A338809 - OEIS

%I #23 Dec 07 2020 01:45:39

%S 12,8,120,108,756,704,3384,3340,11880,10032,33800,32312,82440,78656,

%T 182172,144540,365712,350600

%N 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.

%C 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.

%C See A338825 for the number and images of the k-faced polyhedra in each bipyramid dissection.

%C The author thanks _Zach J. Shannon_ for assistance in producing the images for this sequence.

%H Hyung Taek Ahn and Mikhail Shashkov, <a href="https://cnls.lanl.gov/~shashkov/papers/ahn_geometry.pdf">Geometric Algorithms for 3D Interface Reconstruction</a>.

%H Scott R. Shannon, <a href="/A338809/a338809.png">5-bipyramid, showing the 16 plane cuts on the external edges and faces</a>.

%H Scott R. Shannon, <a href="/A338809/a338809.jpg">5-bipyramid showing the 120 polyhedra post-cutting and exploded</a>. 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.

%H Scott R. Shannon, <a href="/A338809/a338809_1.png">12-bipyramid, showing the 103 plane cuts on the external edges and faces</a>.

%H Scott R. Shannon, <a href="/A338809/a338809_1.jpg">12-bipyramid, showing the 10032 polyhedra post-cutting</a>. The 4,5,6,7 faced polyhedra are colored red, orange, yellow, green respectively. The 8-faced polyhedra are not visible on the surface.

%H Scott R. Shannon, <a href="/A338809/a338809_2.jpg">12-bipyramid, showing the 10032 polyhedra post-cutting and exploded</a>.The 8-faced polyhedra colored blue can be seen.

%H Scott R. Shannon, <a href="/A338809/a338809_2.png">20-bipyramid, showing the 331 plane cuts on the external edges and faces</a>.

%H Scott R. Shannon, <a href="/A338809/a338809_3.jpg">20-bipyramid, showing the 350600 polyhedra post-cutting</a>. 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.

%H Scott R. Shannon, <a href="/A338809/a338809_4.jpg">20-bipyramid positions vertically, showing the 350600 polyhedra post-cutting</a>.

%H Scott R. Shannon, <a href="/A338809/a338809_5.jpg">20-bipyramid, showing the 350600 polyhedra post-cutting and exploded</a>. The 10-faced and 12-faced polyhedra, colored black and white, can also be seen.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Dipyramid.html">Dipyramid</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bipyramid">Bipyramid</a>.

%e a(3) = 12. The 3-bipyramid is cut with 4 internal planes resulting in 12 polyhedra, all 12 pieces having 4 faces.

%e a(5) = 120. The 5-bipyramid is cut with 16 internal planes resulting in 120 polyhedra, all 120 pieces having 4 faces.

%e 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.

%e 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.

%Y Cf. A338825 (number of k-faced polyhedra), A338571 (Platonic solids), A333539 (n-dimensional cube), A007678 (2D n-polygon).

%K nonn,more

%O 3,1

%A _Scott R. Shannon_, Nov 10 2020