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Its easy enough to see poles, but if you have 2 poles, those points are equidistant along one dimension, but the other two are missing (im trying not to get off topic by imagining the two points as projecting a north and south hemisphere), while 3 points creates a plane triangle, which like the 2 pole object occupies infinite superpositions along one dimension of rotation.

But, we get to tetrahedron, octohedron, cube, dodecahedron, etc., and we have stable equidistant points relative to the sphere surface.

But what about an object with five, six, or seven vertices? Are such solids even possible for equidistant values on the surface of a sphere? Would such objects have to necessarily have superpositioned vectors approximating the space where they might be?