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
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.
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
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.
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.
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
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.
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
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.
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
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.
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
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.
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