A063722 -id:A063722 - cp's OEIS frontend

A053016 Numbers of vertices of Platonic solids in the order tetrahedron, octahedron, cube, icosahedron, dodecahedron. Equally, numbers of faces of Platonic solids in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron.

4, 6, 8, 12, 20

Offset: 1

Jeffrey Keller (jeff(AT)auctionflow.com), Feb 24 2000

It appears that the stereographic projection of the Platonic solids requires respectively 4, 6, 8, 6, 10, different colors to represent them. - Eric Desbiaux, Feb 15 2009

H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover, NY, 1973.

S. Alejandre, Studying Polyhedra
P. Alfeld, The Platonic Solids
D. M. Cahir, Platonic Solids
M. Chaplin, The Platonic Solids
D. Eberly, Platonic Solids
T. Eveilleau, Les solides de Platon
W. Fendt, The Platonic Solids (in German)
F. Fernandez, Platonic Solids
Z. Fiedorowicz, V-E+F=2
D. A. Fontaine, The Five Platonic Solids
FreeDictionary.com, Platonic solid
Polyhedra.mathmos.net, The Platonic Solids
E. S. Rowland, Regular Polyhedra
A. Ruediger, The Platonic Solids
L. Stemkoski, Platonic Solids
G. Tulloue, Plato's Polyhedra
University of St. Andrews, The five Platonic solids
Utah State University, Platonic Solids
Visual Geometry Pages, Platonic Solids
Eric Weisstein's World of Mathematics, Platonic Solid.
D. Wells, Regular Polyhedra
Wolfram Research, Polyhedron Explorer
W. Wu, The Platonic Solids
Anonymous, Polyhedra
Anonymous, The Platonic Solids

Cf. A063927, A063722, A063723.

Definition expanded by N. J. A. Sloane, Nov 06 2020

A063723 Number of vertices in the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

Original entry on oeis.org

4, 8, 6, 20, 12

Offset: 1

Henry Bottomley, Aug 14 2001

The preferred order for these five numbers is 4, 6, 8, 12, 20 (tetrahedron, octahedron, cube, icosahedron, dodecahedron), as in A053016. - N. J. A. Sloane, Nov 05 2020
Also number of faces of Platonic solids ordered by increasing ratios of volumes to their respective circumscribed spheres. See cross-references for actual ratios. - Rick L. Shepherd, Oct 04 2009
Also the expected lengths of nontrivial random walks along the edges of a Platonic solid from one vertex back to itself. - Jens Voß, Jan 02 2014

			a(2) = 8 since a cube has eight vertices.

Cf. A053012, A053016, A060296, A060852.
Cf. A165922 (tetrahedron), A049541 (octahedron), A165952 (cube), A165954 (icosahedron), A165953 (dodecahedron). - Rick L. Shepherd, Oct 04 2009
Cf. A234974. - Jens Voß, Jan 02 2014

a(n) = A063722(n) - A053016(n) + 2.

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

Scott R. Shannon, Nov 04 2020

See A338571 for further details and images of this sequence.

The author thanks Zach J. Shannon for producing the images for this sequence.

			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.

Cf. A338571 (total number of polyhedra), A333539 (n-dimensional cube), A053016, A063722, A063723, A098427, A333543.

Sum of row n = A338571(n).

A063924 Number of 3-dimensional cells in the regular 4-dimensional polytopes.

Original entry on oeis.org

5, 8, 16, 24, 120, 600

Offset: 1

Henry Bottomley, Aug 15 2001

Sorted by number of 3-dimensional cells.

Also, number of vertices of the regular 4-dimensional polyhedra. - Douglas Boffey, Aug 12 2012

			a(2) = 8 since a 4D hypercube contains eight cubes.

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.

Cf. A053016, A060296, A063722, A063723.

a(n) = A063925(n) - A063926(n) + A063927(n).

Corrected faces to cells by Douglas Boffey, Aug 12 2012

A063927 Number of vertices in the regular 4-dimensional polytopes.

Original entry on oeis.org

5, 16, 8, 24, 600, 120

Offset: 1

Henry Bottomley, Aug 15 2001

Sorted by number of 3-dimensional faces.

			a(2) = 16 since a 4D hypercube contains sixteen vertices.

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.

Cf. A060296, A053016, A063722, A063723.

a(n) = A063926(n) - A063925(n) + A063924(n).

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

Scott R. Shannon, Nov 03 2020

For a Platonic solid create all possible planes defined by connecting any three of its vertices. For example, in the case of a cube this results in fourteen planes; six planes between the pairs of parallel edges connected to each end of the face diagonals, and eight planes from connecting the three vertices adjacent to each corner vertex. Use all the resulting planes to cut the solid into individual smaller polyhedra. The sequence lists the numbers of resulting polyhedra for the Platonic solids, ordered by number of vertices: tetrahedron, octahedron, cube, icosahedron, dodecahedron.
See A338622 for the number and images of the k-faced polyhedra in each dissection for each of the five solids.
The author thanks Zach J. Shannon for producing the images for this sequence.

			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.

Cf. A338622 (number of k-faced polyhedra in each dissection), A333539 (n-dimensional cube), A053016, A063722, A063723, A098427.

A063925 Number of 2-dimensional faces in the regular 4-dimensional polytopes.

Original entry on oeis.org

10, 24, 32, 96, 720, 1200

Offset: 1

Henry Bottomley, Aug 15 2001

Sorted by number of 2-dimensional faces.

Also the number of edges in the regular 4-dimensional polytopes [Douglas Boffey, Aug 12 2012]

			a(2) = 24 since a 4D hypercube contains twenty-four faces.

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.

Cf. A053016, A063722, A063723.

a(n) = A063924(n)+A063926(n)-A063927(n).

A063926 Number of edges in the six regular 4-dimensional polytopes.

Original entry on oeis.org

10, 32, 24, 96, 1200, 720

Offset: 1

Henry Bottomley, Aug 15 2001

Sorted by number of 3-dimensional faces.

			a(2) = 32 since a 4D hypercube contains thirty-two edges.

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.

Cf. A053016, A063722, A063723, A063924, A063925, A063927.

a(n) = A063927(n) + A063925(n) - A063924(n).

A198861 The number of ways to paint the faces of the five Platonic solids using exactly n colors where n is the number of faces of each solid.

Original entry on oeis.org

2, 30, 1680, 7983360, 40548366802944000

Offset: 1

Geoffrey Critzer, Oct 30 2011

Let G, the group of rotations in 3 dimensional space act on the set of n! paintings of each Platonic solid having n faces. There are n! fixed points in the action table since the only element in G that leaves a painting fixed is the identity element. The order of G is A098427/2. So by Burnside's Lemma a(n)=n!/|G|.

David Broughton's Puzzles & Programs, Colouring The Platonic Solids

Cf. A053016 (number of faces), A063722 (number of edges).

lista() = {ve = [6, 12, 12, 30, 30 ]; vf = [4, 6, 8, 12, 20 ]; for (i=1, 5, nb = vf[i]!/(2*ve[i]); print1(nb, ", "););} \\ Michel Marcus, Aug 25 2014

a(n) = A053016(n)!/(2*A063722(n)) (see link). - Michel Marcus, Aug 24 2014

A359201 Number of edges of regular m-polytopes for m >= 3.

Original entry on oeis.org

6, 10, 12, 15, 21, 24, 28, 30, 32, 36, 40, 45, 55, 60, 66, 78, 80, 84, 91, 96, 105, 112, 120, 136, 144, 153, 171, 180, 190, 192, 210, 220, 231, 253, 264, 276, 300, 312, 325, 351, 364, 378, 406, 420, 435, 448, 465, 480, 496, 528, 544, 561, 595, 612, 630, 666

Offset: 1

Marco Ripà, Dec 20 2022

In 3 dimensions there are five (convex) regular polytopes and they have 6, 12, or 30 edges (A063722).
In 4 dimensions there are six regular 4-polytopes and they have 10, 24, 32, 96, 720, or 1200 edges (A063926).
In m >= 5 dimensions, there are only 3 regular polytopes (i.e., the m-simplex, the m-cube, and the m-crosspolytope) so that we can sort their number of edges in ascending order and define the present sequence.

			6 is a term since a tetrahedron has 6 edges.

Mathematics StackExchange, What are the formulas for the number of vertices, edges, faces, cells, 4-faces, ..., n-faces, of convex regular polytopes in n≥5 dimensions?
Wikipedia, List of regular polytopes and compounds

Cf. A000217, A001787, A046092, A063722, A063926.

Cf. A359202 (faces), A359662 (cells).

{a(n), n >= 1} = {{30, 96, 720} U {A000217} U {A001787} U {A046092}} \ {0, 1, 3, 4}.

cp's OEIS Frontend

A053016 Numbers of vertices of Platonic solids in the order tetrahedron, octahedron, cube, icosahedron, dodecahedron. Equally, numbers of faces of Platonic solids in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron.

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A063723 Number of vertices in the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

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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.

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A063924 Number of 3-dimensional cells in the regular 4-dimensional polytopes.

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A063927 Number of vertices in the regular 4-dimensional polytopes.

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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.

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A063925 Number of 2-dimensional faces in the regular 4-dimensional polytopes.

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A063926 Number of edges in the six regular 4-dimensional polytopes.

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A198861 The number of ways to paint the faces of the five Platonic solids using exactly n colors where n is the number of faces of each solid.

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