Boxing Pythagoras

Philosophy from the mind of a fighter

Archive for the tag “pi”

On Wronski’s Definition of π

 

Joseph Nebus has recently written a couple of posts (here and here) in which he discusses an interesting attempt by Józef Maria Hoëne-Wronski to create a purely numerical definition of the mathematical constant π which is independent of the classical, geometric definition of “the ratio of the circumference of a circle to its diameter.” This has been a goal of many mathematicians, since the idea of π seems like it is more fundamental to mathematics than a definition based on circles would make it seem– as evidenced by the fact that it shows up in areas of mathematics which are seemingly unrelated to circles. Wronski’s idea, to this end, was the following formula:

\pi = \frac{4\infty}{\sqrt{-1}}\left\{ \left(1 + \sqrt{-1}\right)^{\frac{1}{\infty}} - \left(1 - \sqrt{-1}\right)^{\frac{1}{\infty}} \right\}

At first glance, the formula seems inherently nonsensical. After all, \infty is not a number, and therefore cannot be utilized in numerical operations in this way. However, one can get a sense of what Wronski may have intended by this equation. It appears that Wronski wanted to utilize \infty to represent an infinite number, and modern mathematics actually gives us several tools for handling this sort of idea. One which might be of particular use, here, is Non-Standard Analysis with its infinite and infinitesimal Hyperreal numbers. In NSA, we have the ability to perform calculations with and upon infinite numbers perfectly consistently and reasonably.

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Proof that π=2√3

There is an inherent danger attached to blindly accepting the word of someone who sounds like they are presenting a rational, scientific claim. Too many people are willing to accept a proposition solely because they’ve heard it from someone who bears the appearance of intelligence. The line of thought seems to be, “Well, he’s smarter than me, so he must be right!” Unfortunately, this sort of fallacious reasoning goes largely unchecked, and often becomes formative in the common understanding of entire groups of people.

For almost the entirety of your mathematical education, you have been taught that the ratio of a circle’s circumference to its diameter, which we affectionately refer to as π, is something close to 3\frac{1}{7}, or about 3.14; however, today I’m going to show you that your math teachers were wrong. In actuality, the value of π is exactly 2√3, or about 3.46.

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Be smart. Use tau.

For anyone who didn’t know, this past Saturday was Tau Day, a celebration of the proper circle constant!

A couple weeks ago, I told all of you about how π is stupid, and urged everyone to be smart and use τ, instead. However, you might be surprised to learn that this is not the end of the debate, when it comes to angles. While I argue that people should measure angles in terms of τ, many traditionalists argue that they should be measured in terms of π, our grammar schools are still intent on teaching the incredibly archaic degrees of arc, and if you’ve ever fiddled with a scientific calculator, you might have learned that some backwards people prefer gradians. But that’s still not the end of the debate. According to a video by Dr. David Butler of the University of Adelaide, “π may be wrong, but so is τ!”

I’m going to celebrate Tau Day, belatedly, by rebutting Dr. Butler’s presentation. I’m going to show that degrees, gradians, η, and π are all stupid, and that the only smart choice in this debate is τ.

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Pi is Stupid

I teach Brazilian Jiu-Jitsu to people of all ages, from preschoolers to middle-aged parents. While BJJ, in itself, is not necessarily the most academic of pursuits, I also happen to be a huge nerd. So while teaching some of my 8 to 13 year-old students, it sometimes happens that I overhear them talking about their math classes, often to complain about ideas that they’re struggling to grasp. Being a huge nerd, and also a delighted teacher, I do my best to help them through these issues. If I can teach a kid how to find the length of the hypotenuse of a right triangle at the same time as teaching her how to finish a Triangle Choke, I become pretty much the proudest martial arts instructor you could hope to meet.

One of the things that my kids often use in their math classes, but almost never really understand, is the constant π (pi). They are taught π in class to help learn things like how to calculate the area of a circle, but they usually don’t really know what π actually is. They just think of it as some number that they have to memorize, never thinking about where the number comes from, or why it is what it is. Sometimes, I’ll tell the kids that they can earn their way out of doing push-ups if anyone can tell me what π is. Most often– after the jokes about desserts are made– I’ll hear someone say, “Coach, π is three-point-one-four!” Every now and again, one of the kids is clever enough to say, “Coach, π is three-point-one-four-on-into-infinity!” They get confused when I tell them that’s the value of π, but that is not what π actually is. It’s not their fault that they get confused by this; they were usually taught about π all wrong. I don’t even blame their math teachers, because most of the time, those math teachers were also taught about π in the wrong way. For a very long time, math classes have been teaching that π is a number, instead of teaching that π is the relationship between a circle’s circumference and its diameter. There is a reason it has been taught this way.

Ladies and gentlemen, π is just plain stupid.

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