Mathematics > The Relationship between a Square and a Circle

The Relationship between a Square and a Circle

By LUKE MCGOVERN
Published: October 11, 2011

An old set of ideas has come flooding back. It centers around the idea of Pi (π). Well, 2π actually - also know as Tau (τ). See when travelling around Central America and chilling out in Bocas Del Toro I spent two days playing with a set of ideas. In essence I wanted to find the relationship between a square and a circle - fun times.

Now, I'm far from a mathematician. In fact my math is pretty average but the beauty of maths is that even if you are average you can still have a crack at it and you'll either be right or you will be wrong. So even if you suck at maths you can still be perfectly correct.

The reason I spent time in a hammock thinking about this square/circle relationship and not doing more touristy things is because - I could. That is the beauty of being on holiday - you can do what ever you want because you are on holidays! That's what a holiday is.

So how far did I get in my two days of number crunching? Not very far. You see squares and circles have such and amazing relationship in the fact that they are like chalk and cheese. Trying to avoid the exact idea of "squaring the circle" http://en.wikipedia.org/wiki/Squaring_the_circle - my goal was to try and find a way to calculate the area of both shapes using the same equation. A circle is πr squared and a square is just one of its sides squared. I thought of trying to pretend that the square had some kind of radius and that would allow me to use something based on πr squared. In the end there were too many rational numbers at play and nothing would meet up.

I then moved on to a different idea. Can the relationship between the perimeter of both shapes be related? I failed to come up with anything decent in Panama but as I mentioned earlier - the idea resurfaced. This time with with τ. Now you may be thinking - well that just 2π so what. Well, it makes the math easier or at least it helps me see things a bit better now.

So, take a square with a side of 2 units and match it to a circle with a diameter of 2 units (or a radius of 1 unit). This is the biggest circle that the area of the square can contain. What is the difference in their perimeters? The square has the value of 8. The circle has the exact value of τ. Now, consider a method for calculating τ:

τ = 8/1 - 8/3 + 8/5 - 8/7 + 8/9 ... etc

When I first learned about http://en.wikipedia.org/wiki/Leibniz_formula_for_pi (which also works for τ) it blew me away. I have yet to see it explained as to how or why it works. I understand the math but I do not understand the greater meaning behind it. How could τ equal such a series? And why does it start with 8 (as 8/1 = 8).

So there it is from the start. A relationship between the perimeter of a square and the number τ. We start off in the first instance with the number 8. Then we exercise a pattern on it and then it equals τ. That is a direct relationship - square = 8 and circle = τ. There is only one problem...

Here is the series again:

τ = 8/1 - 8/3 + 8/5 - 8/7 + 8/9 ... etc

The pattern or rather, the series, is infinite. And yes we can calculate it with calculus but we have an infinity involved in the relationship between a square and a circle. This pretty much sums up the relationship between these two shapes. They are separated by an infinite divide. That's a sad relationship.

The reason for the sadness is that Infinity in maths tends to cause headaches. For instance, if you add it in as a value to an equation, the equation will break and no longer give you an accurate result.

My Take on the whole shebang

In my mind, if the relationship is infinitely opposed like I have suggested - then it appears to be something like an inflection point between the two. As one passes from one shape-state to the other (say by cutting off corners of the square to become a circle - which would need an infinite number of passes) it appears like it goes through an inflection point - almost like a singularity where all rules break down and new ones come in to play and nothing can be done to change it.

Any Comments?


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