I was watching a video today about pi, the mathematical concept for the ratio of a circle’s circumference to its diameter. In the video they noted that the beauty of pi is that it never repeats. I’m not a math wiz or anything of that nature but I wanted to take a stab at why I think this is the case. It is something I’ve enjoyed in my head off and on whenever I think about shapes, specifically any shape more complex than a circle. This can be done with triangles, squares, hexagons, dodecagons, etc, etc. I’m visually going to use a square because it is part of a very silly idea. Namely the square wheel, the idea that if you spin a square fast enough it will always be touching the surface for long enough to keep the vehicle suspended and continue with forward motion.
This belief obviously doesn’t take into account the surface area that all this kinetic energy needs to actually be transferred through (very inefficient) but regardless it is easy to visualize. It is also the basic idea for my understanding of a circle and why pi has no repeats.
A circle is similar to me in mathematics as light is in physics. It is two things at the very same time. A circle is both a one sided object and an infinitely sided object. Much like light is both a wave and a particle. It is something I’ve held in my heart for ages and as far as I’m aware this is not necessarily wrong. For simplicity I’m going to use a square with equally long sides so that this all lines up nicely.
This idea has actually been done in a different fashion by ancient mathematicians and is known in the older concept as squaring the circle. What I want to do however is to literally make a circle from squares themselves
This is obviously a square. But lets say we take a clone of this same square and we tilt it over this one at 45 degrees?
Forgive my image skills I’m using gimp and it is remarkably complicated to do anything with that program. However as you can see we have a square (closer to a rectangle it appears) inside of a clone of itself. 45 degrees is not a random number from the depths of my mind, it is half of 90 degrees. So we will repeat this step and take half of 45 or 22.5.
You might already be seeing where I’m going with this. We’ve created a shape with man different sides but in terms of points it has 12 points, 3 squares x 4 points = 12. I’m going to repeat this process for 11.25 degrees, 5.63 degrees. 2.81 degrees. 1.41, and finally .7 degrees. This should give a good representation of what I mean.
This monstrosity is now featuring 8 squares, which means it has 32 points, 8 squares x 4 points = 32 points. As you can also tell the squares are starting to merge into one another as the halving becomes smaller and smaller. This will however never reach 0 and will continue to spiral in a circle an infinitely many times. Infinity is not technically a number but in this context you would have infinity squares x 4 points = infinity x 4 points on your circle. Additionally this means that between any 2 of these infinitely many squares you have an infinitely small straight line. This infinite collection of infinitely small lines constitutes a collection of infinitely many sides.
Here is that same process repeated with all previous squares an additional 7 times, so that’s 8 squares repeated 7 times or 56 squares x 4 points = 224 points total. Already you can see more easily within than without that a circle is forming.
There is apparently a fractional system used to find pi with computer systems that reads out like this (I didn’t know this till I watched numberphile today).
pi / 4 = 1-(1/3)+(1/5)-(1/7)+(1/9) and so on till infinity.
The longer you do this the more accurate your answer for pi becomes. I believe the reason for this is that those triangles being formed outside of a circle are also being replicated in the structure of the circle itself. A circle is the collection of an infinitely many squares that have been rotated half the distance of their priors tilt. So 90, 45, 22.5, 11.25, 5.625, 2.8125, 1.40625, 0.703125, 0.3515625, 0.17578125, 0.087890625, 0.043945313, 0.021972656, … and so on. Because it has an infinitely many points and infinitely sides and because each side is comprised of two even smaller sides the numbers continue on ad infinitum without ever reaching 0. It’s geometries most popular asymptote…I suppose.
This also, I feel, ties in nicely with String theory. One of the underlying ideas of string theory (and perhaps the most important) is that the entire universe is made up of the same thing. Essentially it is a circle, but depending on how you twang this circle you get a different subatomic particle and those subatomic particles combine into particles and those particles into atoms and those atoms into molecules and those molecules into chemical bonds and those bonds lead inevitably up the chain to everything you taste, see, smell, hear, or touch.
A circle has an infinite amount of sides so it is not unreasonable that it could be “strung” in an infinitely many variants. This means that string theory is not all that far fetched, at least from my perspective, because when your scale of outcomes is infinite then any outcome is not only possible but plausible.
So that’s that, I really like circles, so this was quite a lot of fun to write about. It is also the second time I’ve written an article almost entirely about squares and the nature of angles. Now if only I had any sort of higher level mathematics degree to not make all this mad ramblings.