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I'm trying to run a simulation that shows higher dimensional objects collapsing and expanding. I've run into a slight difficulty with some solid and hypersolid angles. I'm aware that the angle will obey the basic complementary and suplementary laws for n-dimensions, but when a vertex is lined up with multiple vertexes (just 2d vertexes) then I know that said vertexes have to be members of n-gons (where n is prime) in order for a reflection to take place without any other adjacent vertexes passing through each other. It works if it's a solid and it has a prime amount of faces or a hypersolid with a prime amount of hyperfaces. If any of the vertexes are not a member of any n-gon such that n is prime, it doesn't immediately disqualify the possibility of a pass-through as long as there are at least two vertexes that AREN'T adjacent that ARE members of n-gons where n is prime. This is because you take the sum of the n's (or any partial sum) not just each individual one. Now I know that the sum of the n's is always even unless there is a hole in the object. (There won't ever be because I'm just dealing with closed objects with no gaps or self-intersections.) The method I'm using depends on all this working out this way. If it isn't always the case that we're going to find at least two vertexes that belong to prime-gons that have the even sum, then I'll have to find a new brute-force way of doing the rotations. The angles won't inflect properly without whole faces or solids just passing through each other. I've done a brute-force check and verified that this is the case for several million values (starting at 4 and incrementing by 2) we always can find two vertexes that belong to at least one (usually more) n-gons and that for each of those, n is prime. Is there a proof or a way to verify that it will hold for any situation?

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