A338622 Irregular table read by rows: The number of k-faced polyhedra, where k>=4, formed when the five Platonic solids, in the order tetrahedron, octahedron, cube, icosahedron, dodecahedron, are internally cut by all the planes defined by any three of their vertices.

1, 8, 72, 24, 2160, 360, 205320, 208680, 94800, 34200, 7920, 1560, 120

Offset: 1

Author

Scott R. Shannon, Nov 04 2020

Keywords

• nonn
• fini
• full
• tabf

Comments

See A338571 for further details and images of this sequence.

The author thanks Zach J. Shannon for producing the images for this sequence.

Examples

			The cube is cut with 14 internal planes defined by all 3-vertex combinations of its 8 vertices. This leads to the creation of 72 4-faced polyhedra and 24 5-faced polyhedra, 96 pieces in all. See A338571 and A333539.
The table is:
1;
8;
72, 24;
2160, 360;
205320, 208680, 94800, 34200, 7920, 1560, 120;
		

Links

• Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
• Polyhedra.mathmos.net, The Platonic Solids.
• Scott R. Shannon, Tetrahedron, showing the 1 4-faced polyhedra post-cutting. This is the original tetrahedron itself as no internal cutting planes are present.
• Scott R. Shannon, Octahedron, showing the 8 4-faced polyhedra post-cutting. The octahedron has 3 internal cutting planes, each along the 2D axial planes. For clarity in this image, and the two cube images, the pieces are moved away from the origin a distance proportional to the average distance of their vertices from the origin.
• Scott R. Shannon, Cube, showing the 72 4-faced polyhedra post-cutting. The cube has 14 internal cutting planes.
• Scott R. Shannon, Cube, showing the 24 5-faced polyhedra post-cutting. These form a perfect octahedron inside the original cube.
• Scott R. Shannon, Icosahedron, showing the 2160 4-faced polyhedra post-cutting. The icosahedronhas 47 internal cutting planes.
• Scott R. Shannon, Icosahedron, showing the 360 5-faced polyhedra post-cutting.
• Scott R. Shannon, Icosahedron, showing all 2520 polyhedra post-cutting.
• Scott R. Shannon, Dodecahedron, showing the 205320 4-faced polyhedra post-cutting. The dodecahedron has 307 internal cutting planes.
• Scott R. Shannon, Dodecahedron, showing the 208680 5-faced polyhedra post-cutting.
• Scott R. Shannon, Dodecahedron, showing the 94800 6-faced polyhedra post-cutting.
• Scott R. Shannon, Dodecahedron, showing the 34200 7-faced polyhedra post-cutting.
• Scott R. Shannon, Dodecahedron, showing the 7920 8-faced polyhedra post-cutting.
• Scott R. Shannon, Dodecahedron, showing the 1560 9-faced polyhedra post-cutting. None of these polyhedra are visible on the surface of the original dodecahedron.
• Scott R. Shannon, Dodecahedron, showing the 120 10-faced polyhedra post-cutting. None of these polyhedra are visible on the surface of the original dodecahedron.
• Scott R. Shannon, Dodecahedron, showing a combination of the 4-faced and 5-faced polyhedra post-cutting. These two types make up about 75% of all the pieces.
• Scott R. Shannon, Dodecahedron, showing all 552600 polyhedra post-cutting. No 9-faced or 10-faced polyhedra are visible on the surface.
• Eric Weisstein's World of Mathematics, Platonic Solid.
• Wikipedia, Platonic solid.

Crossrefs

Cf. A338571 (total number of polyhedra), A333539 (n-dimensional cube), A053016, A063722, A063723, A098427, A333543.

Formula

Sum of row n = A338571(n).