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

Author

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

Keywords

• nonn
• more

Examples

			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
		

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

• 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.

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.