

A338809


Number of polyhedra formed when an nbipyramid, formed from two ngonal pyraminds joined at the base, is internally cut by all the planes defined by any three of its vertices.


3



12, 8, 120, 108, 756, 704, 3384, 3340, 11880, 10032, 33800, 32312, 82440, 78656, 182172, 144540, 365712, 350600
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OFFSET

3,1


COMMENTS

For a nbipyramid, formed from two ngonal pyraminds joined at the base, create all possible internal planes defined by connecting any three of its vertices. For example, in the case of a 3bipyramid this results in 4 planes. Use all the resulting planes to cut the nbipyramid into individual smaller polyhedra. The sequence lists the number of resulting polyhedra for bipyramids with n>=3.
See A338825 for the number and images of the kfaced polyhedra in each bipyramid dissection.
The author thanks Zach J. Shannon for assistance in producing the images for this sequence.


LINKS

Table of n, a(n) for n=3..20.
Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
Scott R. Shannon, 5bipyramid, showing the 16 plane cuts on the external edges and faces.
Scott R. Shannon, 5bipyramid showing the 120 polyhedra postcutting and exploded. Each piece has been moved away from the origin by a distance proportional to the average distance of its vertices from the origin. All 120 polyhedra have 4 faces, shown in red.
Scott R. Shannon, 12bipyramid, showing the 103 plane cuts on the external edges and faces.
Scott R. Shannon, 12bipyramid, showing the 10032 polyhedra postcutting. The 4,5,6,7 faced polyhedra are colored red, orange, yellow, green respectively. The 8faced polyhedra are not visible on the surface.
Scott R. Shannon, 12bipyramid, showing the 10032 polyhedra postcutting and exploded.The 8faced polyhedra colored blue can be seen.
Scott R. Shannon, 20bipyramid, showing the 331 plane cuts on the external edges and faces.
Scott R. Shannon, 20bipyramid, showing the 350600 polyhedra postcutting. The 4,5,6,7,8,9,11 faced polyhedra are colored red, orange, yellow, green, blue, indigo, violet respectively. The polyhedra with 10 and 12 faces are not visible on the surface.
Scott R. Shannon, 20bipyramid positions vertically, showing the 350600 polyhedra postcutting.
Scott R. Shannon, 20bipyramid, showing the 350600 polyhedra postcutting and exploded. The 10faced and 12faced polyhedra, colored black and white, can also be seen.
Eric Weisstein's World of Mathematics, Dipyramid.
Wikipedia, Bipyramid.


EXAMPLE

a(3) = 12. The 3bipyramid is cut with 4 internal planes resulting in 12 polyhedra, all 12 pieces having 4 faces.
a(5) = 120. The 5bipyramid is cut with 16 internal planes resulting in 120 polyhedra, all 120 pieces having 4 faces.
a(7) = 756. The 7bipyramid is cut with 36 internal planes resulting in 756 polyhedra; 448 with 4 faces, 280 with 5 faces, and 28 with 6 faces.
Note that for a single npyramid the number of polyhedra is the same as the number of regions in the dissection of a 2D npolygon, see A007678, as all planes join two points on the polygon and the single apex, resulting in an equivalent number of regions.


CROSSREFS

Cf. A338825 (number of kfaced polyhedra), A338571 (Platonic solids), A333539 (ndimensional cube), A007678 (2D npolygon).
Sequence in context: A121961 A168386 A338825 * A038334 A101501 A299515
Adjacent sequences: A338806 A338807 A338808 * A338810 A338811 A338812


KEYWORD

nonn,more


AUTHOR

Scott R. Shannon, Nov 10 2020


STATUS

approved



