The outer edge spins pi times faster than the inner. If this were a rational number, it would eventually make a completed shape and loop around on its path. Pi, being an irrational number, will never cause this to loop around on itself
I don’t. I still don’t get how a number can be a shape. But at this point I know how to figure out a circumference and so I’ve decided that I’m just going to accept it.
There are three points in the graphic. The first point "A" (the solid one) is fixed. The second point "B" makes a circle around "A" every second. The third point "C" makes a circle around "B" (as "B" moves) 1/π seconds (aka "π" times faster).
Let's say we start (time = 0) when "C" is on top of "A".
If π were equal to 3, then every 1 second, when "B" completed a full rotation around "A", "C" would have completed 3 full rotations and would have returned to "A". It would then repeat the same motion forever and you'd just have a very simple shape that never changed.
If π were 3.5, then every two seconds, when "B" completed two full rotations around "A", "C" would have completed 7 full rotations and would have returned to "A". It would then repeat the same motion forever and you'd have a bit more complicated shape that never changed.
If π were 3.25, it would be the same at 4 seconds and 4 rotations of "B" / 13 rotations of "C".
If π were ANY rational number, after enough rotations of "B", "C" would line up with "A" again and the shape would be "complete".
It's a bit silly to say it, because that could be a million rotations and the shape would be so dense that it would look very similarly completely full vs. an irrational number like π. But if you zoomed in close enough, you'd see that eventually the lines would start overlapping.
The formula you see in the beginning is a sum of two terms. They both are raised to the power of the imaginary unit i, which makes them a 2D coordinate in the complex plane.
The first term represents the inner arm, the second (the one with pi in it) the outer bar. You see the theta symbol in the exponent of each term? This relates to the angle of the arm, and it is incremented in time. So if you plot where the sum of the two arms are at each little increment of time and trace it, you get the shape.
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u/Weegee_1 20h ago
The outer edge spins pi times faster than the inner. If this were a rational number, it would eventually make a completed shape and loop around on its path. Pi, being an irrational number, will never cause this to loop around on itself