Math Fun

by Joseph Buchmann

Calculate Pi

Pythagoras Proof

Days of the Week

The Golden Ratio

The Name of the Months

Simple Method to Calculate Pi

We all know the value of Pi, the ratio of a circle's circumference to its diameter: 3.14. We confidently use that 3.14 number in circles and spheres calculations. But have you ever tried to actually calculate that number yourself, say to a 100 decimal accuracy? Are you wondering how Archimedes figured it out and calculated Pi almost 2500 years ago? With some basic knowledge of the Pythagorean theorem that we learned in Grade 6, it is quite logical and straightforward.

The realization that six circles fit perfectly around a single same-size circle must have been thought-provoking to ancient mathematicians, and it opened up a new world of geometry. It is the foundation of the Hexagon. The hexagon, as the name implies, is a polygon of six sides, all sides of equal length. It is an amazing figure. Even nature uses it; think of the honeycomb. The hexagon figure encaptures six equilateral triangles, each side the same length as the sides of the hexagon, and, of course, the same length as the hexagon's radius. It is the only polygon where the length of a side has a direct relation to the radius. The hexagon is the perfect base for our Pi calculation.

To demonstrate it differently: Take six beer cans and place them around a same-size can. They fit perfectly. Then, mentally draw lines from the center of each can to the center of each neighboring can, thus three lines emanating from each can. The drawing is a hexagon with six perfect triangles.

The hexagon is a quasi-circle. The ratio of the circumference of that 'circle' (six triangle sides) to its diameter (two triangle sides) is 3. It is the beginning point of our Pi calculation.

We start with a triangle of the hexagon. Running through several iterations, we cut the triangle in half, again and again, and align the half-side to the imaginary circle line. Cutting an equilateral triangle in half creates two triangles each with a 90-degree angle, and this allows us to use the Pythagorean theorem. See the sample below. As the number of triangles doubles with each iteration, from 6 to 12, 24, 48, etc, the outer rim becomes rounder and rounder, smoother and smoother. After only ten iterations, the original hexagon has more than 6000 edges, and it appears virtually round to the eye. The circle's circumference divided by the diameter (double-radius) gives us the new Pi ratio. The circle will never be completely round, and the Pi number can thus be calculated to infinity. The pictures below show the process in more detail.

We cut one of a triangle in half. Between the two halves, we draw a line from the center of the circle, the length equal to a side of the original triangle (combined red RA and blue RB in the illustration), crossing the upper side of the triangle (R2). The line RA+RB is equal to line R. Now, we need to calculate the length of the line section from the center point up to the crossing point (red RA) and the line beyond the crossing point (blue RB)). Here we need to do our first simple Pythagoras calculation. We know the length of the radius (R) and we know the length of R2 (R2 is, of course, exactly half of the radius). RA can be calculated as follows:

RA = Square Root of ((R*R) - (R2 * R2))

We are now ready to refine the circle line further. The initial six sides will be doubled to 12. With our second Pythagoras calculation, we obtain the hypotenuse of the yellow RB and R2 configuration, shown as a green line (S) in the image. The calculation is as follows:

S = Square Root of (( R2*R2) + (RB*RB))

With this calculation, we now have 12 equal sections around our circle, and the Pi stands much improved at 3.13. Better, but it is not good enough.

And we continue. Using the same steps as above, we cut the sections by half again, and again. The Pi improves to 3.139, then 3.141. Once we get to 20 iterations, we have a near-perfect Pi value of 3.14159265358976. Of course, we write a simple program to do this for us. The sample code is shown below.

We can calculate it to infinity; a number is never too small to be cut in half.

Sample code for the Pi calculation in C#:

double radius = 1000;
double sec = 0;
double adjacent = 0;
double opposite = radius / 2;
adjacent = Math.Sqrt((radius * radius) - (opposite * opposite));
sec = Math.Sqrt((opposite * opposite) + ((radius - adjacent) * (radius - adjacent)));
int iterations = 20;
for (int i = 1; i <= iterations; i++)
{
adjacent = Math.Sqrt((radius * radius) - ((sec / 2) * (sec / 2)));
sec = Math.Sqrt(((sec / 2) * (sec / 2)) + ((radius - adjacent) * (radius - adjacent)));
}
double circumference = sec * 6 * (Math.Pow(2, iterations));
double pi = circumference / radius;
MessageBox.Show(pi.ToString());

Each iteration gets closer to the ultimate Pi number, unreachable because the circle is never completely round:
3.10583
3.13263
3.13935
3.14103
3.14145
3.14156
3.14158
...
3.14159265359...

A side notation, a thought.....

A full circle is expressed as 360 degrees. The common belief is that the number 360 is based on the number of days of the full solar cycle, thus each day represents one degree. The Sumerians and Babylonians of Mesopotamia used the Sexagesimal numbering system (Base-60). Since our quasi-circle has six sides, one could also speculate that Sumerians may have originated the 360 degrees based on the six-sided circle.

The Sumerians and Babylonians practiced finger-bone counting. With an open hand, using the thumb, they tapped and counted the finger-bones; four fingers, each with three bones, a count of 12. The other hand tracked the counts of 12, for example, 5 x 12, a total of 60.

The finger-bone counting method is likely the foundation of the Sexagesimal (base 60) numbering system used by the Sumerians and Babylonians, a system that we still use today when referring to degrees, minutes, and seconds. The high divisibility of the number 60 makes it very special. You can divide the number by 2, 3, 4, 5, 6, and more.

Pythagoras Theorem, Simple Proof Method

Here is a simple presentation to prove the Pythagoras Theory which states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

The Origin of the Week-Day Names

The notion of Month and Week existed since the beginning of the human mind. The month was a moon cycle. A week was a logical split of the month cycle: the new moon, the first quarter moon, the full moon, and the third quarter moon. The seven-day periods became times for food stocking, gatherings, prayers, and rest. First, the days had no names, like today when hours are nameless.

The history behind the weekday names and their sequence is simply mind-boggling.

We know that the weekday names are clearly linked to heavenly bodies, the sun, the moon, and the known planets. This is particularly evident in the Roman languages, like Latin and French. In English and German, some weekday names relate to deities associated with these same celestial bodies.

Each weekday stands for a celestial object in the sky. But the names seem set in random order. Why is the day of Mars before the day of Mercury? Why is Tuesday before Wednesday? One day is like the other, each day is like the next. Days have no special character; they are all the same, not like the seasons which have some relation to the weather. So, how did ancient civilizations assign the day names in today's universally established order? The answer is truly eye-opening.

Monday: Named after the Moon. Latin: Dies Lunae. German: Montag. French: Lundi.

Tuesday: Named after Mars. The name Tuesday derives from the Germanic god Tiw, the god of war. God Tiw is associated with the Roman god Martii. German: Dienstag. Latin: Dies Martis. French Mardi.

Wednesday: Named after the Germanic god Wodan. The Romans identified Wodan with their god Mercury. Latin: Dies Mercurii. French Mercredi. (German: Mittwoch, unrelated).

Thursday: Named after the god Thor, the god of thunder, God Thor is associated with the Roman god Jupiter, the god of skies and thunder. Latin: Dies Lovis (Jupiter). French Jeudi.

Friday: Named after the goddess Frigg, associated with the Roman goddess Venus, the goddess of love. German: Freitag. Latin: Dies Veneris. French: Vendredi.

Saturday: The day is named after the planet Saturn. Latin: Dies Saturni. French Samedi. German: Samstag.

Sunday: The day is named after the Sun. Latin: Dies Solis. German: Sonntag. (French: Dimanche, unrelated).

In the ancient world, only seven of the heavenly bodies were known by name and revered. The planets Uranus and Neptune had yet to be discovered. Each of the seven godly objects was sacred and each was the embodiment of a God or Goddess. Each new day was devoted to a heavenly object, in a seven-day rotation.

The seven known sky objects have traditionally been listed in the order of the length of their orbits, being the elapsed time before their reappearance. In ancient times, Earth was the center of the universe, and the sun was believed to circle the Earth, once a year. Here is how long it takes the heavenly object to complete an orbit:

1. Moon: 29 days (dies Lunae, Monday)
2. Mercury: 88 days (dies Mercurii, Wednesday)
3. Venus: 225 days (dies Veneris, Friday)
4. Sun: 1 year, earth's cycle (dies Solis, Sunday)
5. Mars: 2 years (dies Martis, Tuesday
6. Jupiter: 12 years (dies Iovis, Thursday)
7. Saturn: 29 years (dies Saturni, Saturday)

So, we should think that the lineup of the days would be:

Monday, Wednesday, Friday, Sunday, Tuesday, Thursday, Saturday, or in reverse. But they are not.

Let's dig into it further. As early as four thousand years ago, the ancient Egyptians and Babylonians already split the day into two halves, a 12-hour day, and a 12-hour night, a total of 24 hours. Time devices, such as shadow clocks, marked the hours as the sun ascended in the morning and descended in the evening, also allowing for the twilight hours. The number 12 has always been an important number.

Given the seasonal variation of sunlight time, the summer hours were longer than the winter hours. The ancient civilizations, knowing the movement of the zodiacs, had a way to handle the nighttime hours, but that is irrelevant to the topic at hand. Important for now is the fact that each day always had 24 hours.

In ancient Egypt, and other civilized countries, a full hour of time was dedicated to a God. So, 24 times each day, until midnight, in rotation, the next hour was devoted to another God, in the strict order of the orbital lineup. The hour after midnight was paramount. The God of that first hour owned the whole new day, and the day was named after that hour's deity.

There are only 7 God Stars, but there are 24 hours each day; so, every seven hours, the cycle repeats itself. The number 7 does not neatly divide into the number 24; it overlaps by 3. Thus, the pick from the lineup of stars shifts by three names every 24 hours. The animated Gif below explains it better. Incredible! The seven-day, 24-hour run-through creates the exact day order that we know today: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, to Sunday. It's a coincidence, you say, pure chance. The odds of a correct random pick are almost One-in-a-Million.

The Golden Ratio

The golden ratio is one of the most interesting phenomena in nature. It is the space where the universal law of nature is in harmony with the rule of mathematics. No wonder, it is also called the divine proportion and the golden number.

In mathematics and in the arts, a rectangle has a golden ratio if the ratio of the longer side to its shorter side is the same as the ratio of the total of both sides to the longer side. An object created by man or nature, built to a golden ratio, is of the greatest beauty to the human eye. I will now refer to the golden ratio simply as the Ratio. Early artists and architects were well aware of the importance of the Ratio. The architectural design of the Parthenon in Athens has many elements that follow the rules of Ratio, by instinct or by calculation. Most books printed in the 16th to the 18th century had heights and widths exactly in harmony with the Ratio, accurate to within a tenth of an inch. The Mona Lisa painting has Ratio dimensions. The divine proportion of the rectangles gives the artwork the most appealing and pleasing appearance.

The Fibonacci Sequence

The numeric steps that lead to the golden ratio are known as the Fibonacci Sequence. The Fibonacci Sequence can be demonstrated by drawing a snake with a formation of squares. Start with one square, then add a second square, then add an additional square next to it of a size equal to the widest side of the existing assembly, and continue. As more and more squares are added, the squares get larger and larger and the ratio between the short and the wide side of the resulting build gets closer and closer to the Golden Ratio of 1.61803399..., but never exceeds it. A ratio is an irrational number, it is never exhausted. The sizes of the squares that you add represent the Fibonacci Sequence. The number in the Fibonacci Sequence is always the total of the previous two numbers, thus the sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. The size ratio of the last added square to its immediate predecessor represents the Golden Ratio, for example, 89 divided by 55.

It is most fascinating to observe Fibonacci numbers everywhere in nature. These numbers are Nature's way of counting. Consider the numbering of the growing and living things in nature: Children have 5 teeth on each side, 8 teeth for adults, 5 fingers, 5 toes, petals in flowers, seeds on flower heads, and spirals in sunflowers. More often than not they match Fibonacci numbers. Shrimps have five pairs of walking legs and five pairs of swimmerets. Look at your hand and its joints. The three phalanges of the finger have the familiar ratio of Fibonacci, the same extending to the metacarpus of the hand. Some flowers have four or six petals. Oh, these are not Fibonacci numbers. But look at the flowers, and you will see that the petals are in two equal layers, thus each layer being a Fibonacci number.

The value of the golden ratio is 1.618033... and can also be calculated by taking the square root of 5, adding 1, and dividing the result by 2. Or, as mentioned before, pick a number from the Fibonacci Sequence and divide it by the next lower value in the sequence.

The Name of the Months

The names given to the months have nothing in common with mathematics but one can argue that they have some relationship to numbers. It is an interesting subject; we give scarce thought to some weird abnormalities rooted in our calendar. Do you know why in the world the residual leap-year days are adjusted in February instead of December? With the 30-day and 31-day months orderly alternating, why is there an exception for July and August? Why is September the ninth month when the number Seven is in its name, like 'Septem' in Latin? Just like the mystery of the sequence of week names that I explained in the section above, there are good explanations for all of these questions.

We all know, that a month represents a moon cycle. The moon cycles do not calibrate neatly into a 'sun' year. Twelve moon cycles represent about 354 days and a solar cycle is just over 365 days. Over the long span of history, different approaches were used to handle the discrepancy.

In the early years of the Romans, the year started with March and consisted of only ten months. The two moons after the ten-month period were a nameless dead season in the midst of winter when nothing was growing. March, the first month of the year was named after Mars, the God of Wars, aptly named for the month when the soggy soil firmed up, daylight grew longer and young men were called back to war to conquer new lands and bring home the booty. April has its origin in Aprilis, 'the month that opens', perhaps alluding to the buds opening up in spring. It has been argued also that April may be named in honor of Aphrodite, the Goddess of Love and Beauty. May and June were probably named after the Roman Goddesses Maiesta and Juno. The remaining six months were simply named after their respective numeric position. Keep in mind that March was the first month. So, Quintilis (July) was the name of the fifth month, Sextilis (August) of the sixth month, then Septilis, Octilis. Novem and Decem were the root words for the ninth and tenth months.

Around 700 BC the Romans decided to add the two 'dead' winter moon cycles at the beginning of the calendar year. Januarius was named after Janus the God of Gates, Lord of beginning and end, his head appropriately depicted with two faces looking in opposite directions, looking back at the old year and looking forward to the new year. Februarius refers to Februa the festival of purification celebrated in Rome that month. A root word for Februa refers to 'burning' relating perhaps to fever. It makes good sense; we all get a cold or the flu when our defenses are down after a long winter. Adding the two months in front explains why the names of September to December are now out of whack with their embedded sequence number.

The number of days allocated to the months varied. The Romans viewed odd numbers as more propitious than even numbers. An odd number felt more copious because there was some extra after an equal sharing. The left-over or deficit days were handled differently over time but in time they were taken care of in the month of February, the end of the original dead season. Still today, February is the handler of the adjustment days.

Then, the Roman Caesars and Emperors with their big egos arrived on stage. Julius Caesar wanted a month named after him. He ordered that Quintilis, the fifth month, be named July. Not to be outdone, Emperor Augustus claimed the sixth-month Sextilis for him. Julius' month traditionally had 31 days but Augustus' month Sextilis had 30 days. It was not acceptable for Augustus to have his month with a number of days lower and less favorable than that of his uncle Julius, so the number of days in August was increased to 31, and it still is today.

The Ingenious A4 Paper Format

Decades ago, the Europeans developed and propagated paper sizes that are based on the 'Square Root of Two' ratio, now also dubbed the Lichtenberg ratio. It follows the clever Metric system. The ingenuity of the v2 ratio is that when the paper is cut or folded in half lengthwise, the two halves also are of the exact same aspect ratio as their parent.

The scheme has three series, the 'A' series being the most important one. Thus, the paper size A0 has a total area of 1 square Meter. The ratio of width to height is 1 to 1.4142, the value of 1.4142 being the square root of 2. When you cut the A0 paper sheet in half, you have two A1 size sheets, with both pieces having exactly the same width-height ratio as the original A0. The A1 cut in half again creates two A2 sheets; cut again it creates two A3s, then A4s, and so on. The A4 is the most commonly used paper size, the equivalent of the North American Letter Size. The thought behind the ratio is so ingenious, it challenges the human mind.

In addition to the 'A' series, there is a 'B' series. The 'B' paper size uses the same principle. It starts with a width of 1 meter and the length of the Lichtenberg ratio (v2 = 1.4142). The B1, B2, B3, etc are the sizes that descend from the larger size cut in two. Furthermore, there is a 'C' series of paper-size. The 'C' sizes are the arithmetic average of the 'A' and the 'B' sizes. They are thus just slightly larger than the 'A' size and are perfectly suited for envelopes of A-type sheets.

In 1996, realizing the ingenuity of the European standard, the prestigious American National Standards Institute agreed on a mimicked standard based on the 8.5x11 letterhead. The double-size of the now common 8.5x11 inch letterhead became the 'Ledger' size', and if rotated, the same paper got the name 'Tabloid'; all other sizes in the series remained nameless. Not surprisingly, there were not many followers.

Copyright 2009 Josef Buchmann