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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.

%I A338622 #33 Feb 16 2025 08:34:00
%S A338622 1,8,72,24,2160,360,205320,208680,94800,34200,7920,1560,120
%N 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.
%C A338622 See A338571 for further details and images of this sequence.
%C A338622 The author thanks _Zach J. Shannon_ for producing the images for this sequence.
%H A338622 Hyung Taek Ahn and Mikhail Shashkov, <a href="https://cnls.lanl.gov/~shashkov/papers/ahn_geometry.pdf">Geometric Algorithms for 3D Interface Reconstruction</a>.
%H A338622 Polyhedra.mathmos.net, <a href="http://www.srcf.ucam.org/~rjw62/polyhedra/entry/platonicsolids.html">The Platonic Solids</a>.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622.png">Tetrahedron, showing the 1 4-faced polyhedra post-cutting</a>. This is the original tetrahedron itself as no internal cutting planes are present.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_3.png">Octahedron, showing the 8 4-faced polyhedra post-cutting</a>. 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.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_1.png">Cube, showing the 72 4-faced polyhedra post-cutting</a>. The cube has 14 internal cutting planes.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_2.png">Cube, showing the 24 5-faced polyhedra post-cutting</a>. These form a perfect octahedron inside the original cube.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_13.png">Icosahedron, showing the 2160 4-faced polyhedra post-cutting</a>. The icosahedronhas 47 internal cutting planes.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_14.png">Icosahedron, showing the 360 5-faced polyhedra post-cutting</a>.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_15.png">Icosahedron, showing all 2520 polyhedra post-cutting</a>.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_4.png">Dodecahedron, showing the 205320 4-faced polyhedra post-cutting</a>. The dodecahedron has 307 internal cutting planes.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_5.png">Dodecahedron, showing the 208680 5-faced polyhedra post-cutting</a>.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_6.png">Dodecahedron, showing the 94800 6-faced polyhedra post-cutting</a>.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_7.png">Dodecahedron, showing the 34200 7-faced polyhedra post-cutting</a>.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_8.png">Dodecahedron, showing the 7920 8-faced polyhedra post-cutting</a>.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_9.png">Dodecahedron, showing the 1560 9-faced polyhedra post-cutting</a>. None of these polyhedra are visible on the surface of the original dodecahedron.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_10.png">Dodecahedron, showing the 120 10-faced polyhedra post-cutting</a>. None of these polyhedra are visible on the surface of the original dodecahedron.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_11.png">Dodecahedron, showing a combination of the 4-faced and 5-faced polyhedra post-cutting</a>. These two types make up about 75% of all the pieces.
%H A338622 Scott R. Shannon, <a href="/A338622/a338622_12.png">Dodecahedron, showing all 552600 polyhedra post-cutting</a>. No 9-faced or 10-faced polyhedra are visible on the surface.
%H A338622 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlatonicSolid.html">Platonic Solid</a>.
%H A338622 Wikipedia, <a href="https://en.wikipedia.org/wiki/Platonic_solid">Platonic solid</a>.
%F A338622 Sum of row n = A338571(n).
%e A338622 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.
%e A338622 The table is:
%e A338622 1;
%e A338622 8;
%e A338622 72, 24;
%e A338622 2160, 360;
%e A338622 205320, 208680, 94800, 34200, 7920, 1560, 120;
%Y A338622 Cf. A338571 (total number of polyhedra), A333539 (n-dimensional cube), A053016, A063722, A063723, A098427, A333543.
%K A338622 nonn,fini,full,tabf
%O A338622 1,2
%A A338622 _Scott R. Shannon_, Nov 04 2020