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A339468
The number of n-faced polyhedra formed when a truncated tetrahedron is internally cut by all the planes defined by any three of its vertices.
1
4818, 4596, 2454, 816, 246, 60, 0, 0, 9
OFFSET
4,1
COMMENTS
For a truncated tetrahedron 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 <= 12.
LINKS
Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
Scott R. Shannon, Image of the 246 8-faced polyhedra. None of these are visible on the surface.
Scott R. Shannon, Image of the 60 9-faced polyhedra. None of these are visible on the surface.
Scott R. Shannon, Image of the 9 12-faced polyhedra. None of these are visible on the surface.
Scott R. Shannon, Image of all 12999 polyhedra. The polyhedra are colored red, orange, yellow, green for face counts 4 to 7 respectively. The polyhedra with 8, 9 and 12 faces are not visible on the surface.
Scott R. Shannon, Image of all 12999 polyhedra, exploded. Each polyhedron has been moved away from the origin by a distance proportional to the average distance of its vertices from the origin. Some of the 8-, 9- and 12-faced polyhedra can now be seen.
Eric Weisstein's World of Mathematics, Truncated Tetrahedron.
EXAMPLE
The truncated tetrahedron has 12 vertices, 18 edges and 4 faces (4 equilateral triangles and 4 hexagons). It is cut by 82 internal planes defined by any three of its vertices, resulting in the creation of 12999 polyhedra. No polyhedra with 10, 11, or 13 or more faces are created.
KEYWORD
nonn,fini,full
AUTHOR
Scott R. Shannon, Dec 08 2020
STATUS
approved