Original entry on oeis.org
1, 2, 16, 23752
Offset: 1
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
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.
A007847
Number of hyperplanes spanned by the vertices of an n-cube.
Original entry on oeis.org
2, 6, 20, 140, 3254, 252434, 71343208, 86246755608, 448691419804586
Offset: 1
Oswin Aichholzer (oaich(AT)igi.tu-graz.ac.at)
For n=2 there are the four edges of the square and the two diagonals, for a total of 6. - _N. J. A. Sloane_, Apr 14 2020
From _Tom Karzes_, Apr 14 2020: (Start)
The classes of hyperplanes are listed below for d = 2-5. Each class is shown below preceded by the number of instances of that class.
I define two hyperplanes as being in the same class if the vertex set of one can be transformed to the vertex set of the other by some combination of (1) permuting the coordinates and (2) inverting some set of coordinates (1->0 and 0->1).
For d=2 there are 2 classes hyperplanes (i.e., lines):
4: 00 01
2: 00 11
This gives a total of 6. The first class corresponds to the 4 perimeter slices.
For d=3 there are 3 classes of hyperplanes (i.e., planes):
6: 000 001 010 011
6: 000 001 110 111
8: 000 011 101
This gives a total of 20. The first class corresponds to the 6 perimeter slices.
For d=4 there are 6 classes of hyperplanes:
8: 0000 0001 0010 0011 0100 0101 0110 0111
12: 0000 0001 0010 0011 1100 1101 1110 1111
32: 0000 0001 0110 0111 1010 1011
16: 0000 0011 0101 1001
8: 0000 0011 0101 1010 1100 1111
64: 0000 0011 0101 1110
This gives a total of 140. The first class corresponds to the 8 perimeter slices.
For d=5 there are 15 classes of hyperplanes:
10: 00000 00001 00010 00011 00100 00101 00110 00111 01000 01001 01010 01011 01100 01101 01110 01111
20: 00000 00001 00010 00011 00100 00101 00110 00111 11000 11001 11010 11011 11100 11101 11110 11111
80: 00000 00001 00010 00011 01100 01101 01110 01111 10100 10101 10110 10111
80: 00000 00001 00110 00111 01010 01011 10010 10011
40: 00000 00001 00110 00111 01010 01011 10100 10101 11000 11001 11110 11111
320: 00000 00001 00110 00111 01010 01011 11100 11101
32: 00000 00011 00101 01001 10001
32: 00000 00011 00101 01001 10010 10100 10111 11000 11011 11101
80: 00000 00011 00101 01001 10110 11010 11100 11111
160: 00000 00011 00101 01001 11110
160: 00000 00011 00101 01010 01100 01111 10110
320: 00000 00011 00101 01110 10110
320: 00000 00011 00101 01110 11000 11011 11101
640: 00000 00011 00101 01110 11001
960: 00000 00011 01101 10101 11010
This gives a total of 3254. The first class corresponds to the 10 perimeter slices.
(End)
TOPCOM as of versions >= 1.0.0 can now compute these numbers up to n=9 and the same numbers up to symmetry. The computed numbers coincide with the preceding comment for dimensions from 2 through 5. - _Jörg Rambau_, Jun 06 2023
- O. Aichholzer, F. Aurenhammer, Classifying Hyperplanes in Hypercubes, 10th European Workshop on Computational Geometry, Santander, Spain, March 1994.
See
A333539 for the number of pieces formed when the cube is cut along these hyperplanes.
Cf.
A363505 for the same numbers up to symmetry.
Cf.
A363512 for the total numbers dual to these (in the oriented-matroid sense)
Cf.
A363506 for the numbers dual to these up to symmetry (in the oriented-matroid sense)
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
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,
...
- 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.
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
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.
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
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).
A338806
Number of polyhedra formed 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, 195, 834, 6365, 22770, 81769, 271702, 688793
Offset: 3
a(3) = 8. The 3-antiprism is cut with 3 internal planes resulting in 8 polyhedra, all 8 pieces having 4 faces.
a(4) = 195. The 4-antiprism is cut with 16 internal planes resulting in 195 polyhedra; 128 with 4 faces, 56 with 5 faces, 8 with 6 faces, and 3 with 8 faces. Note the number of 8-faced polyhedra is not a multiple of 4 - they lie directly along the z-axis so are not symmetric with respect to the number of edges forming the regular n-gons.
- Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
- Scott R. Shannon, 4-antiprism, showing the 16 plane cuts on the external edges and faces.
- Scott R. Shannon, 4-antiprism, showing the 195 polyhedra post-cutting. The 4-faced polyhedra are colored red, the 5-faced polyhedra are colored orange. The 6 and 8 faced polyhedra are not visible on the surface.
- Scott R. Shannon, 4-antiprism, showing the 195 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 and 8 faced polyhedra are colored yellow and green respectively.
- Scott R. Shannon, 7-antiprism, showing the 91 plane cuts on the external edges and faces.
- Scott R. Shannon, 7-antiprism, showing the 22770 polyhedra post-cutting. The 4,5,6,7,8,9 faced polyhedra are shown as red, orange, yellow, green, blue, indigo respectively. The polyhedra with 10,11,12,14,21 faces are not visible on the surface.
- Scott R. Shannon, 7-antiprism, showing the 22770 polyhedra post-cutting and exploded.
- Scott R. Shannon, 10-antiprism, showing the 280 plane cuts on the external edges and faces.
- Scott R. Shannon, 10-antiprism, showing the 688793 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 11,12,20 faces are not visible on the surface.
- Eric Weisstein's World of Mathematics, Antiprism.
- Wikipedia, Antiprism.
A338809
Number of polyhedra formed 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, 108, 756, 704, 3384, 3340, 11880, 10032, 33800, 32312, 82440, 78656, 182172, 144540, 365712, 350600
Offset: 3
a(3) = 12. The 3-bipyramid is cut with 4 internal planes resulting in 12 polyhedra, all 12 pieces having 4 faces.
a(5) = 120. The 5-bipyramid is cut with 16 internal planes resulting in 120 polyhedra, all 120 pieces having 4 faces.
a(7) = 756. The 7-bipyramid is cut with 36 internal planes resulting in 756 polyhedra; 448 with 4 faces, 280 with 5 faces, and 28 with 6 faces.
Note that for a single n-pyramid the number of polyhedra is the same as the number of regions in the dissection of a 2D n-polygon, see A007678, as all planes join two points on the polygon and the single apex, resulting in an equivalent number of regions.
- Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.
- Scott R. Shannon, 5-bipyramid, showing the 16 plane cuts on the external edges and faces.
- 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, 12-bipyramid, showing the 103 plane cuts on the external edges and faces.
- Scott R. Shannon, 12-bipyramid, showing the 10032 polyhedra post-cutting. The 4,5,6,7 faced polyhedra are colored red, orange, yellow, green respectively. The 8-faced polyhedra are not visible on the surface.
- Scott R. Shannon, 12-bipyramid, showing the 10032 polyhedra post-cutting and exploded.The 8-faced polyhedra colored blue can be seen.
- Scott R. Shannon, 20-bipyramid, showing the 331 plane cuts on the external edges and faces.
- Scott R. Shannon, 20-bipyramid, showing the 350600 polyhedra post-cutting. The 4,5,6,7,8,9,11 faced polyhedra are colored red, orange, yellow, green, blue, indigo, violet respectively. The polyhedra with 10 and 12 faces are not visible on the surface.
- Scott R. Shannon, 20-bipyramid positions vertically, showing the 350600 polyhedra post-cutting.
- Scott R. Shannon, 20-bipyramid, showing the 350600 polyhedra post-cutting and exploded. The 10-faced and 12-faced polyhedra, colored black and white, can also be seen.
- Eric Weisstein's World of Mathematics, Dipyramid.
- Wikipedia, Bipyramid.
A333544
Irregular triangle read by rows, formed from the triangle A333543 by dividing the terms in row n by n!.
Original entry on oeis.org
1, 2, 12, 4, 6784, 4024, 4936, 2704, 1912, 936, 824, 496, 360, 352, 256, 72, 48, 16, 16, 16
Offset: 1
Triangle begins:
1,
2,
12, 4,
6784, 4024, 4936, 2704, 1912, 936, 824, 496, 360, 352, 256, 72, 48, 16, 16, 16
...
A347753
Number of polyhedra 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
96, 2968, 42384, 319416
Offset: 1
a(1) = 96. A single cube, with eight vertices, has 14 internal cutting planes resulting in 96 polyhedra. See A333539 and A338571.
a(2) = 2968. Two adjacent cubes, with twelve vertices, have 51 internal cutting planes resulting in 2968 polyhedra.
a(3) = 42384. Three adjacent cubes, with sixteen vertices, have 124 internal cutting planes resulting in 42384 polyhedra.
a(4) = 319416. Four adjacent cubes, with twenty vertices, have 245 internal cutting planes resulting in 319416 polyhedra.
- Scott R. Shannon, The 245 cutting planes on the surface of 4 adjacent cubes.
- Scott R. Shannon, The surface of the 4 adjacent cubes after cutting. The 4-, 5-, 6-, 7-, 8-, and 9-faced polyhedra created by the planes are colored red, orange, yellow, green, blue, and indigo, respectively. The 10-, 11-, and 12-faced polyhedra are not visible on the surface. See also A347918.
- Scott R. Shannon, The 4 adjacent cubes after cutting exploded. Each of the 319416 polyhedra is moved away from the center of the solid a distance proportional to the average distance of its vertices from the center.
Showing 1-10 of 11 results.
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