So it seems the question is, does a relatively simple mathematical example portray this beauty that the learned mathematician speaks so highly of? Is there some “simple” math that also is deep in this special quality that aestheticians or epistemologists (or metaphysicians) so elegantly long for? I think the answer is yes. This is a little bit of math that was introduced to most students in the 6th grade level have conquered this principle and have a good understanding of … at the very least … some basic facts about pi (π).
And what are those basic facts? I’m thinking about the ratio of the Circumference of a circle (the distance around a circle) to the Diameter of a circle (the length of a line across a circle that passes through its center). These sixth graders know that it is symbolized by the Greek letter π called “pi” that rhymes with Apple Pie. It is approximately equal to 3.14 or 22/7. And that is another fact they typically have conquered: there is no precise value for pi. It is an irrational number, although sixth graders may not use the term “irrational.” But they mostly know you can’t write the number down. It just goes on, and on, and on: 3.1415926535897932385 … That’s the value to 20 places, but there are more, and they don’t repeat, they appear to be random and non-ending … because they are.
Many mathematicians celebrate Pi Day each year on March 14 or 3/14. This year (2015) was especially interesting pi day because the year plus the hour, minutes, and seconds, extended the number to more decimals than will occur for over a 1,000 years. March 14 is also Albert Einstein's birthday!
Of course, these same sixth graders will likely use pi in various equations and may even physically measure a few circles just to confirm, as best simple measurements can confirm, that the number is in the neighborhood of a 3.14. The formula for the area of a circle will be introduced and concepts such as “squaring” take on practical mathematical usefulness. But there is so much more. What else could we teach these sixth graders, or perhaps someone with a bit more math education like a High School graduate? Or we could just state that cake are square … not pie … old joke!
From the basic definition that the Circumference equal pi times the Diameter, we can write this as an equation: C = π D. Further, we know that the radius, r, is half of the Diameter, so this would be: C = 2πr.
The next thing we learned in the sixth grade math class was how to calculate the area of a circle. That too used this magic number pi. I still recall how it was explained to me so long ago at Garfield Elementary school. You take the circle and divide it up into sections (called segments). You then rearrange the segments into this nearly rectangular shape.
The shape on the right is a parallelogram and the formula for the area of a parallelogram is A = b x h, or base times height. The base is half of the circumference, since the other half makes up the top. If the total circumference is 2πr, then half of the circumference would be πr. Since height of each segment is the radius, then height would be r. So the area of the circle is πr times r which equals πr2.
When I first saw this example I had a problem. It isn’t exactly a parallelogram. The top and the bottom are “wavy” and the wave part reduces the area from a “real” parallelogram. So the formula seemed bogus. Then I thought about what the teacher had said about there being an infinite number of numbers in pi and that got me thinking about infinity. What if, instead of cutting the circle into 8 segments, you cut it into 16 and made the parallelogram out of the 16. The waviness would be less and the formula would seem more accurate.
Keep up dividing the circle into smaller and smaller pieces and building the parallelogram. As the number of segments goes to infinity (I now know the correct description is “approaches” infinity), the waviness would disappear and the formula would be exactly right. If you make the parallelogram out of the the tiniest slices imaginable, then the curvature would disappear. And there you have it … as a sixth grade student I’d just invented the Calculus. Well, sort of.
Are you starting to see the beauty of mathematics? How you can play games in your mind and do impossible things with thought. Einstein called these little mental exercises (in German, of course) gedanken experiments.
Early mathematicians realized pi’s usefulness in calculating areas, which is why they spent so much effort trying to dig its digits out. Archimedes used 96-sided polygons to painstakingly approximate the circle and showed that pi lay between 223/71 and 22/7.
By calculating the area of the polygon drawn within the circle you get the lower bound for pi. The polygon that circumscribes the circle is a bit larger and gives the upper bound. If you increase the size of the polygon … the number of sides … you get values closer and closer to the actual value of pi. But this method is very tedious and better methods were soon found.
By the time Madhava (in India, around 1400) calculated pi to over 10 decimal places using his groundbreaking infinite series (which regrettably bears Leibniz’s name), it was already more than accurate enough to address all practical applications. Pursuing pi further had essentially become a mathematical challenge.
The equation we now call “Leibniz’s Equation” is very interesting for its simplicity. It is an infinite equation, which means it has an infinite number of terms. After all, if an equation existed that didn’t have an infinite number of terms then that would imply pi has an exact or “algebraic” value, and we know it does not.
This interesting equation is simply the sum and difference of all the odd numbers written as fractions. Actually it is the formula for one quarter of pi, so you have to multiply the result by 4.
One, minus a third, plus a fifth, minus a seventh, etc. is the calculation. Here, in the terse symbols of math is what I just said.
Here it is in summation notation.
Leibniz's formula converges extremely slowly: it exhibits sublinear convergence. Calculating π to 10 correct decimal places using direct summation of the series requires about five billion terms. There are much better formulas that converge on the correct answer with less terms. These “better” equations are now used to calculate pi. Two mathematicians, Yasumasa Kanada and Daisuke Takahashi from the University of Tokyo, calculated pi to 206,158,430,000 decimal places in 1999. And that’s not the record.
Still, imagine the power and beauty of mathematics. This simple yet difficult to write down fully number that comes from the simple forms of geometry and measuring a circle also shows up in an infinite series of the odd numbers. How could this be? But wait, even more amazing “coincidences” (or are they) will appear as we dig into the simple, yet complex number pi. This is part of the beauty of mathematics I keep talking about. Are you starting to “grok” it?
With the advent of computers, pi offered a proving ground for successively faster models. But eventually, breathless headlines about newly cracked digits became less compelling, and the big players moved on. Recent records (currently in the trillions of digits) have mostly been set on custom-built personal computers. The history of pi illustrates how far computing has progressed, and how much we now take it for granted.
So what use have all those digits been put to? Statistical tests have suggested that not only are they random, but that any string of them occurs just as often as any other of the same length. This implies that, if you coded this monograph, or any other article or book, as a numerical string, you could find it somewhere in the decimal expansion of pi. One could argue that all knowledge of man, both current and future discoveries, is hidden somewhere in the digits of pi. Don't be too amazed, that's just the concept of probability extended to the infinite.
Of course, that’s relatively useless, since you don’t know where to find the material you want … exactly where in the decimal expansion of pi would you look. An apt metaphor for an age when we are being asphyxiated by mushrooming clouds of information.
But pi’s infinite randomness can also be seen more as richness. What amazes me, then, is the possibility that such profusion can come from a rule so simple: Circumference divided by Diameter. This is characteristic of mathematics, whereby elementary formulas can give rise to surprisingly varied phenomena. For instance, the humble quadratic can be used to model everything from the growth of bacterial populations to the manifestation of chaos. Pi makes me wonder if our universe’s complexity emerges from similarly simple mathematical building blocks. That is a basic belief of science encoded in Occam's Razor, the philosophical concept that the simplest explanation is the correct one. Without that belief, science would probably be impossible. Did you know that Science involves "faith"? Yes it does. That's rather beautiful in itself.
Oh, I’ve got a lot more to say about pi. There is so much more wonder and mystery in this simple concept known since the beginning of civilization. I’ll let my humble readers digest this morsel before I pour on more syrup and add some butter. So, until the next installment, happy trails.