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A338809
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Number of polyhedra formed when an n-bipyramid, formed from two n-gonal pyraminds joined at the base, is internally cut by all the planes defined by any three of its vertices.
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3
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12, 8, 120, 108, 756, 704, 3384, 3340, 11880, 10032, 33800, 32312, 82440, 78656, 182172, 144540, 365712, 350600
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OFFSET
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3,1
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COMMENTS
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For a n-bipyramid, formed from two n-gonal 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 3-bipyramid this results in 4 planes. Use all the resulting planes to cut the n-bipyramid 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 k-faced polyhedra in each bipyramid dissection.
The author thanks Zach J. Shannon for assistance in producing the images for this sequence.
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LINKS
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Scott R. Shannon, 20-bipyramid, showing the 350600 polyhedra post-cutting. 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.
Eric Weisstein's World of Mathematics, Dipyramid.
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EXAMPLE
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a(3) = 12. The 3-bipyramid is cut with 4 internal planes resulting in 12 polyhedra, all 12 pieces having 4 faces.
a(5) = 120. The 5-bipyramid is cut with 16 internal planes resulting in 120 polyhedra, all 120 pieces having 4 faces.
a(7) = 756. The 7-bipyramid 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 n-pyramid the number of polyhedra is the same as the number of regions in the dissection of a 2D n-polygon, see A007678, as all planes join two points on the polygon and the single apex, resulting in an equivalent number of regions.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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