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.

1, 4, 96, 570048

Offset

1,2

References

N. J. A. Sloane, Cutting Up a Cube, Math Fun Mailing List, Apr 10 2020; with replies from Tom Karzes, Tomas Rokicki, Veit Elser, and others.

Links

Table of n, a(n) for n=1..4.

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.

Example

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

Crossrefs

For the number of hyperplanes see A007847.

Cf. A333540, A338571 (number of pieces for the 3D Platonic solids).

For a more detailed count, see A333543 and A333544.

Sequence in context: A323818 A027638 A309483 * A356808 A041275 A024384

Adjacent sequences: A333536 A333537 A333538 * A333540 A333541 A333542

Keyword

nonn,more

Author

N. J. A. Sloane, Apr 14 2020, in response to a question raised by Scott R. Shannon.

Status

approved