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A338825 Irregular table read by rows: The number of k-faced polyhedra, where k >= 4, created 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.

%I A338825 #20 Jan 09 2021 02:11:54
%S A338825 12,8,120,84,24,448,280,28,368,256,48,32,1332,1440,540,72,1160,1380,
%T A338825 500,220,40,40,2992,5280,2816,748,44,3288,4272,1608,672,192,7176,
%U A338825 14040,8684,3120,624,156,8120,12460,7084,2968,1064,532,84,14820,34020,22620,7560,2580,720,120
%N A338825 Irregular table read by rows: The number of k-faced polyhedra, where k >= 4, created 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 A338825 See A338809 for further details and images for this sequence.
%C A338825 The author thanks _Zach J. Shannon_ for assistance in producing the images for this sequence.
%H A338825 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 A338825 Scott R. Shannon, <a href="/A338825/a338825.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 A338825 Scott R. Shannon, <a href="/A338825/a338825_1.jpg">5-bipyramid, seen from above, showing the 120 polyhedra post-cutting and exploded</a>.
%H A338825 Scott R. Shannon, <a href="/A338825/a338825_2.jpg">20-bipyramid, showing the 69160 4-faced polyhedra</a>.
%H A338825 Scott R. Shannon, <a href="/A338825/a338825_3.jpg">20-bipyramid, showing the 123040 5-faced polyhedra</a>.
%H A338825 Scott R. Shannon, <a href="/A338825/a338825_4.jpg">20-bipyramid, showing the 86240 6-faced polyhedra</a>.
%H A338825 Scott R. Shannon, <a href="/A338825/a338825_5.jpg">20-bipyramid, showing the 46080 7-faced polyhedra</a>.
%H A338825 Scott R. Shannon, <a href="/A338825/a338825_6.jpg">20-bipyramid, showing the 17600 8-faced polyhedra</a>.
%H A338825 Scott R. Shannon, <a href="/A338825/a338825_7.jpg">20-bipyramid, showing the 5920 9-faced polyhedra</a>.
%H A338825 Scott R. Shannon, <a href="/A338825/a338825_8.jpg">20-bipyramid, showing the 320 11-faced polyhedra</a>
%H A338825 Scott R. Shannon, <a href="/A338825/a338825_9.jpg">20-bipyramid, showing the 1920 10-faced and the 320 12-faced polyhedra</a>. These are colored white and black respectively. These are not visible on the surface of the 20-bipyramid.
%H A338825 Scott R. Shannon, <a href="/A338825/a338825_10.jpg">20-bipyramid, showing all 350600 polyhedra</a>.
%H A338825 Scott R. Shannon, <a href="/A338825/a338825.png">20-bipyramid from above, slightly exploded, showing the 69160 4-faced polyhedra</a>.
%H A338825 Scott R. Shannon, <a href="/A338825/a338825_1.png">20-bipyramid from the side, slightly exploded and colored white, showing the 69160 4-faced polyhedra</a>.
%H A338825 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Dipyramid.html">Dipyramid</a>.
%H A338825 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bipyramid">Bipyramid</a>.
%F A338825 Sum of row n = A338809(n).
%e A338825 The 4-bipyramid (an octahedron) is cut with 3 internal planes defined by all 3-vertex combinations of its 6 vertices. This leads to the creation of 8 4-faced polyhedra. See A338622.
%e A338825 The 7-bipyramid is cut with 36 internal planes defined by all 3-vertex combinations of its 9 vertices. This leads to the creation of 448 4-faced polyhedra, 280 5-faced polyhedra, and 28 6-faced polyhedra, 756 polyhedra in all.
%e A338825 The table begins:
%e A338825      12;
%e A338825       8;
%e A338825     120;
%e A338825      84,     24;
%e A338825     448,    280,     28;
%e A338825     368,    256,     48,    32;
%e A338825    1332,   1440,    540,    72;
%e A338825    1160,   1380,    500,   220,    40,   40;
%e A338825    2992,   5280,   2816,   748,    44;
%e A338825    3288,   4272,   1608,   672,   192;
%e A338825    7176,  14040,   8684,  3120,   624,  156;
%e A338825    8120,  12460,   7084,  2968,  1064,  532,   84;
%e A338825   14820,  34020,  22620,  7560,  2580,  720,  120;
%e A338825   18528,  28480,  18560,  9024,  2592, 1024,  384,  64;
%e A338825   32028,  66708,  51136, 22372,  7956, 1836,  136;
%e A338825   35280,  53028,  37080, 14364,  4104,  360,  180, 144;
%e A338825   57380, 131480, 104576, 50616, 17328, 4256,   76;
%e A338825   69160, 123040,  86240, 46080, 17600, 5920, 1920, 320, 320;
%Y A338825 Cf. A338809 (number of polyhedra), A338622 (Platonic solids), A333543 (n-dimensional cube).
%K A338825 nonn,more,tabf
%O A338825 3,1
%A A338825 _Scott R. Shannon_, Nov 11 2020