A338801 -id:A338801 - 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).

A339348 The number of n-faced polyhedra formed when a rhombic dodecahedron is internally cut by all the planes defined by any three of its vertices.

Original entry on oeis.org

8976, 8976, 3936, 1440, 672

Offset: 4

Scott R. Shannon, Dec 01 2020

For a rhombic dodecahedron create all possible internal planes defined by connecting any three of its vertices. Use all the resulting planes to cut the polyhedron into individual smaller polyhedra. The sequence lists the number of resulting n-faced polyhedra, where 4 <= n <= 8.

See A339349 for the corresponding sequence for the cubooctahedron, the dual polyhedron of the rhombic dodecahedron.

			The rhombic dodecahedron has 14 vertices, 12 faces, and 24 edges. It is cut by 103 internal planes defined by any three of its vertices, resulting in the creation of 24000 polyhedra. No polyhedra with nine or more faces are created.

Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
Scott R. Shannon, Image showing the 103 internal plane cuts on the external edges and faces.
Scott R. Shannon, Image of the 8976 4-faced polyhedra.
Scott R. Shannon, Image of the 8976 5-faced polyhedra.
Scott R. Shannon, Image of the 3936 6-faced polyhedra.
Scott R. Shannon, Image of the 1440 7-faced polyhedra.
Scott R. Shannon, Image of the 672 8-faced polyhedra.
Scott R. Shannon, Image of the 672 8-faced polyhedra from directly above a vertex.
Scott R. Shannon, Image of all 24000 polyhedra. The colors are the same as those used in the above images.
Eric Weisstein's World of Mathematics, Rhombic Dodecahedron.
Wikipedia, Rhombic dodecahedron.

Cf. A339349, A338622, A338801, A338808, A338825.

A339349 The number of n-faced polyhedra formed when a cuboctahedron is internally cut by all the planes defined by any three of its vertices.

Original entry on oeis.org

2304, 3000, 944, 408, 48, 24

Offset: 4

Scott R. Shannon, Dec 01 2020

For a cuboctahedron create all possible internal planes defined by connecting any three of its vertices. Use all the resulting planes to cut the polyhedron into individual smaller polyhedra. The sequence lists the number of resulting n-faced polyhedra, where 4 <= n <= 9.

See A339348 for the corresponding sequence for the rhombic dodecahedron, the dual polyhedron of the cuboctahedron.

			The cuboctahedron has 12 vertices, 14 faces, and 24 edges. It is cut by 67 internal planes defined by any three of its vertices, resulting in the creation of 6728 polyhedra. No polyhedra with ten or more faces are created.

Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
Scott R. Shannon, Image showing the 67 internal plane cuts on the external edges and faces.
Scott R. Shannon, Image of the 2304 4-faced polyhedra.
Scott R. Shannon, Image of the 3000 5-faced polyhedra.
Scott R. Shannon, Image of the 944 6-faced polyhedra.
Scott R. Shannon, Image of the 408 7-faced polyhedra.
Scott R. Shannon, Image of the 48 8-faced polyhedra. None of these are visible on the surface of the cuboctahedron.
Scott R. Shannon, Image of the 24 9-faced polyhedra. None of these are visible on the surface of the cuboctahedron.
Scott R. Shannon, Image of all 6728 polyhedra. The colors are the same as those used in the above images.
Eric Weisstein's World of Mathematics, Cuboctahedron.
Wikipedia, Cuboctahedron.

Cf. A339348, A338622, A338801, A338808, A338825.

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).

A339528 The number of n-faced polyhedra formed when an elongated dodecahedron is internally cut by all the planes defined by any three of its vertices.

Original entry on oeis.org

153736, 177144, 106984, 44312, 12120, 2464, 304, 24, 0, 8

Offset: 4

Scott R. Shannon, Dec 08 2020

For an elongated dodecahedron create all possible internal planes defined by connecting any three of its vertices. Use all the resulting planes to cut the polyhedron into individual smaller polyhedra. The sequence lists the number of resulting n-faced polyhedra, where 4 <= n <= 13.

			The elongated dodecahedron has 18 vertices, 28 edges and 12 faces (8 rhombi and 4 hexagons). It is cut by 268 internal planes defined by any three of its vertices, resulting in the creation of 497096 polyhedra. No polyhedra with 12 faces or 14 or more faces are created.

Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
Scott R. Shannon, Image showing the 268 internal plane cuts on the external edges and faces
Scott R. Shannon, Image showing the 153736 4-faced polyhedra.
Scott R. Shannon, Image showing the 153736 4-faced polyhedra, viewed from above.
Scott R. Shannon, Image showing the 12120 8-faced polyhedra, viewed from above.
Scott R. Shannon, Image showing the 2464 9-faced polyhedra, viewed from above.
Scott R. Shannon, Image of all 497096 polyhedra. The polyhedra are colored red,orange,yellow,green,blue,indigo,violet for face counts 4 to 10 respectively. The polyhedra with face counts 11 and 13 are not visible on the surface.
Eric Weisstein's World of Mathematics, Elongated Dodecahedron.
Wikipedia, Elongated dodecahedron.

Cf. A339348, A339349, A338622, A338801, A338808, A338825.

A347918 Irregular table read by rows: The number of k-faced polyhedra, where k >= 4, formed when a row of n adjacent cubes are internally cut by all the planes defined by any three of their vertices.

Original entry on oeis.org

72, 24, 1472, 912, 416, 128, 32, 0, 8, 16192, 14952, 6832, 2816, 1288, 184, 80, 32, 8, 118800, 112904, 55088, 21064, 8920, 1560, 736, 232, 112

Offset: 1

Scott R. Shannon, Sep 19 2021

See A347753 for an explanation of the sequence and additional images.

See A333539 and A338622 for images of the single cube.

			The single cube, row 1, is internally cut with 14 planes which creates seventy-two 4-faced polyhedra and twenty-four 5-faced polyhedra. See also A333539.
The table begins:
      72,     24;
    1472,    912,   416,   128,   32,    0,   8;
   16192,  14952,  6832,  2816, 1288,  184,  80,  32,   8;
  118800, 112904, 55088, 21064, 8920, 1560, 736, 232, 112;

Scott R. Shannon, Image showing the 319416 different k-faced polyhedra for 4 adjacent cubes. The 4-, 5-, 6-, 7-, 8-, and 9-faced polyhedra are colored red, orange, yellow, green, blue, indigo respectively. The 10-, 11-, and 12-faced polyhedra, which are not visible on the surface and are shown together, are colored violet, white, black.

Cf. A347753 (total number of polyhedra), A333539 (n-dimensional cube), A338622 (Platonic solids), A338801 (n-prism), A338825 (n-bipyramid).

Sum of row n = A347753(n)

A339468 The number of n-faced polyhedra formed when a truncated tetrahedron is internally cut by all the planes defined by any three of its vertices.

Original entry on oeis.org

4818, 4596, 2454, 816, 246, 60, 0, 0, 9

Offset: 4

Scott R. Shannon, Dec 08 2020

For a truncated tetrahedron create all possible internal planes defined by connecting any three of its vertices. Use all the resulting planes to cut the polyhedron into individual smaller polyhedra. The sequence lists the number of resulting n-faced polyhedra, where 4 <= n <= 12.

			The truncated tetrahedron has 12 vertices, 18 edges and 4 faces (4 equilateral triangles and 4 hexagons). It is cut by 82 internal planes defined by any three of its vertices, resulting in the creation of 12999 polyhedra. No polyhedra with 10, 11, or 13 or more faces are created.

Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
Scott R. Shannon, Image showing the 82 internal plane cuts on the external edges and faces.
Scott R. Shannon, Image of the 4818 4-faced polyhedra.
Scott R. Shannon, Image of the 4596 5-faced polyhedra.
Scott R. Shannon, Image of the 2454 6-faced polyhedra.
Scott R. Shannon, Image of the 816 7-faced polyhedra.
Scott R. Shannon, Image of the 246 8-faced polyhedra. None of these are visible on the surface.
Scott R. Shannon, Image of the 60 9-faced polyhedra. None of these are visible on the surface.
Scott R. Shannon, Image of the 9 12-faced polyhedra. None of these are visible on the surface.
Scott R. Shannon, Image of all 12999 polyhedra. The polyhedra are colored red, orange, yellow, green for face counts 4 to 7 respectively. The polyhedra with 8, 9 and 12 faces are not visible on the surface.
Scott R. Shannon, Image of all 12999 polyhedra, exploded. Each polyhedron has been moved away from the origin by a distance proportional to the average distance of its vertices from the origin. Some of the 8-, 9- and 12-faced polyhedra can now be seen.
Eric Weisstein's World of Mathematics, Truncated Tetrahedron.
Wikipedia, Truncated tetrahedron.

Cf. A339528, A339348, A339349, A338622, A338801, A338808, A338825.

A339538 Irregular table read by rows: The number of k-faced polyhedra, where k>=4, created when an elongated n-bipyramid, with faces that are squares and equilateral triangles, is internally cut by all the planes defined by any three of its vertices.

Original entry on oeis.org

258, 336, 60, 424, 584, 208, 48, 8, 8830, 16090, 12210, 5040, 1210, 260, 80, 10

Offset: 3

Scott R. Shannon, Dec 08 2020

For an elongated n-bipyramid with faces that are squares and equilateral triangles, formed by joining the two halves of an n-gonal bipyramid by an n-prism, create all possible internal planes defined by connecting any three of its vertices. Use all the resulting planes to cut the polyhedron into individual smaller polyhedra. The sequence lists the number of resulting k-faced polyhedra, where k>=4, for elongated n-bipyramids where 3 <= n <= 5. These three elongated bipyramids are the only possible elongated bipyramids that are Johnson solids, i.e., their faces are all regular polygons.

			The elongated 5-bipyramid has 12 vertices, 25 edges and 15 faces (5 squares and 10 equilateral triangles). It is cut by 112 internal planes defined by any three of its vertices, resulting in the creation of 43730 polyhedra.
The 11 faced polyhedra are unusual in that all 10 are visible on the surface; most polyhedra cut with their own planes have the resulting polyhedra with the most faces near the center of the original polyhedra and are thus not visible on its surface.
No polyhedra with 12 or more faces are created.
The table is:
258, 336, 60;
424, 584, 208, 48, 8;
8830, 16090, 12210, 5040, 1210, 260, 80, 10;

Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
Scott R. Shannon, Elongated 3-bypyramid, showing the 29 plane cuts on the external edges and faces.
Scott R. Shannon, Elongated 3-bypyramid, showing the 258 4-faced polyhedra.
Scott R. Shannon, Elongated 3-bypyramid, showing all 654 polyhedra post cutting. The polyhedra are colored red,orange,yellow for face counts 4 to 6 respectively. No 6-faced polyhedra are visible on the surface.
Scott R. Shannon, Elongated 3-bypyramid, showing all 654 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. The 6-faced polyhedra can now be seen.
Scott R. Shannon, Elongated 4-bypyramid, showing the 40 plane cuts on the external edges and faces.
Scott R. Shannon, Elongated 4-bypyramid, showing the 424 4-faced polyhedra.
Scott R. Shannon, Elongated 4-bypyramid, showing the 48 7-faced polyhedra.
Scott R. Shannon, Elongated 4-bypyramid, showing all 1272 polyhedra post cutting. The polyhedra are colored red,orange,yellow,green,blue for face counts 4 to 8 respectively.
Scott R. Shannon, Elongated 4-bypyramid, showing all 1272 polyhedra post cutting and exploded.
Scott R. Shannon, Elongated 5-bypyramid, showing the 112 plane cuts on the external edges and faces.
Scott R. Shannon, Elongated 5-bypyramid, showing the 8830 4-faced polyhedra. This contains very small polyhedra near the peaks of the pyramids due to the convergence of the cutting lines near these points.
Scott R. Shannon, Elongated 5-bypyramid, showing the 8830 4-faced polyhedra viewed from above.
Scott R. Shannon, Elongated 5-bypyramid, showing the 1210 8-faced polyhedra viewed from above.
Scott R. Shannon, Elongated 5-bypyramid, showing all 43730 polyhedra post cutting. The polyhedra are colored red,orange,yellow,green,blue.indigo,violet,light-blue for face counts 4 to 11 respectively.
Scott R. Shannon, Elongated 5-bypyramid, showing all 43730 polyhedra post cutting and exploded.
Eric Weisstein's World of Mathematics, Elongated Square Dipyramid.
Eric Weisstein's World of Mathematics, Johnson Solid.
Wikipedia, Elongated bipyramid.

Cf. A338825, A339528, A339348, A339349, A338622, A338801, A338808.

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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A339348 The number of n-faced polyhedra formed when a rhombic dodecahedron is internally cut by all the planes defined by any three of its vertices.

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A339349 The number of n-faced polyhedra formed when a cuboctahedron is internally cut by all the planes defined by any three of its vertices.

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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.

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A339528 The number of n-faced polyhedra formed when an elongated dodecahedron is internally cut by all the planes defined by any three of its vertices.

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A347918 Irregular table read by rows: The number of k-faced polyhedra, where k >= 4, formed when a row of n adjacent cubes are internally cut by all the planes defined by any three of their vertices.

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A339468 The number of n-faced polyhedra formed when a truncated tetrahedron is internally cut by all the planes defined by any three of its vertices.

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A339538 Irregular table read by rows: The number of k-faced polyhedra, where k>=4, created when an elongated n-bipyramid, with faces that are squares and equilateral triangles, is internally cut by all the planes defined by any three of its vertices.

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