This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A338809 #23 Dec 07 2020 01:45:39 %S A338809 12,8,120,108,756,704,3384,3340,11880,10032,33800,32312,82440,78656, %T A338809 182172,144540,365712,350600 %N A338809 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 A338809 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 A338809 See A338825 for the number and images of the k-faced polyhedra in each bipyramid dissection. %C A338809 The author thanks _Zach J. Shannon_ for assistance in producing the images for this sequence. %H A338809 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 A338809 Scott R. Shannon, <a href="/A338809/a338809.png">5-bipyramid, showing the 16 plane cuts on the external edges and faces</a>. %H A338809 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 A338809 Scott R. Shannon, <a href="/A338809/a338809_1.png">12-bipyramid, showing the 103 plane cuts on the external edges and faces</a>. %H A338809 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 A338809 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 A338809 Scott R. Shannon, <a href="/A338809/a338809_2.png">20-bipyramid, showing the 331 plane cuts on the external edges and faces</a>. %H A338809 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 A338809 Scott R. Shannon, <a href="/A338809/a338809_4.jpg">20-bipyramid positions vertically, showing the 350600 polyhedra post-cutting</a>. %H A338809 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 A338809 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Dipyramid.html">Dipyramid</a>. %H A338809 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bipyramid">Bipyramid</a>. %e A338809 a(3) = 12. The 3-bipyramid is cut with 4 internal planes resulting in 12 polyhedra, all 12 pieces having 4 faces. %e A338809 a(5) = 120. The 5-bipyramid is cut with 16 internal planes resulting in 120 polyhedra, all 120 pieces having 4 faces. %e A338809 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 A338809 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 A338809 Cf. A338825 (number of k-faced polyhedra), A338571 (Platonic solids), A333539 (n-dimensional cube), A007678 (2D n-polygon). %K A338809 nonn,more %O A338809 3,1 %A A338809 _Scott R. Shannon_, Nov 10 2020