Life of π

Seems that three months have passed since I last wrote anything about anything here.  Turns out that actually participating in science is a time consuming endeavor.  Actually, it’s been a little over 3 months…3.19 months, actually…which is pretty close to…

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Today is also Pi Day in the United states, as well as Albert Einstein’s b-day, so the power of math compelled me to post something.

The question is, why do we even care about π?  Why are people across America eating a slice of banana cream pie right now to celebrate a number?  Do Americans really need ANOTHER reason to eat pie?  Consequently, Pi Day will fall on Wednesday in 2018, the day on which Village Inn offers free pie…imagine the anarchy that will ensue!

What is it about π that has interested people for millennia and given it almost mythical powers?

First off, where did the symbol π come from in the first place?  Previous to the year 1706, one had to be content to use the Latin phrase quantitas, in quam cum multiplicetur diameter, provenient circumferentia”, meaning “the quantity which, when the diameter is multiplied by it, gives the circumference”.  Clearly the essence of convenience.  Also, guess where all of our mathematical terms come from.

In 1706, my bro William Jones busted out the symbol for the first time in print, thereby relieving the hand cramping of his colleagues, choosing the symbol π (presumably) because it was the first letter of the Greek word περιφερεια, meaning “periphery”.

William Jones, Welch mathematician: Thanks Bro!
William Jones.  Thanks bro!

But what’s with the fascination over this particular ratio, the ratio of a circle’s circumference to its diameter?  Humans have long been enamored by numbers, giving symbolism to the digits 1 through 10 and various combinations thereof.  7 is lucky, 13 is unlucky, 1 represents the self, 2 represents unity with another, and so on.  Combine that mysticism with that which already existed with geometry for thousands of years and that gives some sense of the power π held over the ancients.

The Pythagoreans, pretty much the progenitors of the movement that lead to the Golden Age of Greek mathematics and cause of most middle-school student’s nightmares, believed that everything in the universe could be explained with number, specifically ratios of whole numbers.  They created a musical tuning such that all notes were related to each other by whole ratios, 2:1, 3:2, 4:3, and so on.  Everything divine in the universe could be expressed in this way.

The circle is, by most accounts, the simplest geometric figure you can create: a set of points all equidistant from some center point.  Pretty much the only things going on in a circle are the diameter and the circumference.  Needless to say, the ancients really wanted to find the magical ratio between the two.  They appear everywhere in Nature: all of the celestial bodies, bubbles, droplets of liquid.  What secrets could be unlocked finding this divine ratio?

The Greeks were like, well, ok, we can just  find the correct geometric construction and find it.  My bro Eudoxus came up with a pretty cool plan around 360 B.C. or so called the method of exhaustion.

Eudoxus.  Thanks bro!
Eudoxus. Thanks bro!

His plan was pretty simple, but ingenious.  Time to bust out my tools:

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Now, back in the day, the Greeks didn’t have a ruler per se, merely an unmarked straight edge.  Deal with it.  First up, draw a circle:

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Then, go about bisecting, drawing arcs and such…

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…until you can inscribe a polygon inside your circle.  In my case, I inscribed a regular pentagon.

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Then, using your inscribed polygon, bisect some more…

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…and then create a circumscribed polygon around the circle.

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Now, the perimeter of the inscribed shape is clearly less than that of the circle, while the perimeter of the outside shape is clearly greater than that of the circle.  If you find the perimeters of the two pentagons (30cm and 37.5cm, respectively) and then average them (yielding 33.75cm), you have a pretty good approximation of the perimeter of the circle.  Then you take the ratio of this average perimeter and the diameter of the circle (which is 10.6cm) and you get that π is about 3.1839 or so, not bad.  Give the polygons more sides and you get a better approximation.  Here, I’ve inscribed a hexagon and used it to construct a 24-sided polygon.

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Turns out that the ratio of its perimeter to its radius is 3.1385.  Better…

One hundred years later, my bro Archimedes came up with a pretty crazy idea in his work The Method, was was lost until recently.

Archimedes.  Thanks bro!
Archimedes. Thanks bro!

He made the bold observation that if you give the polygon an infinite number of sides, it becomes the circle!  He drew some 96-gons (we run out of cool names pretty quick, not unlike with elements) , came up with some formulae to find their perimeter and nailed down π  to between 3 10/71 and 3 10/70, or between 3.1408 and 3.1428.  Nice work, Archimedes.  If you continued this infinitely, you would get the true value of π.

And then Greece declined, Rome showed up, and Europe in general just stared out the window into the rain catatonically for a millennium and a half.

Then, in 1761, my bro Johann Lambert showed up and proved that π was irrational, screwing up the party for everyone…sorry.

Johann Heinrich Lambert...thanks, bro...
Johann Heinrich Lambert…thanks, bro…

The fact that π is an irrational number means that it cannot be expressed as the ratio in any form.  It is a never-ending, never-repeating decimal number: 3.1415926…  The issue was that the ancients didn’t know this (well, a Pythagorean “discovered” the irrationality of the square root of 2, but he was promptly killed for heresy) and the search went on for centuries to find the pattern.

By the 16th century, algebraic geometry had taken over and people were looking for an equation that could yield the value of π.  That didn’t last long, because my bro Charles Hermite came up with a method, later used by Ferdinand von Lindemann, to prove that π is not just irrational, it’s transcendental, that is, there is no polynomial of which π is a root.

Charles Hermite.  He has clearly had enough of your π bullshit, Europe.
Charles Hermite.  He has clearly had enough of your π bullshit, Europe.

At this point, all we can do is approximate π with infinite series and such, which we do to further and further numbers of decimal places, secretly hoping that math is wrong and we’ll find a pattern.  Actually, it’s mainly to prove processing power, but whatever.  At this point, the world record is held by Shigeru Kondo, who calculated π to 5 trillion digits.

 

Shigeru Kondo.  Infinite digits?  Come at me, bro.
Shigeru Kondo. Infinite digits? Come at me, bro.

So, π is basically the white whale of mathematics, tying together a linage of mathematicians over almost three millennia trying to answer a simple question:  what do I get if I divide this by this?  Indeed, things in the world are not a simple as they seem…except for the carbohydrates in this yummy pie.

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