Is Pi more than just a ratio for circles? Is there a easy way to understand the universality of Pi?
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Pi is related to circles. A circle of radius 1 (called the unit circle) is the set of all points with a distance of 1 to the designated middle point. You may think this is an odd characterisation of a circle, because it doesnt mention a circles “roundness”. But keep in mind that this characterisation still makes sense: its basically why a compass draws a circle. It encapsulates the circles perfect rotational symmetry. This has an important algebraic implication:
*If you traverse the circle, both the x and the y coordinate changes constantly. However, despite that, the distance stays the same.*
This is very important because it allows you to reduce the number of moving parts from 2 to 1 (instead of studying x and y coordinates, you only need to study the angle). Mathematicians LOVE to do that. So anytime a complicated problem has some semblance to a circle, you can try to “find the circle” in it. If you are successful, you are rewarded with a reduction of variable parameters.
One could also say that such problems are “rotationally symmetric” in some way. In math lingo, we would “switch to polar coordinates”: characterising a point by its angle and distance to the origin, instead of an x and y coordinate. If the distance stays constant, you have eliminated 1 variable.
Notice: the intuition obviously is specifically geometric, but reducing the number of variables is a very general and very very useful problem solving tool. Thats why we try to use it whenever we can.
There are many other reasons why circles (and therefore appearances of pi) are ubiquitous in math, but I wanted to give details on one specific reason.
Pi is related to circles. A circle of radius 1 (called the unit circle) is the set of all points with a distance of 1 to the designated middle point. You may think this is an odd characterisation of a circle, because it doesnt mention a circles “roundness”. But keep in mind that this characterisation still makes sense: its basically why a compass draws a circle. It encapsulates the circles perfect rotational symmetry. This has an important algebraic implication:
*If you traverse the circle, both the x and the y coordinate changes constantly. However, despite that, the distance stays the same.*
This is very important because it allows you to reduce the number of moving parts from 2 to 1 (instead of studying x and y coordinates, you only need to study the angle). Mathematicians LOVE to do that. So anytime a complicated problem has some semblance to a circle, you can try to “find the circle” in it. If you are successful, you are rewarded with a reduction of variable parameters.
One could also say that such problems are “rotationally symmetric” in some way. In math lingo, we would “switch to polar coordinates”: characterising a point by its angle and distance to the origin, instead of an x and y coordinate. If the distance stays constant, you have eliminated 1 variable.
Notice: the intuition obviously is specifically geometric, but reducing the number of variables is a very general and very very useful problem solving tool. Thats why we try to use it whenever we can.
There are many other reasons why circles (and therefore appearances of pi) are ubiquitous in math, but I wanted to give details on one specific reason.
Not just circles, curves in general. And like, even things with straight lines have angles and the angles are messured by little curves
Not just circles, curves in general. And like, even things with straight lines have angles and the angles are messured by little curves
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I read to quickly and thought “divorce” instead of diverse. Had to peek at the comments before I realized. I was like damn who’s the third person? The wife’s new boyfriend?
I read to quickly and thought “divorce” instead of diverse. Had to peek at the comments before I realized. I was like damn who’s the third person? The wife’s new boyfriend?
You might as well ask, “Why do I see the most-efficient shape everywhere?”
You might as well ask, “Why do I see the most-efficient shape everywhere?”
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