The circle is a beautiful figure. What is it about the circle that makes this so. From a geometers perspective we examine one of the geometries that mathematicians have invented as they generalize Euclid’s. Perhaps first is Banach’s geometry which foregoes the Pythagorean theorem but maintains the triangle inequality (that the length of an edge of a triangle is no more than the sum of the other two lengths). In that geometry there are not in general rotations that are isomorphisms—no isometries; a wheel can’t turn there. The circle of one Banach space is a square! In general 2D Banach space the circle is merely any centrally symmetric convex shape. Aside from nature liking symmetries, this would be an awkward universe to live in. That there is a physically natural motion that turns some rocks into something that leaves them looking they did before and is strange and unusual of those rocks; they are circular. It is also unusual about our geometries although can we imagine the primitive man sensing that? After all the Pythagorean theorem is necessary to really explain rotation. Rotation seems a bit magical to me. Nature has seldom harnessed it.
The sphere is one step more magical.