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.

12, 8, 120, 84, 24, 448, 280, 28, 368, 256, 48, 32, 1332, 1440, 540, 72, 1160, 1380, 500, 220, 40, 40, 2992, 5280, 2816, 748, 44, 3288, 4272, 1608, 672, 192, 7176, 14040, 8684, 3120, 624, 156, 8120, 12460, 7084, 2968, 1064, 532, 84, 14820, 34020, 22620, 7560, 2580, 720, 120

Offset

3,1

Comments

See A338809 for further details and images for this sequence.

The author thanks Zach J. Shannon for assistance in producing the images for this sequence.

Links

Table of n, a(n) for n=3..54.

Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.

Scott R. Shannon, 5-bipyramid showing the 120 polyhedra post-cutting 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, 5-bipyramid, seen from above, showing the 120 polyhedra post-cutting and exploded.

Scott R. Shannon, 20-bipyramid, showing the 69160 4-faced polyhedra.

Scott R. Shannon, 20-bipyramid, showing the 123040 5-faced polyhedra.

Scott R. Shannon, 20-bipyramid, showing the 86240 6-faced polyhedra.

Scott R. Shannon, 20-bipyramid, showing the 46080 7-faced polyhedra.

Scott R. Shannon, 20-bipyramid, showing the 17600 8-faced polyhedra.

Scott R. Shannon, 20-bipyramid, showing the 5920 9-faced polyhedra.

Scott R. Shannon, 20-bipyramid, showing the 320 11-faced polyhedra

Scott R. Shannon, 20-bipyramid, showing the 1920 10-faced and the 320 12-faced polyhedra. These are colored white and black respectively. These are not visible on the surface of the 20-bipyramid.

Scott R. Shannon, 20-bipyramid, showing all 350600 polyhedra.

Scott R. Shannon, 20-bipyramid from above, slightly exploded, showing the 69160 4-faced polyhedra.

Scott R. Shannon, 20-bipyramid from the side, slightly exploded and colored white, showing the 69160 4-faced polyhedra.

Eric Weisstein's World of Mathematics, Dipyramid.

Wikipedia, Bipyramid.

Formula

Sum of row n = A338809(n).

Example

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.
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.
The table begins:
     12;
      8;
    120;
     84,     24;
    448,    280,     28;
    368,    256,     48,    32;
   1332,   1440,    540,    72;
   1160,   1380,    500,   220,    40,   40;
   2992,   5280,   2816,   748,    44;
   3288,   4272,   1608,   672,   192;
   7176,  14040,   8684,  3120,   624,  156;
   8120,  12460,   7084,  2968,  1064,  532,   84;
  14820,  34020,  22620,  7560,  2580,  720,  120;
  18528,  28480,  18560,  9024,  2592, 1024,  384,  64;
  32028,  66708,  51136, 22372,  7956, 1836,  136;
  35280,  53028,  37080, 14364,  4104,  360,  180, 144;
  57380, 131480, 104576, 50616, 17328, 4256,   76;
  69160, 123040,  86240, 46080, 17600, 5920, 1920, 320, 320;

Crossrefs

Cf. A338809 (number of polyhedra), A338622 (Platonic solids), A333543 (n-dimensional cube).

Sequence in context: A164675 A121961 A168386 * A338809 A038334 A101501

Adjacent sequences: A338822 A338823 A338824 * A338826 A338827 A338828

Keyword

nonn,more,tabf

Author

Scott R. Shannon, Nov 11 2020

Status

approved