A338806 -id:A338806 - cp's OEIS frontend

A338808 Irregular table read by rows: The number of k-faced polyhedra, where k>=4, created when an n-antiprism, formed from two n-sided regular polygons joined by 2n adjacent alternating triangles, is internally cut by all the planes defined by any three of its vertices.

8, 128, 56, 8, 0, 3, 450, 270, 82, 20, 10, 0, 2, 2592, 2376, 972, 204, 168, 48, 0, 0, 5, 7266, 7574, 4550, 2254, 660, 336, 98, 14, 14, 0, 2, 0, 0, 0, 0, 0, 0, 2, 27216, 31088, 15632, 5360, 1904, 432, 128, 0, 0, 0, 0, 0, 9, 68778, 84240, 61272, 33138, 15714, 5400, 1946, 720, 270, 126, 72, 18, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4

Offset: 3

Scott R. Shannon, Nov 10 2020

See A338806 for further details and images for this sequence.

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

			The 4-antiprism is cut with 16 internal planes defined by all 3-vertex combinations of its 8 vertices. This leads to the creation of 128 4-faced polyhedra, 56 5-faced polyhedra, 8 6-faced polyhedra, and 3 8-faced polyhedra, 195 pieces in all. Note the number of 8-faced polyhedra is not a multiple of 4 - they lie directly along the z-axis so need not be a multiple of the number of edges forming the regular n-gons.
The table begins:
8;
128,56,8,0,3;
450,270,82,20,10,0,2;
2592,2376,972,204,168,48,0,0,5;
7266,7574,4550,2254,660,336,98,14,14,0,2,0,0,0,0,0,0,2;
27216,31088,15632,5360,1904,432,128,0,0,0,0,0,9;
68778,84240,61272,33138,15714,5400,1946,720,270,126,72,18,0,0,4,0,0,0,0,0,0,0,0,4;
194580,235880,153620,68580,25240,7460,2560,660,200,0,0,0,0,0,0,0,13;

Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
Scott R. Shannon, 4-antiprism, showing the 56 5-faced polyhedra. See A338806 for an image of the full polyhedra.
Scott R. Shannon, 4-antiprism, showing the 8 6-faced polyhedra
Scott R. Shannon, 4-antiprism, showing the 3 8-faced polyhedra
Scott R. Shannon, 7-antiprism, showing the 7266 4-faced polyhedra. See A338806 for an image of the full polyhedra.
Scott R. Shannon, 7-antiprism, showing the 7574 5-faced polyhedra
Scott R. Shannon, 7-antiprism, showing the 4550 6-faced polyhedra
Scott R. Shannon, 7-antiprism, showing the 2254 7-faced polyhedra
Scott R. Shannon, 7-antiprism, showing the 660 8-faced polyhedra
Scott R. Shannon, 7-antiprism, showing the 336 9-faced polyhedra.
Scott R. Shannon, 7-antiprism, showing the 98 10-faced polyhedra. None of these are visible on the surface.
Scott R. Shannon, 7-antiprism, showing the 14 11-faced, 14 12-faced, 2 14-faced, 2 21-faced polyhedra. These are colored white, black, red, yellow respectively. None of these are visible on the surface.
Scott R. Shannon, 10-antiprism, showing the 13 20-faced polyhedra. See A338806 for an image of the full polyhedra.
Eric Weisstein's World of Mathematics, Antiprism.
Wikipedia, Antiprism.

Cf. A338806 (number of polyhedra), A338801 (regular prism), A338622 (Platonic solids), A333543 (n-dimensional cube).

Sum of row n = A338806(n).

A338783 Number of polyhedra formed when an n-prism, formed from two n-sided regular polygons joined by n adjacent rectangles, is internally cut by all the planes defined by any three of its vertices.

Original entry on oeis.org

18, 96, 1335, 4524, 29871, 65344, 319864, 594560

Offset: 3

Scott R. Shannon, Nov 08 2020

For an n-prism, formed from two n-sided regular polygons joined by n adjacent rectangles, create all possible internal planes defined by connecting any three of its vertices. For example, in the case of a triangular prism this results in 6 planes. Use all the resulting planes to cut the prism into individual smaller polyhedra. The sequence lists the number of resulting polyhedra for prisms with n>=3.
See A338801 for the number and images of the k-faced polyhedra in each prism dissection.
The author thanks Zach J. Shannon for assistance in producing the images for this sequence.

			a(3) = 18. The triangular 3-prism has 6 internal cutting planes resulting in 18 polyhedra; seventeen 4-faced polyhedra and one 6-faced polyhedron.
a(4) = 96. The square 4-prism (a cuboid) has 14 internal cutting planes resulting in 96 polyhedra; seventy-two 4-faced polyhedra and twenty-four 5-faced polyhedra. See A338622.

Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
Scott R. Shannon, 3-prism, showing the 6 plane cuts on the external edges and faces.
Scott R. Shannon, 3-prism, showing the 18 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. Red shows the 4-faced polyhedra, orange the single 6-faced polyhedron.
Scott R. Shannon, 7-prism, showing the 98 plane cuts on the external edges and faces.
Scott R. Shannon, 7-prism, showing the 29871 polyhedra post-cutting. The 4,5,6,7,8,9,10 faced polyhedra are colored red, orange, yellow, green, blue, indigo, violet respectively. The polyhedra with 11,12,13,14 faces are not visible on the surface.
Scott R. Shannon, 7-prism, showing the 29871 polyhedra post-cutting and exploded.
Scott R. Shannon, 10-prism, showing the 275 plane cuts on the external edges and faces
Scott R. Shannon, 10-prism, showing the 594560 polyhedra post-cutting. The 4,5,6,7,8,9 faced polyhedra are colored red, orange, yellow, green, blue, indigo respectively. The polyhedra with 10,11,12,13 faces are not visible on the surface.
Eric Weisstein's World of Mathematics, Prism.
Wikipedia, Prism (geometry).

Cf. A338801 (number of k-faced polyhedra), A338806 (antiprism), A338571 (Platonic solids), A338622 (k-faced polyhedra in Platonic solids), A333539 (n-dimensional cube).

cp's OEIS Frontend

A338808 Irregular table read by rows: The number of k-faced polyhedra, where k>=4, created when an n-antiprism, formed from two n-sided regular polygons joined by 2n adjacent alternating triangles, is internally cut by all the planes defined by any three of its vertices.

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