How many hyperplanes do you get in the 4D case? I believe I see 68. 12 through 8 points (cutting the 4D cube in half), 32 through 6 points, and 24 through 4 points. I'm using a similar approach to yours, so I'm curious why I'm only seeing 18,432 regions for the 4D case. The number I get for the 5D case is 31,610,880. On Sat, Apr 11, 2020 at 9:08 AM Veit Elser <ve10@cornell.edu> wrote:
This series starts 4, 96, … and does not seem to be in OEIS because of a bound I have on the next element (which is being improved as I write this).
Consider an interior point of one of these regions: a d-tuple of real numbers between 0 and 1. If we re-order the d-tuple we get a different region because there are planes that distinguish the order of any consecutive pair of numbers in the d-tuple. We may therefore count the number of regions for a particular ordering and multiply by d!.
Using Mathematica I generate random sorted d-tuples and assign them a binary code giving the side of each of the hyperplanes. Whenever a new code is found it is appended to the codebook of regions. Initially the codebook grows rapidly, and then, mercifully, the “curve flattens”.
Reduced by d! the series starts 2, 16, … Right now the next number is at least 23450, with Mathematica discovering new “cases” at a rate of 21/10^5. Any guesses — insights? — where it will end?
-Veit
On Apr 10, 2020, at 9:59 PM, Neil Sloane <njasloane@gmail.com> wrote:
Take a unit square and cut along the lines joining any two vertices. This cuts the square into 4 pieces.
Now take a unit cube and make plane cuts though any three vertices: how many pieces are produced? (I don't know)
Same question for a unit d-dimensional cube, where the cuts are along hyperplanes through any d vertices.
Best regards Neil
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