Is Pi more than just a ratio for circles? Is there a easy way to understand the universality of Pi?
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Although there are plenty of other decent answers, I’d suggest the most fundamental misunderstanding in your question is your assertion that pi is _only_ related to circles.
Pi is deeply related to _cycles_, of which circles are just one of the simplest examples. Cycles appear virtually everywhere both in math and in nature, hence the frequent appearance of pi.
Although there are plenty of other decent answers, I’d suggest the most fundamental misunderstanding in your question is your assertion that pi is _only_ related to circles.
Pi is deeply related to _cycles_, of which circles are just one of the simplest examples. Cycles appear virtually everywhere both in math and in nature, hence the frequent appearance of pi.
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Triangles and circles. The simplest fundamental shapes and you can draw a circle touching all 3 points of any triangle. The circle is a perfect shape. The best way to think about it isn’t that circles are everywhere so that means pi is everywhere. Pi is a perfect fundamental mathematical constant. A circle is a perfect shape created from that ratio.
So the circle is like a sterile environment for us to discover that mathematical constant.
There is actually a movement among many mathematicians that tau or 2(pi) is actually the more fundamental mathematical constant and gives a more intuitive description of the unity of mathematics.
Triangles and circles. The simplest fundamental shapes and you can draw a circle touching all 3 points of any triangle. The circle is a perfect shape. The best way to think about it isn’t that circles are everywhere so that means pi is everywhere. Pi is a perfect fundamental mathematical constant. A circle is a perfect shape created from that ratio.
So the circle is like a sterile environment for us to discover that mathematical constant.
There is actually a movement among many mathematicians that tau or 2(pi) is actually the more fundamental mathematical constant and gives a more intuitive description of the unity of mathematics.
It isn’t quite only related to circles but instead also spheres and higher-dimensional versions of spheres as well as cylinders (and higher-dimensional versions of cylinders). The real trick is that when pi shows up, you can bet that there’s at least one way to reformat your question to ask about circles. 3Blue1Brown has done a few examples of this because of how satisfying to watch they are including [this recent one which actively acknowledges the oddness of pi showing up](https://youtu.be/cy8r7WSuT1I).
Edit: link formatting fix
It isn’t quite only related to circles but instead also spheres and higher-dimensional versions of spheres as well as cylinders (and higher-dimensional versions of cylinders). The real trick is that when pi shows up, you can bet that there’s at least one way to reformat your question to ask about circles. 3Blue1Brown has done a few examples of this because of how satisfying to watch they are including [this recent one which actively acknowledges the oddness of pi showing up](https://youtu.be/cy8r7WSuT1I).
Edit: link formatting fix
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