We all know the mathematical constant Pi, the ratio of a circle's circumference to its diameter, is approximately 3.14. We confidently use the number 3.14 in many geometry calculations. But have you ever tried calculating Pi to 10 decimal places? Are you wondering how the Babylonians figured it out and calculated it 2500 years ago, and how Archimedes and Euclid refined and proved the theorem? With some basic knowledge of the Pythagorean theorem learned in Grade Six, we can calculate the Pi value easily. And it is fun..
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. They realized that it was the core of the hexagon. The Hexagon, as the name implies, is a six-sided polygon, with all sides of equal length. It is a fundamental figure. Even Nature uses it; think of the honeycomb. Here, Nature meets Mathematics. The hexagon encloses six equilateral triangles, each side equal to a side of the hexagon, and thus the radius of the hexagon. The hexagon is the perfect base for our Pi calculation.
To better demonstrate, take a tin can and place six same-size tin cans tightly around it. They fit perfectly. Then, mentally, draw lines from the center of each can to the center of each neighboring can, 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 starting point of our Pi calculation.
The following outlines a step-by-step calculation of PI. It is for illustration, - no one ever needs to calculate the Pi number. It is a practical, step-by-step, logical, easy-to-follow presentation, rather than a compressed mathematical formulation.
We start with a triangle of the hexagon. We run through several iterations: we slice the triangle in half, again and again, thinner and thinner, aligning the slices to the hexagon's imaginary 'circle' line. Cutting a triangle in half creates two triangles, each with a 90-degree angle vertex. This allows us to use the Pythagorean theorem. The pictures below demonstrate the process. As the number of triangles doubles with each iteration, from 6 to 12, 24, 48, and so on, the imaginary outer rim becomes rounder and rounder. After only ten iterations, the original hexagon has over 6,000 edges and appears virtually round to the eye. The circle's circumference divided by its diameter (double radius) gives us the new Pi ratio. The circle will never be perfectly round, and the Pi number can thus be calculated to infinity.
RA = Square Root of ((R*R) - (R2 * R2))
Now that we know the value of RA, we calculate RB by subtracting RA from the radius. RB = R-RAWe are now ready to refine the circle line. 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 Pi stands much improved at 3.13. It is better, but not good enough.
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());
The results are:
3.10583
3.13263
3.13935
3.14103
3.14145
3.14156
...
Some more interesting facts.....
A full circle is expressed as 360 degrees. Although close, the number 360 is not based on the number of days in the solar cycle, with each day representing one degree. The Sumerians and Babylonians of Mesopotamia used the Sexagesimal Numbering system (Base-60). Since a circle is mathematically closely related to a hexagon (six equilateral triangles, a quasi-circle), 360 degrees represents its six sides, each with a sexagesimal value of 60.
The Sumerians and Babylonians practiced finger-bone counting. With an open hand and using the thumb, they tapped and counted the 'finger-bones': four fingers, each with three bones, amounted to 12. On the other hand, using all five fingers, they tracked the counts of 12, so 12 x 5 = 60.
Most likely, the finger-bone counting method is the foundation of the Sexagesimal (base-60) numbering system used by the Sumerians and Babylonians, a system still used today when we refer to degrees, minutes, and seconds. The high divisibility of the number 60 makes it useful. You can divide the number by 2, 3, 4, 5, 6, and more.
The 'Golden Ratio' is one of the most compelling curiosities in nature. It is the space where nature harmonizes with the rules of mathematics and physics. No wonder it is also called the divine proportion and the golden number.
In mathematics and 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 sum of both sides to the longer side. An object created by man or nature, built to a golden ratio, is of optimum beauty to the human eye. Early artists and architects knew the importance of the golden ratio. Much of the Parthenon in Athens's architectural design abides by the golden ratio, by instinct or by calculation. Most books of the 16th to the 18th century had heights and widths in harmony with the golden ratio, accurate to within a tenth of an inch. The dimensions of the Mona Lisa painting are in the golden ratio. The divine proportion of the rectangles gives the artwork the most appealing and pleasing appearance.
The Fibonacci Sequence is a progression of numbers where each number is the sum of the two preceding numbers.
1
2
1+1 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
8 + 13 = 21
13 + 21 = 34
21 + 34 = 55
...
...
The Fibonacci Sequence can also be demonstrated by building square boxes or drawing a snake. In the example, the ratio of the last set of numbers (34/21) is 1.619... As the Fibonacci Sequence increases, the ratio gets closer and closer to the Golden Ratio of 1.61803399..., but never exceeds it. The ratio is an irrational number; its progression is never exhausted.
We find Fibonacci numbers everywhere in nature. These numbers are nature's way of tallying. We have eight teeth on each side (two incisors, one canine, and five molars). We have five fingers on each hand and five toes on each foot. Petals in flowers, seeds on flower heads, and spirals in sunflowers almost always match Fibonacci numbers. Shrimps have five pairs of walking legs and five pairs of swimmerets. Starfish have five arms. Check the joints, the bones in your fingers. They are aligned in a Fibonacci sequence; the length of each increases by the golden ratio factor of 1.61. Some flowers do have four or six petals. Oh, you say, these are not Fibonacci numbers. But look closely; you will see that the petals may be in two layers, and the count in each layer matches a Fibonacci number. There are exceptions. The probability of a four-leaf clover is 1 in 10,000. No wonder - finding one is considered Good Luck.
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.
All weekday names are related to heavenly bodies: the sun, the moon, and the known planets. This is particularly evident in Romance languages, such as Latin and French. In English and German, some weekday names refer to deities associated with these same celestial bodies.
Each weekday stands for a celestial object in the sky. The names appear set in random order. Why is Tuesday (for Mars) before Wednesday (for Mercury)? One day is like the other; no attribute sets them apart, unlike, for example, how we name the four seasons.
So, how did ancient civilizations assign day names thousands of years ago in the order that is still valid today? The answer is truly eye-opening. Stay with it; it is a long read.
In ancient times, just seven heavenly bodies were known and revered. Each was given a name. The planets Uranus and Neptune were yet to be discovered. Each of the seven objects was sacred and was an embodiment of a God or Goddess. Each new day was devoted to a celestial object in a seven-day rotation.The seven heavenly objects have traditionally been listed in the order of the length of their orbit, meaning the elapsed time before their reappearance. In ancient times, the Earth was the center of the universe, and the sun was assumed to circle the Earth once a year. Here is how long it takes for each to complete an orbit:
Orbit sequence:
1. Moon: 29 days, Dies Lunae, Monday
2. Mercury: 88 days, Dies Mercurii, Germanic God Wodan, Wednesday
3. Venus: 225 days, Dies Veneris, Germanic Goddess Frigg, Friday
4. Sun: (Earth) 1 year, Dies Solis, Sunday
5. Mars: 2 years, Dies Martis, Germanic God Tiw, Tuesday
6. Jupiter: 12 years, Dies Iovis, Germanic God Thor, Thursday
7. Saturn: 29 years, Dies Saturni, Saturday
So, we should think that the lineup of the days would be as follows:
Monday,
Wednesday,
Friday,
Sunday,
Tuesday,
Thursday,
Saturday
or in reverse.
But the days are not in that order ... How come? Let us dig into it further. As early as four or five thousand years ago, the ancient civilizations, the Egyptians and Babylonians, already partitioned the day into time sections: a 12-hour day and a 12-hour night, for 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, allowing for twilight hours. The number 12 has always been important; finger-bone counting may have been its origin.
Each new hour was dedicated to a God. So, 24 times a day, at the turn of the hour, the hour was devoted to another God, in strict order of the established orbit lineup. The first hour after midnight was paramount. The God of the first hour owned the whole new day, and the day was named after that hour's deity.
There were seven God Stars, and the cycle repeats itself every seven hours. But there are 24 hours in a day. The number 7 does not neatly divide into 24. Thus, every midnight, every 24 hours, the pick from the fixed list of Gods shifts by 3. The animated GIF below demonstrates the process. Watch it for the full cycle; it runs over two minutes. It is amazing; the seven-day, 24-hour run-through creates the exact weekday order that we know today: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday.
In the early years of Roman history, the year began two months (moon cycles) after the winter solstice (today's March) when daylight grew longer, and the soggy soil firmed up. Thus, the calendar year consisted of only ten months, and was followed by a nameless two-moon dead winter season. The first month of the year was named after Mars, the God of War, perhaps marking the time when young men were called back to fight and conquer new lands. The month name April derives from Aprilis, meaning 'the month that opens,' perhaps alluding to the buds opening up in spring. April may have been 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 from the sequence numbers in the Romans' language: Quintilis 5, Sextilis 6, Septilis 7, Octilis 8, and so on. Remember, the year started two moons after the winter solstice, so Quintilis (July) was the name of the fifth month.
Around 700 BC, to synchronize the solar year with their calendar, the Romans added 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 is appropriately depicted as two-faced, looking in opposite directions, looking back at the year past 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 means 'burning,' perhaps related to fever. It makes 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, October, November, and December, in Roman/Latin, which refer to the numbers Seven, Eight, Nine, and Ten, are now out of whack. And the leap year adjustment is still in February.Then, Roman Emperors with big egos arrived on stage. Julius Caesar wanted a month named after him. He ordered that Quintilis, the fifth month, be named July, in honor of his name Julius. Not to be outdone, Emperor Augustus claimed the sixth month, Sextilis, for himself. Julius' month traditionally had 31 days, but Augustus' month, Sextilis, had 30 days. Augustus did not accept that his month had fewer days than his uncle Julius's, so the number of days in August was increased to 31, and it still is today. The Romans viewed odd numbers as more propitious than even numbers; odd numbers felt more copious because there was some extra after an equal sharing.
Decades ago, Europeans developed and propagated Metric paper sizes based on the 'Square Root of Two' ratio, now also dubbed the Lichtenberg ratio. The ingenuity of the v2 ratio rests in the premise that when the paper is cut or folded in half lengthwise, the two halves retain the parent's aspect ratio.
There are three size series, the 'A' series being the most prevalent. Thus, paper size A0 has a total area of 1 square Meter. Take that square and reformat it into portrait mode at a height-to-width ratio of 1.4142 (which is the square root of 2). When you cut an A0 paper sheet in half horizontally in portrait mode, you have two A1 sheets. When rotated, both have the same width-to-height ratio as the original A0. Cutting an A1 sheet in half creates two A2 sheets, then two A3 sheets, two A4 sheets, and so on. A4 is the most commonly used paper size, the equivalent of the North American Letter Size. The thought behind the ratio is ingenious; it challenges the human mind!
