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.
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
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
- 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.
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.
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.
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