Greek mathematician, physicist, and inventor Archimedes (287-212BC) had a column topped by a sphere in a cylinder placed above his now lost grave in Syracuse, Italy. During his researches he had found that the volume of a sphere to that of a circumscribing cylinder is precisely 3/2. We modify this problem and show via some elementary calculus that the volume ratio of an ellipsoid relative to  an inscribed and circumscribed cylinder goes as  sqrt(3) and 3/2 respectively, regardless of  the  aspect ratio of the ellipsoid. Go HERE for details of the calculations.

March 26, 2018-How does one construct Large Parabolic Mirrors using a Spin Technique?

We examine a method of spin generating parabolic mirrors ideal for solar experiments requiring solar concentrations of several hundred suns. With an added secondary  concentrator (axicon) at the primary mirror focus one is able to achieve about a thousand sun concentration making such a two reflector configuration ideal for the pumping of small  cylindrical liquid cooled lasers. The construction technique was first developed by a group of us working on a NASA contract in the early 1980s . It involves revolving a preformed near parabolic surface of a stabilized foam pad at constant angular velocity about a vertical axis while  poring liquid epoxy onto this  rotating platfom. After several hours of spinning  the epoxy hardens to form a nearly perfect parabolic surface attached to the underlying foam layer. After coating the resultant mirror with reflective mylar, a large diameter D and long focal length F mirror will have been produced. Go HERE for details of the discussion.

April 5, 2018-How good are the Correlations between various US stock Indexes?

We look at the correlation in price and time of various US stock Indexes including the S&P500 and the Nasdaq Composite.The correlations between the indexes are found to be excellent on both a long term (20year) and short term (3 month) basis allowing one to make accurate predictions of when an index and hence it underlying stocks are in an uptrend (bull market) or a downtrend (bear market).The more speculative stocks present in the Nasdaq index tend to rise more during bull markets and fall more during bear markets relative the the stocks present in the S&P500. Go HERE for the details of the discussion.

April 9, 2018-What are ETFs and how may they be used during Bull and Bear Markets?

ETFs are a new type of investment instrument which relies on the underlying value of a collection of stocks. They have become very popular with investors during the last decade who recognized that it is almost impossible to improve on investment returns exceeding a good market average such as the S&P500. We discuss in this article some of the characteristics of ETFS ( Exchange Traded Funds) including what type of  ETF to use during bull and bear markets. The best known and also largest product of price times daily volume ETFs are SPY and QQQ applicable during up-markets. Their down-market counterparts are SDS and QID, respectively. Go HERE for details of our discussions.

April 24, 2018-What are the Characteristics of Projectiles associated  with Hand-Held Weapons?

We examine the properties of projectiles ranging from thrown rocks to bullets from sniper rifles. The important quantities of velocity, mass, momentum and kinetic energy for these different projectiles are examined and presented in MKS units in both graphic and table form. It is found that all of the five different projectiles considered have momenta slightly above 2 kg-m/s and all can be lethal at different ranges. Go HERE for details of the discussion.

May 10, 2018-How may Lag Curves be used to predict Uptrends an Downtrends in Equities?

We show how using only lag curves for both long and short term time windows one can accurately indicate at what stage an equity finds itself and more importantly whether it is in an uptrend or downtrend. Using only two types of price history graphs, one based on a twenty year time period, and the other on a six month time scale, we can accurately predict the type of market one is in be it a Bull Market (uptrend in prices) or a Bear Market(downturn in prices). The case of GE is discussed in some detail showing how this widely held stock , although in a long term downtrend, still offers opportunities on the long side at shorter time scales. Go HERE to see the rest of the discussion.

May 25, 2018-What is a Caustic and how may it be generated using Ray Tracing?

We use geometrical optics to generate the familiar light pattern often seen at the bottom of coffee or tea cups. Using ray tracing we first find the formula for all straight lines representing reflected light rays of off a circle by an incoming parallel light array.  After plotting a family of these curves, a typical cusp pattern associated with a standard caustic emerges. Comparison of this caustic with the classic nephroid are also made. Go HERE for details of the discussion.

May 26, 2018-What was Operation Hydra?

Operation Hydra was the code name for a 596 heavy British bomber attack on Peenemuende along the Baltic Sea in northern Germany. The raid took place seventy-five years ago on the night of  August 17-18, 1943. The British had just discovered the top secret rocket development facility there and with greatest urgency devised a plan in early 1943  to eliminate as many of the scientists and engineers and their families by saturation bombing of the  housing facilities in Peenemuende. I happened to be living there at the time as a six year old boy so that my family and I were on the receiving end of this bomb attack. At the end of the raid we had escaped with our lives twice that night. Go  to OPERATION-HYDRA to see further details of the story and how it influenced my later life.

June 3, 2018-How do Areas, Connectors, and Vertexes vary for N sided Regular Polygons subject to superimposed Diagonals?

The 2D version of Euler's famous topological formula reads A+V-C=1, where V are the number of points (nodes) connected by C lines to produce A sub-areas. In this article we will be concerned with applying this result to N sided regular polygons connected by D diagonals connecting opposite vertexes. In particular we are interested in how many sub-areas A  are formed by connecting the vertexes of a polygon by the maximum number of possible allowed diagonals. Treating polygons N=4,5,6,7, and 8,we obtain some interesting patterns which when colored make for interesting computer art. Go HERE for the details.

June 6, 2018-What are the Rust Belt Cities, how did They get that Way, and what is the Route to Their Recovery?

Starting in the late 1970s many cities in a swath about the Great Lakes including Detroit, Flint,Youngstown, Cleveland, Pittsburgh , Buffalo and Erie found that their major industries  could no longer compete with foreign manufacturers on quality and price. As a result the manufacturers in these cities went out of business producing massive unemployment , an increase of poverty levels, and a mass exodus. These cities soon became to be known as the Rust Belt Cities. Most even today still have not been able to recover. I detail my own observations of one of these. In addition we give a recipe for eventual recovery of these cities following the Pittsburgh model of jobs created through scientific and technological innovation  produced by a highly educated work force. We show why tariffs are useless  for such recoveries while financial support for industries and services having favorable trade balances with the rest of the world needs to be encouraged and subsidized by the government. Go to RUST-BELT-CITIES for further details.

June 17, 2018-Where is the Danube River located and what are some of the more important Cities lying along its Banks?

We give a brief  travelogue on  the River Danube starting at its source in the Black Forest to its exit into the Black Sea in Romania. In particular we discuss the history of some of the better known cities lying along its banks and also present some of our own  observations acquired through many years of travel in the region. Starting at the source at Donaueschingen in Swabia to its end in Romania at the Black Sea we give some of the historical details of the better known cities along the way including Ulm, Neuburg, Ingolstadt, Regensburg, Passau, Linz, Vienna, Bratislava, Budapest,  and Belgrade. Also we have added some photos taken by us or found on the internet to accompany the dialogue. Go DANUBE for the details.

June 30, 2018-What are the Properties of Regular n-sided Polygons?

We examine the properties of all n-sided regular polygons including their external and internal vertex angles, their area and the length of their diagonals. A generic form for the area of all n-sided polygons is derived and the limit of this area to that of a circle is shown as n approaches infinity. In addition to deriving the Golden Ratio for pentagon diagonals, an infinite number of other irrational numbers are obtained with some of these expressible as functions of the roots of integers. An upper and lower bound for Pi is also derived using polygons with large n. Go HERE for the details of the discussion.

JULY 7, 2018-What is meant by Compound Interest?

The return on capital by compound interest is examined in some detail. Beginning with compounding on a yearly bases, we reduce the payment increments to ever shorter time intervals until the point of continuous compounding is reached where the capital C(n) increases exponentially as  C(n)=Co exp(in) , with n being the years held, i the yearly interest rate and C0 the initial capital invested. A discussion of the Federal Reserve's misguided inflation policy  which has reduced the true value of the dollar by a factor of  over seven in the last hundred years is also discussed. In addition we  emphasize  the fact that (in a paper economy) successful investment requires that the yearly return exceeds losses due to inflation and taxes. Click on COMPOUND-INTEREST to see more details.

July 27, 2018-What are the Properties of Integers?

We discuss the properties of all positive integers, how they first developed plus their even-odd character and prime or composite forms.
After introducing the concept of number fraction f(N), we clearly show that primes correspond to f(N)=0 and super-composites to f(N)>1.4. Mersenne, Fermat, and Perfect Numbers are also discussed. In addition, logarithms of integers to various bases are examined. Go to ALL-ABOUT-INTEGERS.pdf  for further details.

August 14, 2018-How is the Hexagonal Integer Spiral generated and used to classify Prime Numbers?

It is shown how from a class discussion on the point function (1+i)n we developed a new type of integer spiral for which all
primes fall along just two radial lines 6n+1 and 6n-1 provided the primes are greater than three. The vertexes of the resultant Hexagonal Integer Spiral locate all positive integers starting from 1 through N. Mersenne and Fermat primes are shown to be sub-classes of the 6n+1 and 6n-1 primes, respectively. We also introduce a more general prime number generator T[a,b,c]=ac+b. Go to GENERATING-A-HEXAGONAL-SPIRAL.pdf for further details.

August 25, 2018-What is the complete form of the Goldbach Conjecture?

It is well known  that the main part of the Goldbach Conjecture in number theory is that all even numbers can be represented by the sum  of at most two primes. What is less often mentioned is his extension to odd numbers as also conveyed in 1742 letter to Leonard Euler. The second part of Goldbach's Conjecture is that any odd number N can be represented by the sum of at most three primes. We look here in more detail at this second part proving that it is also valid. The proof depends on the fact that all the odd numbers we considered have multiple triplet expansions with the number increasing monotonically with increasing N. Since only one triplet is required to confirm the conjecture, it is clear that it also holds. Go HERE for further details.

September 2, 2018-What are some Additional Features of the Five Platonic Solids?

In an earlier note we discussed the mathematics of the Platonic Solids. In this article we extend these discussions to include
certain additional facts less well known about these five solids, such as the fact that they all make fair die for any gambling game. After deriving their surface area and volume and their corresponding surface to volume ratio,we show how they may be constructed using metal, wood, or cardboard. In particular we concentrate on the dodacahedron which apparently held religious significance to first and second century AD Romans. Go to MORE-PLATONIC-SOLIDS for further details of the discussion.

September 10, 2018-What are some of the 2D Patterns which fold into closed 3D Structures?

We examine the 2D patterns which can be used to form the Platonic Solids when folded. Also other patterns capable of producing other 3D structures of not necessaroily the same faces when folded are considered. In all cases the resultant closed polyhedra satisfy the Euler condition that V+F-E=2 is satisfied. Here V are the number of vertices of the 3D structure while E represents the edges and F the number of faces. Go to FOLDING-2D-PATTERNS.pdf for further details.

September 16, 2018-What is the Twin Prime Conjecture and how can it be Confirmed?

We look at the well known conjecture that the number of twin-primes are infinite. Although no rigorous mathematical proof exists that this is so, a simple extrapolation of the twin prime sum for integers out to about N=20,000 allows us to come up with a power law which states that the twin-prime sum S(N) relates to the integers N=6n as  S(N)=0.19248 N 0.7532.  Thus as N goes to infinity so does S(N) and hence there must be an infinite number of semi-primes. Go to TWIN-PRIME-SUM.pdf for further details.

September 20, 2018-What are the times of Sunrise, Sunset, and Local Noon at any point in the Northern Hemisphere?

We examine the equations for the Celestial Triangle to determine local noon and the times of sunrise and sunset at any latitude between LAT=0 and LAT=67.5deg and any longitude. The precise values for here in Gainesville, Florida (LAT=29.65N, LONG=82.45W) are calculated for several different dates throughout the year especially at the Equinoxes and Solstices.. The time between sunrise and sunset as a function of latitude are also discussed. A graph is created to determine the departure in minutes from a mean value for local noon due to the earth's not quite circular orbit about the Sun. Go to  SUNRISE-SUNSET for further details.

September 26, 2018-What was the Roman Limes?

We describe the location and characteristics of  the ancient Roman defense wall across in southern Germany known as the Limes Germanicus. This was a border wall between the Roman occupied territories west of the Rhine and south of the Danube extending from present day Cologne on the Rhine to Regensburg on the Danube. Its purpose was to prevent the northern Barbarians from entering Roman occupied territory. It existed from 83AD to 260AD, was about 500km long and studded by forts and watch towers at 5km intervals.  The main legions assigned to enforce the border wall were located at Mainz (Moguntiacum) and Regensburg (Castra Regina). Click on ROMAN-LIMES for further details. Also we point out that limes is pronounced ' leemess' and not as the citrus fruit.

October 21, 2018-What is the Trump Market and what produced its Rise and Fall?

We examine the behavior of the Standard and Poors 500 Stock Index over the past two years starting in November of 2016 through today of  October 21, 2018.The reason for its sharp rise until January of this year was mainly due to the election of Donald Trump as president of the United States. Some of the more onerous restrictions placed on the economy by the previous administration and the promise of more of the same had Hillary Clinton been elected made most investors extremely optimistic that she wasn't. By January of 2018, however, the Trump up Market had risen too far compared to historical standards and the new imposition of tariffs promised to restrict profits and hence US stock dividends. Although there was a brief reprieve  from a further downturn by the summer of this year, the sell off continued starting in early October of this year and we are probably now seeing the fall of the Trump Market . Go HERE for further details including reasons for the market actions.

Nov.11, 2018-How does one Construct a regular Icosahedron ?

Of all of the five platonic solids, the twenty faced icosahedron is the most complicated and difficult to analyse. We show in this note
how one can use elementary mathematics to locate all twelve of its vertices and from this information construct an icosahedron by tiling with equilateral triangles. Using polar coordinates, the vertex points are readily calculated. This allows for the construction for a wire-frame map in the shape of an icosahedron. Once this frame is covered by tiles, the desired figure will result. We also show how one can use a 2D Duerer net to produce this structure by a simple folding procedure. Go HERE for details of the discussion.

Nov.15, 2018-How does one easily obtain Accurate Values for Tangent and other Trigonometric Functions?

About a decade ago we discovered a new way to rapidly obtain estimates for certain functions using integrals involving the Legendre polynomials P(n,x). We want in this note to extend these earlier discussions, concentrating on the two integrals  int(P(2n,x)cos(ax),x=0..1) and int(P(2n+1,x)sin(ax),x=0..1). After finding quotient approximations for tan(a) to various degrees of accuracy, we produce a table of tangent values accurate to at least ten digits in the range 0<a<Pi/4. Once this has been accomplished we then use this information to find values for the other trigonometric functions to the same order of accuracy. Go to MORE-ON-TAN for details of the discussion.

Dec.3, 2018-How do Number Sequences relate to Finite Difference Equations?

We examine numerous number sequences S(n) and show how they relate to finite difference equations subjected to certain
starting conditions. Among the sequences considered are the power sequence S(n)=sum(kp,k=1..n) and the Fibonacci Sequence and its variations. We also look at f[n+3]=f[[n+2]+f[n+1]+f[n]  , with f[1]=1, [2]=2 and f[3]=3. Here the resultant sequence reads S(n)=1,2,3,6,11,20,37,68,125,... and the ratio of the elements approach the unique value of  1.839286755... as n gets large. Go HERE
for details of the discussion.

Dec.21, 2018-How does one optimize the return on the Exchange Traded Fund SPY?

We use the long term price history of the popular exchange traded fund SPY versus a drawn by eye lag curve to show how one can trade this fund successfully in both bull and bear markets. A simple buy and hold strategy over the 24 year time period of 1994-2018 would have increased one's investment by a factor of 6x. On the other hand, our lag curve approach would have achieved a far superior multiplication of  22x. Go HERE for details of how the procedure works.

Dec.27, 2018-How can one ride long term Price Waves of the S&P 500 Index for consistent Profits?

We show how the use of long term price data for the S&P 500 Stock Index exhibits a wave like behavior which when used in conjunction with a lag curve can yield consistent profits on invested capital on both the long and short sides of the US stock market. Using a thirty year historic chart of the S&P 500 Index, one can clearly distinguish when the stock market is in an uptrend or a  downtrend. Following these trends produces profits far superior to what can be accomplished by a buy and hold approach. We also show how applying the present wave method to the 1920-1940 time period would have gotten one out of any long position unscathed some two month prior to the October 1929 market crash. Go HERE for details of the discussion.

DEC.31, 2018-How can one use Lag-Curves and Cross-Over Points to determine longer term Trends for any Stock?

We show how uptrends and downtrends can be found for any stock by using long term price data with a running Lag-Curve to find the cross-over points where long or short transactions should be taken. After a detailed discussion of GE, we examine the stocks AMZN, MCD, AAPL and HD plus the global Dow Jones Average $DJWO. In all cases these stocks and the global average have been in a downtrend since September of this year after impressive earlier gains. According to our technical rules one should never hold such formally active up moving stocks in a downtrending market. We also discuss the behavior of one of my earliest ill-timed stock purchases in 1959. We show how , as a naive investor some sixty years ago, I set myself up for some losses by buying during the down trending phase of RTN at $50/sh after it had dropped from a maximum of $72 earlier in the year. Needless to say this action was never repeated by me at any later date for any stock. Go HERE for further details of our overall trend discussion.

JAN.4, 2019-What are the Area Ratios between Generations of Nested Regular Polygons?

We look at the problem of nesting regular polygons such that the (N+1) generation has its vertices just touch each of the centers of the sides of the previous  Nth generation. After looking at the specific cases of equilateral triangles, squares, and hexagons, we develop a general formula for the area of the nth generation of any regular polygon. Although the found generic formulas for the areas of the Nth and (N+1) th generation are quite complicated, the area ratio R assumes the simple form of R=1/ [cos(Pi/n)]2 , where n equals the number of sides of the polygon. Go HERE for details of the discussion.

JAN.7, 2019-Are all Prime Numbers greater than Three of the Form 6n1?

We show that all primes five or greater have the form 6n+1 or 6n-1 without exception. Thus the prime N=14566243 is equivalent to 6(2427707)+1 and N=107507=6(17918)-1. That primes satisfy this form is a neccessary but not sufficient condition since there are also many composite numbers which  have the from 6n1. We also show that all primes fall along just two radial lines in a hexagonal integer spiral representation. Go HERE for the details of the discussion.

FEB.9, 2019-How did Eratosthenes make the first accurate measure of the Earth's Circumference?

The first accurate measurement of the Earth's circumference C was carried out by Eratosthenes of Cyrene in about 246BC. He used the Sun's angle difference at local noon during the Summer Solstice at both Alexdandria (Lat-31.2N) and Syene (Lat-24N). The angle difference was 7.2 degrees. From it, simple geometry showed that the Earth's circumference is C=(360/7.2) S=250,000 stadia, with S being the known distance of 5000 stadia between the two cities. This estimate is very close to the actual Earth circumference although some questions remain about exactly what value of the stadia Eratosthenes was using. We also discuss two simpler alternate measurement routes he could have taken to measure C, but didn't, involving the Alexandria Light-House. Go HERE for the details of the discussion.

FEB.15, 2019-What is Parallax?

When looking at an object by eye one receives two slightly different images on the retina for each eye. The brain merges these images into a single one lying at distance D where the object is located. Parallax represents the angle αα=arctan(d/D), where 2d is the distance between the eyes and D the distance to the object. The concept of parallax explains not only stereosopic vision but is also of great use in finding distances between stellar bodies. We describe its use in determining the earth-sun distance and also the 4.2 light-year distance to Proxima Centauri. Go HERE for details of the discussion.

FEB. 20, 2019-What are the Properties of a Binary Abacus?

We show how to construct  and manipulate an abacus based on the binary number system. After presenting a schematic of such a digital device, we show how the basic operations of addition, subtraction, multiplication, and division are achieved. It is shown that any number in binary form when multiplied by 2n  simply adds n zeros to its sequence. All numbers equal to 2n-1 are represented in binary by n ones while a number 2n+1 has the form 1 followed by n-1 zeros and finally ending in a one. Go HERE for details of the discussion.

MARCH 9, 2019- How does one determine the Mass Centers and Centroids of Bodies?

After deriving the basic equations for mass centers and centroids of bodies, we apply these formulas to carry out detailed calculations making use of symmetry and elementary calculus. Among other problems, we determine the center of mass of a cone with density which varies linearly with height and  the centroid for a standard pyramid. Also we show how different calculations  are simplest to carry out with choice of an appropriate coordinate system. Go HERE for details of the discussion.

MARCH 11, 2019-How can one eliminate the biannual need for changeover between Standard Time-Daylight Savings Time ?

The changeover from standard to daylight savings time (DST) and visa versa has become a real burden to not only homeowners who are required to often reset over a dozen clocks a year but also for the travel agencies and especially the airlines. This morning I saw an interesting article in the local newspaper(the Sun) about Marco Rubio(R-FL who is proposing doing away with standard time entirely and just staying on DST year round. He is on the right track but is meeting opposition from teachers who worry about their you
ng students having to line up for their schools bus while it is still dark. We propsse here an alternative approach which would eliminate both standard and daylight savings time as they are today and instead add one half hour to the local noon as found in the middle of the four time zones within the US. Go HERE for details of the discussion.

APRIL 13,2019-How can one use Lag Curves and Price History to determine Market Tends?

We explain how one can construct and use Lag Curves in conjunction with Price History to determine when a commodity is in an up or down trend. It is shown, through numerous examples,  how one may use such signals to obtain profitable  returns on any stock or stock index. The approach we are using is purely technical and ignores fundamentals and the need for predictions when a trend will end. The points for buy (B) and sell (S) occur when the price curve P(t) and lag curve become equal to each other. Such points are referred to as cross-over points. Go HERE for details of the method.

APRIL 15, 2019-What are the Properties of Electromagnetic Waves?

We examine the properties of the electromagnetic waves predicted by Maxwell's equations and show how frequency varies inversely with wavelength for waves ranging from radio to gamma rays.Also we show that only certain frequencies are transparent to the atmosphere. After classifying things according to wavelength and frequency, their role in wi-fi and cell phone technology is discussed. In addition we look at possible health effects produced on living cells for both non-polarizing and polarizing forms of this em radiation. Go HERE for details of the discussion.

May 7, 2019-How were the Moai moved from their Quarry to the Shoreline of Easter Island?

We examine the megalithic structures termed  Moai on Easter Island by discussing their features, physical properties, quarry and transport techniques. These statues, consisting mainly of large stylized heads equal to half their height, where hewn out of soft volcanic tuff and transported for miles to reach their final position  on top of a wall facing with their backs to the ocean. It is shown that the most logical means of transport available to the natives ( unfamiliar with the wheel and metal working ) was dragging the Moai horizontally. With enough men, some lubrication, and good ropes this should have been a relatively simple task. Go HERE for details of the discussion.

May 20. 2019-How are the Binomial Expansion, Pascal's Triangle, Coin Flipping and the Gaussian related to each other?

We show that the Binomial Coefficient C[n,k) for large n approaches a standard Gaussian. Furthermore a normalized Pascal Triangle can be used to measure the probability of any coin flipping procedure. Also, the time dependent temperature in the vicinity of a local hot spot can be be nicely represented bv a widening but height decreasing Gaussian. The IQ disribution of the US population also has the form of a Gaussian with a mean of 100 and a standard deviation of 15. This fact allows one to directly measure the fraction of the US population which has an IQ above a specified value. Go HERE for details of the discussion.

June1, 2019-How can  Price Time Windows be used in conjunction with Lag Curves to detect Equity Trends?

We show how with the use of different length price windows taken together with sketched in lag curves one can predict price changes into the immediate future. On does not know how long a new trend will last but nevertheless can act accordingly until the trend changes. For the case of exchange traded funds (ETFs)  such as SPY and QQQ we use three time windows of  twenty-five, five and one year duration to indicate whether markets are in an up or down trend. Points where trends change are marked by B or S and indicate what action should be taken. Go HERE for details of the discussion.

June 12, 2019-How does one generate and manipulate Numbers in different Bases?

We examine numbers N expressed in different bases with emphasis on binary and hexadecimal forms. A list of the first 40 decimal numbers are tabulated in their equivalent binary forms given. Observation of how such numbers are generated are discussed in detail. We also show manipulations with a binary abacus. Several additional calculations involving a binary systems are presented.  The hexadecimal system is shown to produce extremely simple presentations for 16 taken to any positive integer power. Go HERE for the details of our discussion.

July 2, 2019-What is a Frieze and how is it Constructed?

A frieze is any base pattern which is repeated continually in a given direction.Good examples of such figures are found in architecture, rug borders, floor mosaics and web page separators. We show how they are constructed using Heaviside Step Functions, Examples derived include square pulses, waves, and arrow friezes. A second method of construction using copy and paste in the microsoft paint program is also discussed, Go HERE for the details.

August 4, 2019-How does one trade ETFs such as SPY using the S&P500 Stock Index and its Lag Curve?

We show how one can use a broad stock market average such as the S&P500 Index with its time-dependent price P(t) and its constructed lag curve λλ(t) to successfully trade and invest in certain Exchange Traded Funds such as SPY and QQQ. Although the return using this approach will not match the possibly higher yield investing in individual stocks, the method is much safer and offers excellent liquidity when dealing  with high price-volume ETFs. We present a long term historical price chart extending back 25 years to help determine the type of stock market one is in at present. The basic rule to adhere to at all times is to be long only if the price P(t)>λλ(t) and short when P(t)<λλ(t). Go HERE for details of the discussion.

August 27, 2019-What are the four main uses of Nuclear Energy?

We examine in some detail the four basic uses of nuclear energy. These are bombs, nuclear reactors, isotopes in medicine , and nuclear propulsion. Schematics of both fission  and thermonuclear bombs are presented and their various components discussed. Basic nuclear reactors are shown two consist of two loops which exchange heat occurring at a common heat exchanger. The use of cyclotrons to create radioisotopes is also discussed as well as a potential nuclear powered cruise missile. Go HERE for details of the discussion.

September 5, 2019-How does one distinguish Prime Numbers from Composite Numbers?

We divide all numbers into either composite numbers characterized by multiple divisors and  primes which are only divisible by themselves and one. This separation is most easily accomplished by evaluating the number fraction f(N)=[sigma(N)-N-1]/N, where sigma(N) is the divisor function of number theory. For all prime numbers the value of f(N)=0 while for composites f(N)>0. We also show from a graph of N versus f(N) that all primes greater than three have the form 6n+1 or 6n-1 without exception. Twin primes and semi-primes are also examined using  points located along a hexagonal integer spiral. Go HERE for details of the discussion.

September 11, 2019-What is the KTL Method and how can it be used to yield very accurate estimates for Tan(x)?

We discuss a new mathematical technique based on the use of even Legendre polynomials to produce highly accurate approximations for all trigonometric functions. The procedure was first introduced on our MATHFUNC web page in 2012. It has since that time been named the KTL method in the literature. By evaluating the integral int[cos(ax)P(2n,x),x=0..1], where P(2n.x) are the 2nth order even Legendre Polynomials, one finds a quotient approximation for tan(a) accurate to any desired number of digits provided one makes n large enough. Since tan(a) relates directly to sin(a) and cos(a), we are able to easily generate a table with elements accurate to 50 places in 0<a<Pi/4. Go HERE for details of the discussion.

September 14, 2019-How can dual Price Charts distinguish between Bull Markets and Bear Markets?

We create dual price charts for a variety of different commodities including stocks, bonds, and home prices. These allow us to clearly distinguish between Bull(uptrend) and Bear (downtrend) Markets leading to profitable transactions on both the long and short side of markets. The dual curves  present both price and inverse price as a function of time typically over ten year time frames. Go HERE for details of the procedure.

October 3, 2019-How does one construct Fractal Curves starting with a Unit Square?

We examine the construction of several fractal curves using different base elements and starting with a unit square. The results for the first few iterations allow one to predict both the enclosed area A(n) and the boundary length L(n) for each generation. One finds that A(n) remains one for all iterations n (for the particular generating elements chosen) while the bounding curve  increases continually in length following the rule L(n)=4(b/a)n. Here 'b' is the the number of length increments making up the generating element while 'a' is the direct distance in increments between the ends of the generating element. That is, the fractal curve will have a Hausdorff Dimension equal to d=ln(b)/ln(a). Go HERE for details of the discussion.

November 11, 2019-How does one determine when a Stock is a Buy or a Sale?

We show that the basic idea of buying low and selling high can be applied to stocks by use of long term historical price charts coupled to running lag curves. As a result of such a technical approach one can better understand when any equity is expensive or inexpensive and thus when a transaction should take place. Numerous examples of past and future buying and selling opportunities for better known US stocks are analyzed and clear buy(B) and sell(S) signals given. At the moment many of these exhibit upward moving trends indicating a Bull Market.Go HERE for details of the discussion.

November 18, 2019-What were the buy and sell points for the Dow Jones Industrial Average over the past 100 years?

We use five long term(20 year) price charts together with lag curves to determine when one is in a longer term uptrend (bull market) or longer term downtrend (bear market). The beginning of such trends are designated by B for being long the market and S for being short the market.Typically in a twenty year range one has about three up markets and three down markets. By heeding the B and S signals one will always remain on the right side of a trend. Detailed discussions of each of the five long term graphs show how one would have been out of any long position several months before the 1929 market crash and been able to take advantage of the factor of ten rise in stocks during the period 1980-2000. Also one would have gotten out of any long positions shortly before the beginning of the dot.com bubble collapse(2000) and the great recession(2008). Go HERE for details of the discussion.

November 28, 2019-How are all the Positive Integers categorized into their various Sub-Groups?

We show how all positive integers N may be classified into subgroups of even, odd, prime, semi-prime,twin-prime, composite,and super-composite. Both of our earlier found concepts of hexagonal integral spiral and number fraction f(N) are employed to do this. Results show that f(N)=0 produces primes, 0<f(N)<1 yields composites, and f(N)>1 produces
super-composites with  numerous divisors.Twin primes exist only if their mean valueequals 6n, with n=1,2,3,. .Go HERE for details of the discussion.

December 12, 2019-How do Microwave Ovens work?

Microwave ovens, their configuration, power source, and cooking ability are discussed in detail. We first present a schematic of a typical microwave oven, then look at the magnetron
power source, and next discuss how the generated microwaves produce heating of the water molecules in foods. The microwaves used are typically in the 2450MHz range
 meaning the wavelength is about 12 cm. The microwaves used are non-ionizing radiation since their photon energies are orders ofmagnitude smaller than that required
for tearing molecules apart as occurs with x-rays. Go HERE for details of the discussion. 


March 1, 2020-How Does a Vapor Compression Refrigerator work?

A standard vapor compression refrigerator functions
by withdrawing heat from a cold reservoir  and depositing it at the hot side of the device. The typical liquid-gas mixture
used in this heat exchanger is tetrahflouromethane alsoknown as R-314a. A typical refrigerator has just four basic components. These are the compressor, the condenser, the
expansion-valve, and the evaporator. Such machines are extremely reliable and no longer use toxic or other harmful gases.Go HERE for details of the discussion. 

March 12, 2020
-What are my Thoughts on the Corona Pandemic?

We discuss the Corona Pandemic and show how existing data can be used to make predictions about the number of infections and deaths to be expected during the
 coming months. We also show what makes this virus such an effective killer by keeping the infection rate low but he means of virus transmission easy. Go
 HERE for details of the discussion.

March 21, 2020-How does one Extrapolate the total Number of expected Corona Deaths from existing Data?

We use data published between March 10th and March 20th to obtain estimates for the number of expected deaths due to the Corona Virus. Using an exponential model, our extrapolation formula reads D=250exp(0.062t), where D represents the number of worldwide deaths and t the time in days from Jann22 when the pandemic began. The number D reaches about one million at day t=140. Assuming an equilibrium distribution,  puts the US numbers at 1/20 th of the world values. Go HERE for details of the discussion.

MARCH 27, 2020-How can one estimate the  Cumulative Number of CoronaVirus Deaths thirty Days from Now?
We use existing data on the total number of Corona Virus deaths to extrapolate these numbers thirty days into the future. A straight line extrapolation curve relating the logarithm of the total number of deaths D(t) to the number of days t since January 22, allows us to do this. We expect the world wide deaths to reach 2.5 million a month from now. Go HERE for details of the discussion.

April 7, 2020-What are the running Fatality Numbers for the Corona Virus Pandemic?

We use the reported daily world-wide Corona Virus death data valid over the past 77 days to construct two basic curves for this pandemic. The first presents  D(t) versus t , where D(t) equals the number of cumulative deaths from Jan,22, 2020 (t=1) through the latest date at t=77. The second curve gives the daily change D(t)-D(t-1). Both curves were rising rapidly in value until about two days ago when the daily difference curve indicated a possible peak. We will need to wait a week or so to conform that this D(t)-D(t-1) peak has indeed been reached. It needs to be remembered that the final plateau for D(t) will equal to about twice the value at the difference peak. Go HERE for details of  the discussion.

APRIL25, 2020-What are the Latest Cumulative  World-Wide Deaths from the Corona Virus Pandemic?

We look at the latest data (t=95) on world-wide deaths D(t) caused by the Corona Virus and also present an asymptote for the number of expected deaths over the next few months. The linear asymptote reads  D(t)=-465,200 +7010 t , so we can expect a quarter million deaths world-wide when t=102 (May 2). This is a week from now. We will be studying this latest death trend very carefully looking for the point where D(t) falls significantly below the asymptote. Also some new observations on the effect of mitigations on fatalities will be discussed. The largest death rates are expected for those individuals in confined spaces under continued exposure to corona viruses from infected and asymptomatic neighbors. Go HERE for details of the discussion.

MAY 16, 2020-How long will the Corona Pandemic Last?

We examine the latest chart of both the cumulative  worldwide number of fatalities D(t) due to the CoronaVirus and the daily changes D(t)-D(t-1) starting from Jan.22 of this year(t=1) through today(t=116). It was already shown earlier on this web page that D(t) has an exponential behavior  for the first t=70 days or so, then it transitions into a linear behavior mode, and finally gives an indication of a  leveling off ending near D(t) of one half million fatalities worldwide by July 20.(t=180). The corresponding final numbers for the USA at t=180 will be around 143,000.Go HERE for details of the discussion. 

JUNE 1,2020- What did Nathan Rothshild mean by the saying "Buy Sheep and Sell Deer"?

Back in the early 18 hundreds a reporter asked the banker and speculator Nathan Rothshild how he became so rich. His reply, in his heavy foreign accent,  was to buy sheep and sell dear. What he meant was that one should buy things when they are inexpensive and sell them when they have become overvalued. This is in essence the fundamental law of capitalism easy to state but extremely hard to apply successfully. It is our purpose here to show how one can follow this rule with considerable success by studying long term graphs of any commodity of interest and then marking observed turning points on such graphs by buy(B) and sell(S) signals. Go HERE for details of the discussion.


We look at the latest (t=162) world-wide deaths due to the Corona-Virus. The deaths continue to rise with the US now responsible for 25% of the world total. After three months into the pandemic the cumulative trend has gone from an exponential form to a linear phase represented by D(t)=-244,868+4,709 t. Until the D(t) data falls  significantly below this upward linear trend the hope for a pandemic cessation seems unlikely. Even after a departure point is reached the final worldwide totals will equal about twice the value at  this yet unknown departure point. Go HERE for further details of the discussion.


We examine the properties of the hexagonal spiral  to factor large semi-primes N=pq. Knowing that N=6k1 , we must have that p and q have the restrictive  forms  6n+1and 6m∓1, respectively. We also note that nm >>(n+m).This last inequality allows us to construct a radical whose positive integer solution produces  a factoring of N. In addition we show that the simplest way to factor N=pq is via [p,q[=Ssqrt(S^2-N), where S=[sigma(N)-N-1]/2 is the mean value of p and q. Go HERE for further discussion.

JULY 29, 2020-What is the Latest on Factoring Large Semi-Primes?

We show two ways to solve the equation [p,q]=Ssqrt{S^2-N} to find the prime components of any Semi-Prime N=pq. Here S=(p+q)/2 represents the mean value of the primes p and q, both of which must be greater than three. One way to produce a quick factoring is to use the parameter S in the form sigma(N)-N-1, where sigma(N) is the sigma function given by most advanced mathematics programs for Ns as large as twenty digits. Go  HER
E for details of the discussion.


We discuss a new approach for quickly generating large primes of the form P=6n
+1 or 6n-1 based on finite series produced by
 the product of known functions. One starts with a string of chosen length N and then adjusts things with the term m+6n.The integers m ranges from 0 to 5 and n is produced by a search technique which finds P=N+n+6n to be a prime. Typical primes generated include -
= 32325577209502409472248395634473389900503015251751 and P= 9902401238146805287835557. Go HERE for details of the discussion.

August 18, 2020-What are the latest Facts concerning the CoronaPandemic?

We present the cumulative number of world-wide deaths D(t) due to the Covid19 over the past seven months. The data shows that for the first 60 days after Jan.22 the numbers grew exponentially but took on a near linear upward trend after that with no slow down noted by day t=207. An arctan model indicates that we are about half way through the pandemic with coming decreasing death rates in the US over the coming months but expected large increases for India, Brazil, Mexico, South Africa and Iran. Our model expects D(t) to reach 1.5 million world -wide  by the end of the Pandemic next February. A vaccine for the virus and possible herd immunity against the virus are also discussed. For more details go to HERE

September 23, 2020-How does one define the Roman Numerals and how are they Manipulated?

We first define the basic seven symbols used in defining Roman Numerals. These are I  (one),V (five), X (ten), L (fifty), C (one hundred), D (five hundred) and M (one thousand). A bar across the top of any of these symbols indicate a factor of 1000.We begin by showing how any combination of poitive integers in Arabic can be converted to Roman or visa versa.Typical conversions included 1234=MCCXXXIV  and  3754=MMMDCCLIV. Addition, subtraction, and multiplication of Roman Numbers are also demonstrated. Since the system lacks a zero and  uses fractions expressed in  a duodecimal system , division of numbers in Roman can become quite complicated and is usually ignored. Go HERE for further details of the discussion.

September 26, 2020-What does the Covid 19 data look like on Day 246 into the Corona Pandemic?

We look at the latest statistics on the Corona Virus and summarize the on-line data into a single graph giving the world cumulative death rate D(t) as a function of days t since January 22, 2020. After day 60 one notices that the D(t) trend is going upward in a nearly linear manner approximated by D(t)=-320000+5300t. So far there seems to be no slowing down of theD(t) trend. Using the given linear approximation formula, one can make predictions about the near future. A table giving some of these future D(t) values for the US population only is presented. Go HERE for further details of the discussion.

October 25, 2020-What are the Wave Characteristics of Stock Prices?

When looking at prices of individual stocks or commodities over a long period of time one notices a wavelike structure ranging from buy(B) to sell(S) points along the way. Each stock seems to have its own pattern but one does note that one is on the uward portion of a wave during bull markets and the reverse during bear markets. We discuss eight different stocks in detail  presenting graphs indicating the upward and downward portion of a stock wave. The best time to hold Long is when the price lies between a B and S signal. One should be Short during bear phases where the price satisfies S>P>B. GO HERE for details of the discussion.

November 9, 2020-How does one produce 2D Curves by solving certain first order Differential Equations ?

We find the solution to numerous first order differential equations of the form dy/dx=f(x,y) or dr/[r d(theta)]=g(r,theta) when the solutions(curves) contain a single specified initial point. Among the curves generated we include the Archimedes Spiral, the Rhodonia, the Astroid, the Exponential Spiral, the Cardioid, the Lemniscate, the Circle, and the Batman Curve. In obtaining closed form solutions one usually employs either a Cartessian or polar coordinate system depending upon which is easier to use for producing a solution. Go HERE for details of the discussion.

November 12, 2020-What is Potosi and Cerro Rico?

We look at the town of Potosi, its location in Bolivia, its historical role in silver production and coins minted there.This town lies in the Andes at an elevation of 13,420 ft above sea-level with a maximum population of 200,000 reached in 1650. It sits next to a mountain (Cerro de Potosi)  which has produced over 60,000 tons of silver since its founding some 475 years ago. Today Zin and Tin are mined there but only very little of the remaining silver. The mining conditions at Potosi have been extremely harsh throughout its history with several million local natives and imported slaves having perished via working conditions, mountain sickness, typhoid, pneumonia, and mercury poisoning. Go HERE for further details of the discussion.

November 14, 2020-What are SBS Patterns for Stock Prices?

We introduce a new approach for telling when the price of stocks or other commodities are expensive or inexpensive by looking at long term price graphs and marking on them when one should have bought (B) or sold (S). This is accomplished by drawing a set of progressive Vs with the top two endings being Sell points  and the lower point a buy point.We give explicit graphs and concatenations of SBS Vs. Go HERE for details of the discussion.

December 9th, 2020-How do things look on DAY 321 of the Corona Virus Pandemic?

We look at the latest graph of the Corona 19 Virus deaths on day 21 of the pandemic. So far 1.556 million have died worldwide from the virus with the US numbers standing at 291 thousand. The US numbers are particularly bad compared to the rest of the world believed due to the politicization of the  disease here and the failure by many US residents to wear masks at all times. We also discuss why facial masks are effective in slowing down the death rates. Fortunately the introduction of vaccines in the next few weeks should reduce the increase of virus deaths. Go HERE for further details of the discussion.

December 28, 2020-How does one construct modified Pascal Triangles?

We extend the standard Pascal Triangle associated with the binomial theorem to several other cases including one which allows excellent approximations for
n!. These trianglar arrays can be constructed using summations involving earlier elements. One of the new modified Pascal Triangles reads 1;2 2;3 4 3;4 6 6 4;
. Another one reads 1;1 1;1 6 1;1 16 16 1; . Go HERE for further aspects of the discussion.



January 15, 2021-How does the Corona 19 Pandemic look on Day 358?

We present a graph of the cumulative world deaths D(t) from the Corona Virus  up through day t=358 of the Pandemic. It clearly shows two linear variations in D(t) with the latest portion going as D(t)=-1909090.9+10909.09t. This means that today we have reached a cumulative world value of D(358)=1.996 million deaths. Although vaccines have now been introduced as of about December 1, so far there is no sign of a reduction in the ever growing death rates shown. Go HERE for further details. Wearing masks by everyone appears to be the best way to presently slow down the ever increasing virus threat.

January 19, 2021-How does one factor any Semi-Prime using the S Function?

We show two distinct methods for factoring any semi-prime N=pq into its two prime components p and q using the new function S=(p+q)/2=[sigma(N)-N-1]/2, where sigma(N) is the sigma function of number theory. One approach uses a direct computer evaluation valid for Ns of magnitude 10^20 or less. A second approach finds S by a construction method involving an initial choice of alpha where p=alpha sqrt(N). A 38 digit long semi-prime is factored into its components in a little over one minute. Go HERE for details of the discussion.

January 22,2021-What are the Properties of the S Function and its role in factoring Semi-Primes?

We examine the point function S=[sigma(N)-N-1]/2=(p+q)/2 used in factoring of a semi-prime N=pq into its two components p and q. Two approaches are described and then applied to the special case of N=26578351 to yield the prime factors p=3797 and q=5683. When N is smaller than about 10^40 one's home PC can be used directly to find the sigma function of  sigma(N)=21587832. For still larger Ns above 10^40 one needs an alternate approach starting with a guess for alpha=p/sqrt(N). Go HERE for details of the discussion.

January 23, 2021-What is Scientific Notation  and the Meaning of Greek Prefixes in Number Designation?

Scientific Notation for  numbers involving base ten powers is explained and several examples are given for several specific cases. Such notations are used to express the time for light to travel from the sun to the earth. Also the light year is expressed in kilometers and the travel time to the nearest star is discussed. The meaning of the prefixes tera, giga mega, kilo, milli, micro, nano and pica among others are given. Go HERE for details of the discussion.

February 11, 2021-How does one factor
large Semi-Primes using the point function S=[sigma(N)-N-1]/2?

We examine the formula [q,p]=Ssqrt(S^2-N) which has the ability to factor any semi-prime N=pq into its two prime components p and q. Here S=[sigma(N)-N-1]/2 with sigma(N) being the standard sigma function of number theory. For the small value of N=77 we have sigma(N)=96 and S=9. So that [p,q]=[7,11]. For values as high as N=10^40, values for sigma(N) can be obtained in split seconds on my PC and hence N=pq is factored. It is also shown that the forty digit semi-prime
1912492750926191821089996842096354214449 factors into its components  [p,q]=  [36715327486723625473, 52089764189567832113 ]   in 82 seconds of compqureuter run time. Go HERE for further details of the discussion.  

Febuary 18, 2021- How are the two Basic Laws of Trigonometry derived?

Many of us were first introduced to trigonometry  during 11h grade in high school.   In this course we first were exposed to the important Law of Cosines and Law of Sines which allow one to relate the vertex angles A,B,and C of any oblique triangle to its sides of length a,b, and c.  We derive these two important Laws here by using a square and a rotated smaller inner square. This geometric construction allows one to derive not only c^=a^+b^-2abcos(theta)  but also sin(A)/a=sin(B)/b=sin(C)/c . It is also shown how combining these two laws allows us to express sin(A) as (1/2bc)sqrt((2ab)^2-(a^2+b^2-c^2)^2]. That is , if we know the three side-lengths of a triangle, the sine of any vertex will be known. This value of sin(A) can also be used to derive the Heron Formula. Go HERE for details of the calculations.    

February 21, 2021-What is an SBS Stock Channel?

We examine some long term graphs of Exchange Traded Funds(ETF) modified by the superposition of newly defined SBS Channels. In particular we look at SPY, IWM, and QQQ and show on these graphs when one should be long(Bull Market) and should be short(Bear Market). At the moment these channels indicate that we are near the end of a bull market phase to be followed shortly by a bear market of unknown duration depending on the Federal Reserve reactions. Go HERE for details of the discussion. 
March 5, 2021-What are the Stock Channels for the Golden Five?

Although I use  my  SDS technical approach for stock trading and investing mainly for ETF stock collections, the method also works for individual stocks or a collection of just a few stocks, but usually with higher risk. We wish in this article to apply the method to what I have called the Golden Five. These represent a package off the following five very active stocks   GOOG, MSTF, TSLA, AAPL, and AMZN. By summing the share prices of these stocks adjusted to 1k for each on March 22, we come up with the new portfolio index--

                                   GFI=0.4828 GOOG+4.2868 MSFT+1.4309 TSLA+7.9453 AAPL +0.3130 AMZN

On February 22 it had a value of $5000. By the market close on  March 4 it stood at 4737. A  decline of 5%. Long term graphs for these five stocks are presented and price channels drawn in. At the moment all five have entered a bearish phase which will end when B signals occur again. Go HERE for details of the discussion.

March 8, 2021-How does one generate large Primes using combinations of Infinite Series?

We show how one can use a collection of infinite series evaluated at specified points to generate large prime numbers of chosen size. Specific examples for primes of 10, 30, and 90 digit length are given. Large primes are shown to be transmissible electronically in extremely small packages. Go HERE for further details.

March 17, 2021-What are some further Properties of Number Fractions and Hexagonal Integer Spirals?

We reexamine primes, semi-primes, super-composites, and twin primes based on earlier derived properties of the number fraction f(N) and the hexagonal integer spiral. It is shown that a number N  is prime if Nf(N^2)=[sigma(N^2)-N^2-1]/N equals to one. Super-composites occur when N=(2^a)*(3^b)*5^c)... with a>b>c. Twin primes occur when f[ 6n+1]=f[6n-1]=0. Thus
a twin prime is found at N=63700991 and N=63700993. The number in between is a super-composite. Go HERE for details of the discussion.

April 1, 2021-How are the Roots of Quadratic and Cubic Algebraic Equations found using their Depressed Form?

We solve in closed  form the equations ax^2+bx+c=0 and ax^3+bx^2+cx+d=0 using their depressed representation containing no powers of x^(n-1). The results for the depressed cubic equation takes on the form t^3+mt=n and can be solved via the substitution t=u+v with v=-m/3u. The conditions when all solutions are real plus those where complex conjugate forms arise, are discussed. The specific cubic x^3-2x^2-x+2=0 are discussed in detail. Go HERE for futher discussions. 

April 5, 2021-How does one find the Roots to a Complex Function f(z)? 

We use a modified Newton-Raphson Method together with countour plots of the square of the absolute value of f(z) to find all roots of f(z)=0 by iteration. Three specific functions considered in detail are f(z)=z^2+i, f(z)=z^3+2z-3, and f(z)=z^6-3iz^2+5. By placing the initial starting point within the smallest local contour, one finds very rapid convergence to the zero contained within this contour. Go HERE for details of the discussion.    

April 12, 2021-How Effective are the new Corona 19 Virus Vaccines?

We look at the accelerating number of worldwide Corona 19 virus deaths since the release of vaccines by Moderna, Pfizer, Johnson& Johnson, AstraSeneca, and Sputnik  about three months ago. The data shows that these vaccines are not reducing the promised death rates. This co
uld be due to either new vaccine resistant strains of the Corona 19 virus or the fact that the pharma companies over-promised their vaccines effectiveness just to  get on the CDC approved list. A true measure of vaccine effectiveness would be seeing the worlds cumulative Corona 19 deaths to start slowing down as indicated by a bending to the right of the D(t) versus t curve.  Go HERE for further details of the discussion.

April 16, 2021-How does one get Bounds on Pi using the Archimedes Method of inner and outer Polygons for a unit Circle?

We look at the area of  a large n sided regular polygon inscribed or circumscribed inside a unit circle. This generates a bound on Pi given by 2^(m-1)
sin(180/2^(m-1))<Pi<2^m tan(180/2^m), with n=2^m equal the number of sides of the polygon. By this approach Archimedes was able to prove the bound 3+10/71<Pi<3+1/7 using a 96 sided regular polygon. Later Ludolph van Ceulen(1540-1610) was able to show that  n=2^62 yields a Pi value good to 35 places .  Go HERE for details of the discussion.

April 20,2021-Is there a new Way to quickly factor large Semi-Primes into their Prime Components?

We show how the formula [p,q]=Ssqrt(S^2-N) can be used to factor large semi-primes N=pq into their components. Here S=[sigma(N)-N-1]/2=[(1+alpha^2)/(2 alpha)]sqrt(N) and p=alpha sqrt(N) with q=(1/alpha) sqrt.  A second variable appearing in the calculations is epsilon which must be adjusted to make R=sqrt(S^2-N) an integer. Several specific examples for semi-primes as high as eight digits are looked at in detail. Go HERE for further details.

April 24, 2021-How does one Generate Integer Spirals?

We discuss the Ulam and Hexagonal Integer Spiral and the location of primes within them. After showing the rather random distribution of primes in the Ulam Case, we derive the much simpler and more informative  Hexagonal Integer Spiral. In particular we, show that in this secnd case all prime greater than N=3 lie along just two lines 6 n1. In this new format,  both the location of twin primes and Goldbach primes become clear. Go HERE for further details of the discussion.

April 30, 2021-How did Newton  manipulate the Binomial Theroem to generate numerous  additional Identities?

While playing around with the Binomial Theorem in the late 16 hundreds, Isaac Newton showed that by relaxing  restrictions  on the Theorem one can generate numerous additional identities including  the Gregory Formula and some new formulas for evaluating values of Pi. His generalized form originated from [1+f(x)]^n, where f(x) is any continuous function of x and n can take one fractional and negative values. It is our purpose here to generate some of these new formulas and also to show how values of integrals can be found by expanding differentials in the form of Binomial Series. Go HERE for further details.

May 4, 2021-How can one use the Diophantine Equation to obtain highly accurate Expansions for the Square Roots of Positive Integers?
We look at the higher integer solutions of certain nonlinear Diophantine Equations to generate rapidly convergent series for the square root of integers. Specifically, the roots of two and three are calculated yielding results good to above forty digits using just fours  terms of a Binomial Expansion. Also a partial expansions of a certain continued fraction is used to again find highly accurate expansion for  roots of positive integers. Go HERE for details of the discussion.

MAY 12, 2021-How does one construct A Hexagonal Integer Spiral?
We start with a standard Archimedes Spiral intersected by six equally spaced radial lines. Positive integers are then added to the intersection points where the radial lines cross the spiral. Finally connecting neighboring integers by straight lines, a standard hexagonal integer spiral is created. The similarity to a typical spider web is noted. Some six important observations are made concerning the property of such spirals. The most important observation is that for all primes greater than three, a necessary but not sufficient condition is that all primes greater than three must have the form 6n+1 or 6n-1. Go HERE for further details of the discussion.

May 23, 2021-What are the properties of  sum(x^n, n=0..N)?

We look at the properties of the function F(x,N)=[1-x^(N+1)]/(1-x) , which can also be written as a finite series of N+1 terms. This function includes the classic geometric series as a special case. Also we show that the Mersenne Numbers M=2^(N+1)-1 can always be written as the  sum(2^n, n=0..N). Identities involving complex z=rexp[i theta]
are also presented. Go HERE for the details of the discussion.

May 28, 2021-What is a simple way to construct a Hexagonal-Integer-Spiral?

Taking a circle of radius r=6 one first creates six equal pie shapes by drawing six radial lines. Next going in a counter-clockwise manner around the circle one creates radial lines of  length r=N whose tips are found in polar coordinates at increasing values [r, theta]=[N, N Pi/3]. Finally by connecting these radial distances r by straight red lines, a Hexagonal Integer Spiral is formed. The Spiral shows very nicely that prime numbers must lie along just two radial lies 6n+1 or 6n-1 provided N is geater than three. That is, for a number to be prime, we must have  N mod(6)=1 or N mod(6)=5. Go   HERE  for further details of the discussion.

June 5, 2021-What is a Formula for detecting Weight Trends in Individuals?

It is well known that the majority of adult Americans are overweight as shown by their BMI index. This is caused by the overeating of junk foods while working under conditions which are sedentary. Clearly weight is gained if the food intake I exceeds the expenditure E. Weight is lost when the reverse is true. A simple formula which desrcribes this weight change is dW/dt=const{I-E}. Although there is little the average citizen can do concerning weight loss by increasing the expenditure E because of time factors, it is always possible to decrease I by eating less. In my case the equilibrium I is 2100 kilocaloris/day. So to loose about a pound of fat a week will require a food deficit of 3500 C or 500 C/day. This is doable considering that any food can be consumed as long as the daily deficit stays below 500 C. Go HERE for further discussion. 

June 8, 2021-What are the Properties of Semi-Primes?

We examine semi-primes N=pq, where p and q are prime numbers. It is shown that the prime factors satisfy p=S-R and q=S+R. Here S represents the mean of p and q. It can also be written as S=[sigma(N)-N-1]/2, where sigma(N) is the sigma or divisor function of Number Theory. R represents half the difference between q and p. It can alternatively be written as R=sqrt(S^2-N). Five specific semi-primes and their accompanying functions sigma(N), R and S are presented. On my computer I can generate  sigma(N) directly for semi-primes Ns as large as forty digits and hence can factor this size N or smaller in a split second.. Go HERE for details of the discussion.

June 13, 2021-What is the LCDS food diet?

We recently have been looking at the numerous diets for weight reduction found on the internet. Usually these diets require the purchase of expensive food additives yet they are ineffective in the long run in reducing weight yet cost customers milions of dollars. In thinking about this overweight problem from a mathematical point of view, I  come to the obvious conclusion  that any successful weight reduction diet must have as its basic rule that daily food intake exceeds the daily expenditure (basal metabolism). Expressed mathematically this reads dW/dt=const.{I-E}. Here dW/dt is weight loss per time, I the daily food intake in kilocalories and E the daily expenditure of an individual . Since most of us are involved in sedentary occupations and there is little time to raise E via additional exercise,  the only reasonable diet (and one which is free unless you use Weight Watchers) is to reduces one's calorie intake I. This is doable if the diet intake deficit is not much above 500 Cal per day. We call this diet  the LCDS diet referring to Low-Calorie-Diet-Sedentary. Go HERE for details of the diet.

July 1,2021-What are some Further Properties of Semi-Primes?

We examine further the properties of semi-primes N=pq using the new variables S=(p+q)/2 and R=(q-p)/2. The governing equation for finding the prime factors p and q is given by S^2-R^2=N. The factors are found at one point along this hyperbola in the S-R plain. Also we generate a new semi-prime triangle defined by sides sqrt(N) and R and the hypotenuse S=(p+q)/2=(sigma(N)-N-1)/2. Here sigma(N) is the sigma function of number theory. Most advanced computer programs give the value of sigma(N) in split seconds when N is less than about forty digit length. So the factoring of semi-primes up to 40 digit size are readily factored on one's PC. Go HERE for details of the discussion.

July 9, 2021-What are the basic manipulations involved in Mental Arithmetic?

With a little practice one can carry out many mathematical operations of summation, subtraction, multiplication and division  without the need for external tools. We demonstrate this in his article for several different cases. Among other results we show how to find powers and roots of numbers. Also we show how capital grows under compound interest and how to quickly calculate tips mentally. Go HERE for details of the discussion.

July 15, 2021-What are the SBS Channels for the S&P 500 and the Dow Jones Industrial Average ?

We discuss the channels formed by two of the most important stock market averages, namely, the S&P 500 and the Dow Jones Industrial 30. These channels clearly indicate whether  the stock market finds itself in a Bull or a Bear phase. By buying and holding long only in a bull phase and being short during a bear phases, guarantees a net positive return on one's investment.  Go HERE for further details of the discussion.

July 24, 2021-How does one Deconstruct a Semi-Prime into its Prime Components ?

We demonstrate a new way to factor
any semi-prime N=pq into its prime components p and q using two new variables S=[p+q]/2 and R=[q-p]/2. These allow us to write p=S-R and q=S+R. The variable S can also be given by (1/2)[sigma(N)-N-1]] , with sigma(N) being the summation formula of Number Theory. The value of sigma(N) is given by most advanced mathematics programs out to Ns of about 40 digit length. Three specific Ns are evaluated with the largest of these having 30 digit length. Go HERE for details of the discussion.

JULY 30, 2021-What are the Properties of a new Modified Pascal Triangle?

We examine a new type of Pascal Triangle whose elements in the nth row always sum up to n! All elements in this triangle are defined by a unique summation formula. Also it is shown how the elements in the sequences {1,4,11,26,57,...} and {1,11,66,302,1191,..} can be expressed by simple unique formulas. By looking at the sum of only a few elements near their row maximum, good approximations to n! are obtainable. Go HERE for further details.

Augut 10, 2021-What is the Sigma Function and some of its Related Forms?

We examine the sigma function and some of its related forms. Starting with  sigma(N)=sum of all divisors of integer N including 1 and N, it is shown that sigma[(p^n)(q^m)]=sigma(p^n) x sigma(q^m) for all primes p and q. From this equality one can quickly calculate the sigma function for any large values of N. Also Mersenne Numbers (2^p)-1a are shown to be generated by sigma[(2^(p-1)]. We also determine the value of a new parameter S=(1/2)[sigma(N)-N-1] which can be used to factor any semi-prime N=pq. Go HERE for further details.

August 12, 2021-What does one mean by B followed by p>lambda and S followed by p<lambda?

Over the years we have developed some technical signals which allow one to stay on the right side of market movements. The method uses long term historical price curves to indicate overvalued (S) and undervalued conditions(B). When this information is combined with lag curves, one can clearly see when it is time to hold long or be short a stock.We look specifically at 25 year SBS channels and associated lag curves for GE and SPY to see when they are a buy or a short. Go HERE for further details of the discussion.

August 15, 2021-What is meant by the Double-Kite Structure of the Human Face?

We examine the computer averaged frontal face image of a collection of college aged students and note the location of the eyes, nose and mouth. The averaged pattern  connecting the various facial points are found to be presentable by a single unique Double Kite Grid. Attractive individuals are shown to closely follow the pattern suggested by the averaged double kite. This means that our brain is programmed to find attractive those individuals not departing far from this ideal facial pattern. Go HERE for details of the discussion.

August 20, 2021-What is the Silk Road?
The Silk Road(or Route) was a 4000 mile long trade route between China and Western Europe from 114BC until 1453. We trace here its path connecting Xian China with the Mediterranean Sea. Some of the cities along the route are mentioned in more detail as are the desert and mountains along the way. The goods traded are mentioned and contributions by the Chinese  to western society due to the Silk Road are mentioned. Go HERE for details of the discussion.

August 23, 2021-What are the better known Spirals?

We examine a variety of differnt spirals including those of Archimedes, Bernoulli, and Fermat. They are all generated by very simple formulas. In addition we also look at spirals with discontinuities in their derivatives  or curvature. Also pictures of spirals as they occur in nature are presented. These include hurricanes, star galaxies, and sea-shells. Go HERE for further discussions.

August 27, 2021-How does one construct Angles using only Compass and Ruler?

We show how one can construct angles theta=Pi/n using just a compass and ruler. Starting with Pi/2 and Pi/4 we go on to angles Pi//3 and Pi/6. This is followed by triangles of smallest angle Pi/8, Pi/12, and Pi/24. Manny of these angles are constructed starting with a unit circle which is crossed by radial lines separated from each other by angle Pi/n. Explicit forms for trignometric functions of Pi/n defined only as combination of integers and roots  are also obtained. For example, sin(Pi/10)=[sqrt(5)-1]/4. Go HE
RE for further details of the discussion.

September 2, 2021-What Angles have their Trigonometric Forms expressed as the Square Roots of  3, 4,or 5?
We examine angles Theta=Pi/n to determine  which yield trigonometric forms for sin, cos, and tan involving square rots of n. It turns out that there are an infinite number of these angles having the form Pi/(nk) provided n=3,4, or 5  and k=1,2,3,4,.. . For example at n=5 and k=4  we find cos(Pi/20)=
sqrt[(1/2+(1/4)sqrt((5+sqrt(5))/2))] = 0.9876883.. . This means 9 degrees has trig functions  involving only roots of five.  We find no such root solutions when n is an odd number seven or greater.  Go HERE for futher details of the discussion.

September 7, 2021-How does one apply the L'Hospital Rule to evaluate certain Quotients at limiting values of x?

We examine quotients F(x)= f(x)/g(x) in the limit where the quotient has either value 0/0 or infinity/infinity. Specific examples considered are  F(x)=(x^3-2x+1)/(x-1)^2 and F(x)=sin(x)^2/x^3 at x=1 and x=0, respectively. Also we look at the value of (1+1/n)^n as n approaches infinity. Go HERE for further details of the discussion.

September 11, 2021-How does one use historical Price Behavior versus a Lag Curve to determine when an Equity should be Bought or Sold?

We summarize our approach to stock and commodity investing using historical prices P(t) and their corresponding  running lag curves Lambda(t). Graphs are presented showing this long and short behavior for a variety of different equities including GE stock, Brent Crude Oil and Real Estate. One of the safest and profitable approaches in using our buy and sell signal method is to deal with exchange traded funds, especially SPY. Go HERE for details of the discussion.

 Setptember 16, 2021                                 
Today is my 85th Birthday and an appropriate time to formally end          
                                                        Kurzwegs's TECH BLOG
Many of the ideas discussed on this Web Page over the past eleven years can be expanded further specifically those involving semi-prime factoring, market movements, hexagonal integer spirals, the modified PASCAL triangle, KTL method for approximating any trigonometric function, and generation of large primes using products of irrational numbers.

Once in a while I will be coming up with some new articles falling under the auspices of this Tech Blog Page. These will come at a slower  and more irregular rate since my original September 16,  2021 sign-off. A few word descriptions and their URL addresses follow-

          (1)-Relaton between Polynomials and Infinite Sequences found at-

          (2)-Appoximating Trig Functions using the KTL Method-

   (3)-More on the number fraction f(N)-


(4)-Generating Primes from Irrationals-

          (5)-Barcharts and Barchart Readers-


(6)-Corona Virus on day 651-

          (7)-Five year ETF stock price windows-

          (8)-Solving the Brahmagupta-Pell Equation-

          (9)-Some Equations involving n!- 

         (10)-Determining Market Trends-

         (11)-Deriving Four Trigonometric Formulas-


          (12)-Location of Primes and Semi-Primes along a Hexagonal Integer Spiral
          (13)-Tessellations using Oblique Triangles-  

(14)-Price of Stocks and Bonds versus Yield-

(15)-Latest on the Number Fraction-

              (16)-Properties of a Modified Pascal Triangle-

               (17)-Leibniz Rule for Differentiating under an Integral Sign-


              (18)-Solving simultaneos Diophantine Equations-

               (19)-How does a Thermobaric Bomb work?

                (20)-Constructing an Integer Sandwich

                (21)-Bull and Bear View-

                (22)-Prime Generation by a Quadratic Formula-

                (23)-Iteration Methods

                (24)-Manipulating Logarithms-


                (26)-Heron Formula Derivation-

                (27)-Two Circles-

                (28)-Centroid Irrational Polygons-            

                (29)-Inflation History-

                (30)-TCT Scans-

                (31)-Factoring using Diophantine-

                (32)-Ntuple Algebraic Formulas-

                (33)-Christmas Tree Construction-

               (34)-Number Sandwich-

               (35)-Financial Bubbles-

               (36)-Number Systems for Base b-

              (37)-Sigma Function Variations-

              (38)-Sub-Area Determination-

              (39)-Hole through Sphere-

              (40)-CT Scans for SPY and QQQ-

              (41)-Large Twin Primes-

             (42)-More Hex Spiral-

            (44)-New Factorization Formula

            (45)-Factoring without Sigma

            (46)-Asymptotes for Continuous Functions
            (47)-Correlation of Stock Averages

            (48)-Latest on the Integer Spiral

            (49)-Solving the H(x) Equation


            (51)-All About Triangles





            (54)-Area of Regular Polygons

            (55)-Latest-on Polygons

            (57)-Solving N(n,a)=(2^n) + a


            (59)-Areas of Polygons via Sub-Triangles

            (60)-Unconvential Methods of Factoring N=pq

            (61)-Generating 2D Curves

            (62)-Tau and Sigma-Functions

            (63)-Trends in Equities


            (65)-Quad Numbers and Semi-Primes



            (68)-Stock Windows


           (69)-More Diophantine



         (73)-Tricks for Integral Evaluations



        (76)-Pressure at 12500 ft

        (77)-Factoring Using Sigma

        (78)-More on Sigma









      (88)-Integration by Parts


     (90)-Using a Diophantine Equation to factor a Semi-Prime

     (91)-Properties of a modified Pascal Triangle

     (92)-Mathematical Induction

     (93)-Semi-Prime Factoring using F(x)





     (98)-Use of Windows to Predict Price Futures

     (99)-Properties of the Gamma Function


     (101)-Sigma, Tau, and Number Fraction for p^n

    (102)-Price in the Future


   (104)-Function Approximation



   (107)-Hyperbolic Aproximation

   (108)-Properties of Legendre Polynomials


   (111)-Number Symmetry for N=pq

   (112)-Price Filtering

   (113)-Necessary and Sufficient Condtion for Primes


  (115)-Finding Primes

 (116)-Stock Futures using Past Time Windows


  (117)-Factoring N=1122973

  (118)-Evaluating N Using Sigma

 (119)-A Necessary and Sufficient Condition That a Number is Prime

  (120)-Modular Arithmetic


  (122)-Inverse Functions

                                          Das Ende