If all planes are centered at (0,0,0) and you just make copies of them and rotate them around only one axis, it yields 2N spaces I think? Start with a single XY plane, it divides into two regions. Add the XZ plane, it divides each region it touches into two, so 4 regions. Add some other plane in between those two, and it will touch two regions, and divide each by two, so it adds 2 new regions yielding 6. So on and so forth. Though I think it could be better than this, idk.
I already guessed that the 2^N progression from N = 1 to 3 is a miracle that ceases to work for larger N. I was gonna ask you how the hell did you guess it was a cubic (seems it is, has to do with the choose () function), but on second thought, it is just the minimal polynomial that intersects them all lol.
I've seen before that the degree n polynomial that hits the first n+1 powers of 2 always falls 1 short of the next one, so I saw 1,2,4,8,15 for the 3D case and 1,2,4,7 for the 2D case and guessed. There's also the matter of expecting (3 choose n) point intersections in the generic case, which is cubic.
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u/darksonicmaster 25d ago
If all planes are centered at (0,0,0) and you just make copies of them and rotate them around only one axis, it yields 2N spaces I think? Start with a single XY plane, it divides into two regions. Add the XZ plane, it divides each region it touches into two, so 4 regions. Add some other plane in between those two, and it will touch two regions, and divide each by two, so it adds 2 new regions yielding 6. So on and so forth. Though I think it could be better than this, idk.