@Approximation said in #15:
> Circles, with radius r, should have the same distance to each "side" (or point on the circumference of the circle from the center) - thats what makes a Circle.
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> The problem with any regular polygon is that the shape of a regular polygon side is different than the shape of a circular arc connecting the points on the polygon, whether it has 3 sides, 4 sides, 8 sides, 100 sides, even as the number of sides approach infinity.
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> Therefore, an infinite sided polygon isn't the same shape as a circle because of the infinite straight line segment shortcuts between two points close together on it, which is different than a circular arc connecting the same points on the circle.
Yes, but a curve and therefore a circle is a theoretical geometric construction, as the fundamental attribute of a curve is having an infinite number of sides (or having no sides at all), which is not possible in the real world. At a granular level its a bunch of geometry mashed together that gives the illusion of a curve.