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.
Original entry on oeis.org
1, 2, 16, 23752
Offset: 1
A363505
Number of hyperplanes spanned by the vertices of an n-cube up to symmetry.
Original entry on oeis.org
2, 3, 6, 15, 63, 623, 22432, 3899720
Offset: 2
For n = 2, it can be seen that there are only two non-equivalent hyperplanes spanned by vertices of the square: one spanned by a boundary edge having all remaining points on one side and one spanned by a diagonal separating the remaining points.
For n = 3, we again have a hyperplane parallel to a coordinate plane spanned by a boundary square having all the remaining points on one side; moreover, a hyperplane spanned by the four points on the opposite axis-parallel parallel boundary edges of two opposite boundary squares leaving two remaining points on either side, and a skew hyperplane spanned by the three neighbors of a single point separating that point from the remaining points.
A007847 gives the total numbers (not up to symmetry). Related to
A363506 (and
A363512, resp.) by oriented-matroid duality.
A363506
The number of affine dependencies among the vertices of the n-cube up to symmetry.
Original entry on oeis.org
1, 3, 15, 186, 12628, 3591868, 3858105362
Offset: 2
For n = 2, all vertices of the square constitute the only affine dependence.
For n = 3, there is an affine dependence in each boundary square all of which are equivalent; moreover, there is one affine dependence in each square cutting the cube in half all of which are equivalent; the remaining affine dependence with five elements contains a triangle spanned by all neighbors of a point together with that point and the point opposite to it in the 3-cube.
Cf.
A363512 for the total numbers (not up to symmetry). Related to
A363505 (and
A007847, resp.) by oriented-matroid duality.
A363512
The number of affine dependencies among the vertices of the n-cube.
Original entry on oeis.org
1, 20, 1348, 353616, 446148992, 2118502178496, 38636185528212416
Offset: 2
For n = 2, there is only one affine dependence among the vertices of the square involving all points.
For n = 3, since there are 6 embeddings of the square into the boundary and 6 embeddings of the square into the interior of the 3-cube, there are 12 affine dependences on squares; moreover, there is an affine dependence for each of the 8 vertices of the 3-cube coming from the intersection of the line from that vertex to the vertex opposite in the 3-cube with the triangle spanned by the neighbors of that vertex; this adds up to a total of 20 affine dependencies.
Cf.
A363506 for the same numbers up to symmetry. Related to
A007847 (and
A363505, resp.) by oriented-matroid duality.
A155745
a(n) = number of distinct (n+1)- nonnegative integer vectors describing, up to symmetry, the hyperplanes of the real n-dimensional cube.
Original entry on oeis.org
1, 1, 2, 3, 7, 21, 143
Offset: 1
Ilda P. F. da Silva (isilva(AT)cii.fc.ul.pt), Jan 26 2009
For n=3 a(3)=2 because the 2 vectors (0,0,1,1) and (1,1,1,1) describe all the real planes spanned by the points of {-1,1}^3.
- Ilda P. F. da Silva, Recursivity and geometry of the hypercube, Linear Algebra and its Apllications, 397(2005),223-233
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