A339348 The number of n-faced polyhedra formed when a rhombic dodecahedron is internally cut by all the planes defined by any three of its vertices.

8976, 8976, 3936, 1440, 672

Offset: 4

Author

Scott R. Shannon, Dec 01 2020

Keywords

• nonn
• fini
• full

Comments

For a rhombic dodecahedron create all possible internal planes defined by connecting any three of its vertices. Use all the resulting planes to cut the polyhedron into individual smaller polyhedra. The sequence lists the number of resulting n-faced polyhedra, where 4 <= n <= 8.

See A339349 for the corresponding sequence for the cubooctahedron, the dual polyhedron of the rhombic dodecahedron.

Examples

			The rhombic dodecahedron has 14 vertices, 12 faces, and 24 edges. It is cut by 103 internal planes defined by any three of its vertices, resulting in the creation of 24000 polyhedra. No polyhedra with nine or more faces are created.
		

Links

• Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
• Scott R. Shannon, Image showing the 103 internal plane cuts on the external edges and faces.
• Scott R. Shannon, Image of the 8976 4-faced polyhedra.
• Scott R. Shannon, Image of the 8976 5-faced polyhedra.
• Scott R. Shannon, Image of the 3936 6-faced polyhedra.
• Scott R. Shannon, Image of the 1440 7-faced polyhedra.
• Scott R. Shannon, Image of the 672 8-faced polyhedra.
• Scott R. Shannon, Image of the 672 8-faced polyhedra from directly above a vertex.
• Scott R. Shannon, Image of all 24000 polyhedra. The colors are the same as those used in the above images.
• Eric Weisstein's World of Mathematics, Rhombic Dodecahedron.
• Wikipedia, Rhombic dodecahedron.

Crossrefs

Cf. A339349, A338622, A338801, A338808, A338825.