A338622 -id:A338622 - cp's OEIS frontend

A338801 Irregular table read by rows: The number of k-faced polyhedra, where k>=4, created 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.

17, 0, 1, 72, 24, 575, 450, 232, 60, 15, 0, 3, 1728, 1668, 948, 144, 24, 12, 8799, 10080, 6321, 3052, 898, 490, 161, 14, 35, 14, 7, 22688, 24080, 12784, 4160, 1248, 272, 80, 32, 78327, 101142, 70254, 39708, 19584, 6894, 2369, 1062, 351, 54, 27, 18, 27, 36, 11, 165500, 203220, 134860, 62520, 21240, 5720, 1080, 300, 100, 20

Offset: 3

Scott R. Shannon, Nov 10 2020

See A338783 for further details and images for this sequence.

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

			The triangular 3-prism is cut with 6 internal planes defined by all 3-vertex combinations of its 6 vertices. This leads to the creation of seventeen 4-faced polyhedra and one 6-faced polyhedra, eighteen pieces in all. The single 6-faced polyhedra lies at the very center of the original 3-prism.
The 9-prism is cut with 207 internal planes leading to the creation of 319864 pieces. It is noteworthy in creating all k-faced polyhedra from k=4 to k=18.
The table begins:
17,0,1;
72,24;
575,450,232,60,15,0,3;
1728,1668,948,144,24,12;
8799,10080,6321,3052,898,490,161,14,35,14,7;
22688,24080,12784,4160,1248,272,80,32;
78327,101142,70254,39708,19584,6894,2369,1062,351,54,27,18,27,36,11;
165500,203220,134860,62520,21240,5720,1080,300,100,20;

Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
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 seventeen 4-faced polyhedra, orange the single 6-faced polyhedron.
Scott R. Shannon, 7-prism, showing the 8799 4-faced polyhedra.
Scott R. Shannon, 7-prism, showing the 10080 5-faced polyhedra.
Scott R. Shannon, 7-prism, showing the 6321 6-faced polyhedra.
Scott R. Shannon, 7-prism, showing the 3052 7-faced polyhedra.
Scott R. Shannon, 7-prism, showing the 898 8-faced polyhedra.
Scott R. Shannon, 7-prism, showing the 490 9-faced polyhedra.
Scott R. Shannon, 7-prism, showing the 161 10-faced polyhedra.
Scott R. Shannon, 7-prism, showing the 14 11-faced, 35 12-faced, 14 13-faced, 7 14-faced polyhedra. These are colored white, black, yellow, red respectively. None of these are visible on the surface.
Scott R. Shannon, 7-prism, showing all 29871 polyhedra. The 4,5,6,7,8,9,10 faced polyhedra are colored red, orange, yellow, green, blue, indigo, violet respectively. The 11,12,13,14 faced polyhedra are not visible on the surface.
Scott R. Shannon, 10-prism, showing all 594560 polyhedra.
Eric Weisstein's World of Mathematics, Prism.
Wikipedia, Prism (geometry).

Cf. A338783 (number of polyhedra), A338808 (antiprism), A338622 (Platonic solids), A333543 (n-dimensional cube).

Sum of row n = A338783(n).

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.

Original entry on oeis.org

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).

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.

Original entry on oeis.org

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

Scott R. Shannon, Nov 11 2020

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.

			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;

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.

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

Sum of row n = A338809(n).

A333543 Irregular triangle read by rows: T(n,k) (n >= 1, k >= n+1) is the number of cells with k vertices in the dissection of an n-dimensional cube by all the hyperplanes that pass through any n vertices.

Original entry on oeis.org

1, 4, 72, 24, 162816, 96576, 118464, 64896, 45888, 22464, 19776, 11904, 8640, 8448, 6144, 1728, 1152, 384, 384, 384

Offset: 1

N. J. A. Sloane, Apr 21 2020

Rows 1 through 4 computed by Veit Elser, later confirmed by Tom Karzes.

The row sums give A333539.

			The two diagonals of a square cut it into four triangular pieces, so T(2,3) = 4.
Triangle begins:
1,
4,
72, 24,
162816, 96576, 118464, 64896, 45888, 22464, 19776, 11904, 8640, 8448, 6144, 1728, 1152, 384, 384, 384,
...

N. J. A. Sloane, Cutting Up a Cube, Math Fun Mailing List, Apr 10 2020; with replies from Tom Karzes, Tomas Rokicki, Veit Elser, and others.

Veit Elser, Rows 1 through 4
Scott R. Shannon, Illustration for a(2) = 4.
Scott R. Shannon, Illustration for a(3) = 72. This shows the 4-faced cells in the 3D cube dissection. The 72 pieces have been moved away from the origin a distance proportional to the average distance of their vertices from the origin.
Scott R. Shannon, Illustration for a(4) = 24. This shows the 5-faced cells in the 3D cube dissection. The 24 pieces have been moved away from the origin a distance proportional to the average distance of their vertices from the origin. These polyhedra form a perfect octahedron inside the original cube with its points touching the cube's inner surface.

Cf. A333539, A333540, A333544, A338622 (number of k-faced polyhedra for the 3D Platonic solids).

For the number of hyperplanes see A007847.

A338571 Number of polyhedra formed when the five Platonic solids, in the order tetrahedron, octahedron, cube, icosahedron, dodecahedron, are internally cut by all the planes defined by any three of their vertices.

Original entry on oeis.org

1, 8, 96, 2520, 552600

Offset: 1

Scott R. Shannon, Nov 03 2020

For a Platonic solid create all possible planes defined by connecting any three of its vertices. For example, in the case of a cube this results in fourteen planes; six planes between the pairs of parallel edges connected to each end of the face diagonals, and eight planes from connecting the three vertices adjacent to each corner vertex. Use all the resulting planes to cut the solid into individual smaller polyhedra. The sequence lists the numbers of resulting polyhedra for the Platonic solids, ordered by number of vertices: tetrahedron, octahedron, cube, icosahedron, dodecahedron.
See A338622 for the number and images of the k-faced polyhedra in each dissection for each of the five solids.
The author thanks Zach J. Shannon for producing the images for this sequence.

			a(1) = 1. The tetrahedron has no internal cutting planes so the single polyhedron after cutting is the tetrahedron itself.
a(2) = 8. The octahedron has 3 internal cutting planes resulting in 8 polyhedra.
a(3) = 96. The cube has 14 internal cutting planes resulting in 96 polyhedra. See also A333539.
a(4) = 2520. The icosahedron has 47 cutting planes resulting in 2520 polyhedra.
See A338622 for a breakdown of the above totals into the corresponding number of k-faced polyhedra.
a(5) = 552600. The dodecahedron has 307 internal cutting planes resulting in 552600 polyhedra. It is the only Platonic solid which produces polyhedra with 6 or more faces.

Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
Polyhedra.mathmos.net, The Platonic Solids.
Scott R. Shannon, Tetrahedron, showing the one polyhedra pre and post-cutting. The tetrahedron has no internal cutting planes so remains unaltered.
Scott R. Shannon, Octahedron, showing the 8 polyhedra post-cutting. All pieces have 4 faces. The plane cuts are along the edges of the octahedron and thus only 3 internal cutting planes exist, each along the three 2D axial planes.
Scott R. Shannon, Cube, showing the 14 plane cuts on the external edges and faces.
Scott R. Shannon, Cube, showing the 96 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 5-faced polyhedra. The later form a perfect octahedron inside the original cube, the points of which touch the cube's surface. See A338622.
Scott R. Shannon, Icosahedron, showing the 47 plane cuts on the external edges and faces.
Scott R. Shannon, Icosahedron, showing the 2520 polyhedra post-cutting. Red shows the 4-faced polyhedra, orange the 5-faced polyhedra.
Scott R. Shannon, Icosahedron, showing the 2520 polyhedra post-cutting and exploded. Red shows the 4-faced polyhedra, orange the 5-faced polyhedra.
Scott R. Shannon, Dodecahedron, showing the 307 plane cuts on the external edges and faces.
Scott R. Shannon, Dodecahedron, showing the 552600 polyhedra post-cutting. The 4,5,6,7,8 faced polyhedra are colored red, orange, yellow, green and blue respectively. The 9 and 10 faced polyhedra are all internal.
Scott R. Shannon, Dodecahedron, showing the 552600 polyhedra post-cutting and exploded. Zooming in shows the vast array of polyhedra.
Scott R. Shannon, Dodecahedron, close-up of the post-cutting and exploded image.
Zach J. Shannon, Animation showing the 96 polyhedra for the cube and the 2520 polyhedra for the icosahedron.
Eric Weisstein's World of Mathematics, Platonic Solid.
Wikipedia, Platonic solid.

Cf. A338622 (number of k-faced polyhedra in each dissection), A333539 (n-dimensional cube), A053016, A063722, A063723, A098427.

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)

cp's OEIS Frontend

A338801 Irregular table read by rows: The number of k-faced polyhedra, where k>=4, created 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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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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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.

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A333543 Irregular triangle read by rows: T(n,k) (n >= 1, k >= n+1) is the number of cells with k vertices in the dissection of an n-dimensional cube by all the hyperplanes that pass through any n vertices.

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A338571 Number of polyhedra formed when the five Platonic solids, in the order tetrahedron, octahedron, cube, icosahedron, dodecahedron, are internally cut by all the planes defined by any three of their 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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