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
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).
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.
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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