A333539
Number of pieces formed when an n-dimensional cube is cut by all the hyperplanes defined by any n of the 2^n vertices.
Original entry on oeis.org
1, 4, 96, 570048
Offset: 1
The two diagonals of a square cut it into four pieces, so a(2) = 4.
For the cube the answer is 96 regions. There are 14 cuts through the cube: six cut the cube in half along a face diagonal, and eight cut off a corner with a triangle through the three adjacent corners. The cuts through the center alone divide the cube into 24 regions, and then the corner cuts further divide each of these into four regions. - _Tomas Rokicki_, Apr 11 2020
- Veit Elser, The values of a(1) - a(4)
- Scott R. Shannon, Image of the 3-dimensional cube showing the 96 pieces. The 4-faced polyhedra are shown in red, the 5-faced polyhedra in yellow. The later form a perfect octahedron inside the cube with its points touching the cube's inner surface. The pieces are moved away from the origin a distance proportional to the average of the distance of all its vertices from the origin.
For the number of hyperplanes see
A007847.
A333543
Irregular triangle read by rows: T(n,k) (n >= 1, k >= n+1) is the number of cells with k vertices in the dissection of an n-dimensional cube by all the hyperplanes that pass through any n vertices.
Original entry on oeis.org
1, 4, 72, 24, 162816, 96576, 118464, 64896, 45888, 22464, 19776, 11904, 8640, 8448, 6144, 1728, 1152, 384, 384, 384
Offset: 1
The two diagonals of a square cut it into four triangular pieces, so T(2,3) = 4.
Triangle begins:
1,
4,
72, 24,
162816, 96576, 118464, 64896, 45888, 22464, 19776, 11904, 8640, 8448, 6144, 1728, 1152, 384, 384, 384,
...
- Veit Elser, Rows 1 through 4
- Scott R. Shannon, Illustration for a(2) = 4.
- Scott R. Shannon, Illustration for a(3) = 72. This shows the 4-faced cells in the 3D cube dissection. The 72 pieces have been moved away from the origin a distance proportional to the average distance of their vertices from the origin.
- Scott R. Shannon, Illustration for a(4) = 24. This shows the 5-faced cells in the 3D cube dissection. The 24 pieces have been moved away from the origin a distance proportional to the average distance of their vertices from the origin. These polyhedra form a perfect octahedron inside the original cube with its points touching the cube's inner surface.
For the number of hyperplanes see
A007847.
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