
Facts of the Matter
Richard Brill

Circle holds endless fascination
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FACTS OF THE MATTER
Richard Brill
MATHEMATICS is the study of order, patterns and relationships, and it is considered by most to be an exact science.
But there are certain numbers that mystically defy order, and which have fascinated philosophers and mathematicians for thousands of years.
The ancient Greeks considered the circle to be the perfect shape because of its simplicity and its perfect symmetry. Yet the circle contains a number that is transcendental, precise yet random, calculable but limitless.
It is pi, the apparently simple ratio of the circumference of a circle to its diameter and the ratio of the area of the circle to its radius squared.
But what is the value of pi?
Its complexity in contrast to the simplicity of the circle has made this seemingly simple number one of the most interesting and sought after in the universe.
Long before there was a concept of the number that we call "pi," people attempted to "square the circle," that is to find a circle that has the same area as a given square.
The earliest known record of squaring the circle comes from Egypt around 1650 B.C. on a document known as the Rhind Papyrus.
A scribe, Ahmes, wrote that a square with sides equal to eightninths of the diameter of a circle would have the same area as the circle.
This works out to be equal to 256 divided by 81, which is only about sixtenths of 1 percent different from our modern value.
For those who find amusement in the coincidence of numbers, this ratio is 16 squared divided by nine squared: (a) 16:9 is the aspect ratio of widescreen televisions; (b) "p" is the sixteenth letter of the alphabet, and "i" is the ninth letter.
The Egyptians apparently did not yet have a concept of a single number that related circumference and diameter. Their interest was of a practical nature, to measure land and buildings as accurately as possible.
The work of Ahmes the scribe did not spread. For more than 1,000 years, civilizations around the Mediterranean simply used the value of 3 for the ratio of circumference to diameter.
In the fourth century B.C. the Greeks came to believe that the circle was the perfect figure because of its symmetry and simplicity, so they assumed that the geometry should be equally simple.
The ratio of circumference to diameter was not the most important topic of thought, but it did occupy some of the greatest minds in ancient history.
Around 250 B.C., Archimedes doubled the sides of a hexagon four times, producing a polygon with 96 sides. He calculated the perimeter of the polygon and found a value for pi that was less than eight tenthousandths of 1 percent off.
In 125 A.D., Ptolemy, author of "The Almagest," had improved it by a factor of four.
Around 425 A.D., Zu Chongzh used a polygon of 24,576 sides to calculate a value that was accurate to eight onemillionths of 1 percent, the most accurate for nearly a millennium.
In India around 1400, a mathematical genius by the name of Madhava made a breakthrough in mathematical thought that would revolutionize all of mathematics and allow one to calculate pi to as many decimals as time and patience would allow.
Madhava invented converging infinite series and power series.
An example of an infinite series is: one plus onehalf plus onefourth plus oneeighth plus  the series continuing indefinitely with each term equal to onehalf of the preceding one.
By the tenth term it will have converged to greater than 0.999. The largest value the series can sum to with an infinite number of terms is the integer "one," which is equivalent to 1.000 ... with an infinite number of zeros.
Madhava calculated pi to 13 decimals.
There is little practical use for such precision. Given Earth's diameter and only 10 decimals of pi, you can calculate its circumference to onetrillionth of an inch, and 30 decimals would make the error imperceptibly small for a circle the size of the known universe.
Yet people kept trying to find more ways to add decimals to pi. In Europe, where the ways of the Madhava had not yet trickled in, one calculation that used polygons of more than 32 billion sides to calculate 35 decimals of pi took years to complete.
From the 16th through the 19th century, European mathematicians discovered increasingly ingenious infinite series for calculating pi, competing to see how accurately and to how many decimals they could calculate it.
They also discovered that pi is irrational and does not appear to have any repeating patterns of number sequences.
By the 20th century, the goal had shifted to finding more efficient ways to calculate pi.
A breakthrough came in 1947 when ENIAC, the first largescale, programmable electronic, digital computer, took 70 hours to compute 2,037 decimals of pi.
More efficiency and more powerful computers then led to more digits.
The current record holder is Yasumasa Kanada of the University of Tokyo, who used a supercomputer to calculate pi to 1.2411 trillion digits in only 600 hours.
The Chudnovsky brothers, David and Gregory, who built their first supercomputer at home in the 1980s from mailorder parts, are looking for a deeper understanding of pi.
What they care about are numbers that are beautiful, perfect, more complicated and arguably more real than the physical world.
They are looking for some rules to distinguish pi from other numbers. They wonder if someone gave you a million digits from somewhere in pi, would there be any way to determine if it was from pi.
If there is a design hiding in the digits of pi, no one knows.
Mathematicians generally feel that there is no system in the digits of pi.
The Chudnovskys say that pi is more random than the pseudorandom numbers generated by a computer.
We look for patterns in numbers as in nature because our brains are programmed to recognize and respond to order.
We understand language and recognize music and shapes because we hear patterns of tone and rhythm and see familiar patterns of lines and shapes.
We see faces in clouds, in the surface features of Mars or in the Heike crabs of Japan.
If we assign the digits of pi to musical intervals, there is no melody, but you might start to think you hear one if you listen long enough.
The digits of pi cannot be distinguished from a random string of numbers, yet if you change a single digit, the string of numbers is no longer pi, but just a random number.
The simplicity of the circle described by the complexity of pi is perplexing and feeds the human spirit as it fuels our curiosity.
My birthday begins at digit 1,820,975, and my cell number starts at digit 535,350,159.
If you ever need to get in touch with me, just get out your pi calculator and find my number. It's there in the pi phone book.
Try your own numbers at www.pisearch.de.vu.
Richard Brill, professor of science at Honolulu Community College, teaches earth and physical science and investigates life and the universe. Email questions and comments to
rickb@hcc.hawaii.edu