Is Pi more than just a ratio for circles? Is there a easy way to understand the universality of Pi?
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If you wanna describe the universe around you, you need to build up from the simple things to the more complicated.
Start with points, then connect them with lines. You can measure lines to get lengths and compare them to get Ratios, Fractions, etc.
When you go to 2D shapes, triangles can be used to help describe any polygon, which is why trigonometry is so important (literally Triangle Measure).
But since circles aren’t a polygon (any shape with straight sides), you need to describe them differently. Whilst the most important feature of a circle is its radius, it’s easier to measure a circle’s diameter by hand. Because of this, historically, mathematicians have typically used a circle’s diameter as a reference point.
When this diameter is compared to the circumference (also easily measured physically), you always find the circumference as being 3.14159… times bigger than the diameter. With such an important ratio being so important to circles, they instead called it Pi for accuracy.
Bringing this full circle (pun very much intended) to your original question, “Why does Pi keep popping up in maths?” This model of the universe we’re building up starts with lines that give us basic numeracy, then build to Triangles and Circles. As you go further in maths, you keep using your previous work so as to keep it consistent, and so circles and Pi end up being used all the time, even if not directly in a circle.
TL;DR – Points are basic, Lines are useful, Triangles and Circles are extremely useful. You need Pi to describe Circles.
Source: A very nerdy maths teacher (me)
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