A333543 -id:A333543 - cp's OEIS frontend

A333544 Irregular triangle read by rows, formed from the triangle A333543 by dividing the terms in row n by n!.

1, 2, 12, 4, 6784, 4024, 4936, 2704, 1912, 936, 824, 496, 360, 352, 256, 72, 48, 16, 16, 16

Offset: 1

N. J. A. Sloane, Apr 21 2020

Rows 1 through 4 computed by Veit Elser, later confirmed by Tom Karzes.
Row sums are A333540.
See A333543 for further information.

			Triangle begins:
1,
2,
12, 4,
6784, 4024, 4936, 2704, 1912, 936, 824, 496, 360, 352, 256, 72, 48, 16, 16, 16
...

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.

Cf. A333539, A333540, A333543.

A338622 Irregular table read by rows: The number of k-faced polyhedra, where k>=4, 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, 72, 24, 2160, 360, 205320, 208680, 94800, 34200, 7920, 1560, 120

Offset: 1

Scott R. Shannon, Nov 04 2020

See A338571 for further details and images of this sequence.

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

			The cube is cut with 14 internal planes defined by all 3-vertex combinations of its 8 vertices. This leads to the creation of 72 4-faced polyhedra and 24 5-faced polyhedra, 96 pieces in all. See A338571 and A333539.
The table is:
1;
8;
72, 24;
2160, 360;
205320, 208680, 94800, 34200, 7920, 1560, 120;

Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
Polyhedra.mathmos.net, The Platonic Solids.
Scott R. Shannon, Tetrahedron, showing the 1 4-faced polyhedra post-cutting. This is the original tetrahedron itself as no internal cutting planes are present.
Scott R. Shannon, Octahedron, showing the 8 4-faced polyhedra post-cutting. The octahedron has 3 internal cutting planes, each along the 2D axial planes. For clarity in this image, and the two cube images, the pieces are moved away from the origin a distance proportional to the average distance of their vertices from the origin.
Scott R. Shannon, Cube, showing the 72 4-faced polyhedra post-cutting. The cube has 14 internal cutting planes.
Scott R. Shannon, Cube, showing the 24 5-faced polyhedra post-cutting. These form a perfect octahedron inside the original cube.
Scott R. Shannon, Icosahedron, showing the 2160 4-faced polyhedra post-cutting. The icosahedronhas 47 internal cutting planes.
Scott R. Shannon, Icosahedron, showing the 360 5-faced polyhedra post-cutting.
Scott R. Shannon, Icosahedron, showing all 2520 polyhedra post-cutting.
Scott R. Shannon, Dodecahedron, showing the 205320 4-faced polyhedra post-cutting. The dodecahedron has 307 internal cutting planes.
Scott R. Shannon, Dodecahedron, showing the 208680 5-faced polyhedra post-cutting.
Scott R. Shannon, Dodecahedron, showing the 94800 6-faced polyhedra post-cutting.
Scott R. Shannon, Dodecahedron, showing the 34200 7-faced polyhedra post-cutting.
Scott R. Shannon, Dodecahedron, showing the 7920 8-faced polyhedra post-cutting.
Scott R. Shannon, Dodecahedron, showing the 1560 9-faced polyhedra post-cutting. None of these polyhedra are visible on the surface of the original dodecahedron.
Scott R. Shannon, Dodecahedron, showing the 120 10-faced polyhedra post-cutting. None of these polyhedra are visible on the surface of the original dodecahedron.
Scott R. Shannon, Dodecahedron, showing a combination of the 4-faced and 5-faced polyhedra post-cutting. These two types make up about 75% of all the pieces.
Scott R. Shannon, Dodecahedron, showing all 552600 polyhedra post-cutting. No 9-faced or 10-faced polyhedra are visible on the surface.
Eric Weisstein's World of Mathematics, Platonic Solid.
Wikipedia, Platonic solid.

Cf. A338571 (total number of polyhedra), A333539 (n-dimensional cube), A053016, A063722, A063723, A098427, A333543.

Sum of row n = A338571(n).

A333539 Number of pieces formed when an n-dimensional cube is cut by all the hyperplanes defined by any n of the 2^n vertices.

Original entry on oeis.org

1, 4, 96, 570048

Offset: 1

N. J. A. Sloane, Apr 14 2020, in response to a question raised by Scott R. Shannon

			The two diagonals of a square cut it into four pieces, so a(2) = 4.
For the cube the answer is 96 regions.  There are 14 cuts through the cube: six cut the cube in half along a face diagonal, and eight cut off a corner with a triangle through the three adjacent corners. The cuts through the center alone divide the cube into 24 regions, and then the corner cuts further divide each of these into four regions. - _Tomas Rokicki_, Apr 11 2020

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, The values of a(1) - a(4)
Scott R. Shannon, Image of the 3-dimensional cube showing the 96 pieces. The 4-faced polyhedra are shown in red, the 5-faced polyhedra in yellow. The later form a perfect octahedron inside the cube with its points touching the cube's inner surface. The pieces are moved away from the origin a distance proportional to the average of the distance of all its vertices from the origin.

For the number of hyperplanes see A007847.
Cf. A333540, A338571 (number of pieces for the 3D Platonic solids).
For a more detailed count, see A333543 and A333544.

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.

Original entry on oeis.org

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).

cp's OEIS Frontend

A333544 Irregular triangle read by rows, formed from the triangle A333543 by dividing the terms in row n by n!.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

A338622 Irregular table read by rows: The number of k-faced polyhedra, where k>=4, 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A333539 Number of pieces formed when an n-dimensional cube is cut by all the hyperplanes defined by any n of the 2^n vertices.

Views

Author

Keywords

Examples

References

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula