In this drawing Jason is showing how perfect circles do not exist...anywhere in the universe. Circles are made from triangles as are all shapes and are deeply interconnected with the number Pi. The more triangles you squeeze within a certain perimeter the closer and closer you approach a circle but you never reach it...you just get closer and closer. This is WHY Pi goes to infinity. Pi is the circumference of a circle (how far around the perimeter) divided by its diameter (the distance from one side of the circle through the center and to the other side). When you add more triangles, the circumference gets slightly larger each time you add more triangles (but by an ever decreasing amount), but the diameter stays the same. A circle with 180 right triangles when you divide it's circumference by it diameter you get 3.141433 (Pi accurate to three decimals). A circle with 360 right triangles, one right triangle every one degree, gives you 3.141552 (Pi accurate to four decimals). This picture is 720 right triangles, which when you divide it's circumference by it's diameter gives you 3.141582 (Pi still accurate to four decimals but almost to five). If we could draw a circle with 25,000 right triangles (and nature does this and far more everyday) it would be 3.141592645 (Pi accurate to seven digits). What is so amazing about Pi is that everything in the universe breaks down into it. All shapes (including circles, and 'perfect circles' don't actually exist..they are made from triangles) break down into straight lines and triangles. Every curve breaks down into microscopic straight lines that change angle by very small amounts (just like this picture looks like a curve but is all perfectly straight lines). In math (trigonometry) they teach how to convert measurements made in degrees (360 degrees is a circle), into measurements made with Pi (this measurement is called Pi radians). To convert degrees into Pi radians you multiply the degrees by Pi/180 (the fraction of pi over 180). All straight lines are 180 degrees (two right triangles make a straight line 90 + 90 =180) and every triangle that exists when you add up all three inner angles always equals 180 degrees (the same as the straight line). Since all shapes that exist break down into straight lines and triangles lets see what happens when we convert the degrees into Pi radians. 180 times Pi/180= 180Pi/180 (in words 180 times pi divided by 180). The 180's cancel out because 180 divided by 180 equals one and one times Pi equals Pi! So every shape in the universe breaks down into pieces of Pi interacting with other pieces of Pi, literally. And some say nature doesn't have a sense of humor! This picture is drawn with straight lines only. It is 720 right triangles rotating 1/2 degree at a time. There is no curve but as you can see the more triangles that are squeezed into the perimeter the more it APPEARS to be a circle or curve. Just like when you look at a mountain and think, that is not a curve or circle, but if you zoom out into outer space and look at the earth it LOOKS like a perfect sphere. But when you're on the earth it looks completely different. In other words, what is a circle is relative. The mountain is not a circle when you're looking at it one the earth, but it looks like a circle (or part of one) from outer space. This is one of the many beautiful illusions of relativity. For those who like the equation aspect, graph f(x)=xsin(Pi/x) and see what happens. As x (the number of sides to a circle) approaches infinity, f(x) will approach Pi. Last but most important is that we have been taught that Pi never ends. That is true when looking at this in pure equation (or mathematical) form...but it is wrong when looking at a 'circle' in reality. This is the really amazing part....Pi ends (physically or more descriptively, in an observational sense). The smallest measurement in the universe is something called a Planck length. The Planck length arises from the limits of observation. In order to observe anything you bounce photons off of it and they bounce back and you observe the geometry of what you are observing. To observe smaller things, you must hit what you are observing with more photons (or more energy) to get a finer reading (or to be able to observe it). As you observe smaller and smaller things you have to hit them with more and more energy to observe them. You eventually reach a physical "limit" to observation. When you try to observe something smaller than a Planck length you must hit it with so much energy that the point that you are trying to observe collapses into a black hole, absorbs the photons and there is nothing left to observe. So while points smaller than the Planck length exist, they CANNOT be observed by us and are therefore not 'RELATIVE'. It was discovered by Maxwell Planck and he won the Nobel prize for it. It's accepted and used worldwide by mathematicians, nuclear physicists etc. What this means is that you cannot physically add an infinite number of triangles in a 'circle'. You can keep adding triangles only until each "side" to the circle (microscopic sides) equals one Planck length and this is where Pi physically (or relativistically) ends without question. The reason mathematicians have missed this (I think) is because of the lack of viewing math from a purely artistic (purely geometric) point of view. When you do equations of circles you never see with your own eyes like you do when you look at this drawing. We are visual creatures and like they say, a picture is worth a thousand words, or in this case, a thousand million equations. One picture shows visually what most would never find in an equation alone.
Currently Pi is calculated as a limit. Mathematicians say that as the number of triangles within the 'circle' approach infinity the value of the circumference divided by the diameter approaches the number Pi.
This is mathematically correct but is incorrect when trying to examine real shapes in the real universe. Pi as a limit is describing a flawless circle. But we know that there is no such thing as a perfect circle because of the Planck constant.
Remember that Pi is a shape (not a number). Numbers only describe geometry (always). So when you use the equation f(x)=xsin(Pi/x), each value of x has one output (numerically) and more importantly, each value of x has one output (geometrically). The input for x always equals the number of 'sides' to the circle. So the output as x changes is always describing a specific shape. When you take into account that each smallest piece of the shape is made of whole multiples pf Planck lengths then you can use this equation to describe every shape that actually exists (relativistically). Whereas Pi as a limit describes nothing that is able to be observed by us and is therefore not relative. If something is not relative (cannot be observed by us in any way) then for all purposes it does not exist relative to us.
Finally, a 'circle' is the ultimate fractal. The definition of a fractal is a shape that when decomposed into pieces is the same or similar to the whole. Every piece of a circle decomposes into a triangle...or Pi.