A198861 -id:A198861 - cp's OEIS frontend

A244951 Minimum number of colors needed to color the faces of the Platonic solids such that no two faces meeting at a common edge share the same color (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

4, 3, 2, 4, 3

Offset: 1

Felix Fröhlich, Jul 08 2014

			a(1) = 4, since in the tetrahedron any face shares a common edge with any other face, so each face needs a distinct color.
a(2) = 3, since the cube has three sets of opposite faces. Any two faces that are not opposite share a common edge, so only opposite faces can have the same color.
a(3) = 2, since cutting the octahedron along its "equator" results in two square pyramids. The triangular faces of a single pyramid can be colored using two colors in an alternating fashion. Then the two pyramids are reassembled such that at the "equator" differently colored faces meet.
a(4) and a(5) are shown in illustration in the links.

Felix Fröhlich, Illustration of colorings via Schlegel diagrams
Martin Gardner, The Five Platonic Solids, in Origami, Eleusis, and the Soma Cube: Martin Gardner’s Mathematical Diversions, Cambridge University Press, (see page 6).

Cf. A098112, A198861, A158478 (analog for sides of polygons).

with(GraphTheory): with(SpecialGraphs):
map(ChromaticNumber @ PlaneDual, [TetrahedronGraph(), HypercubeGraph(3), OctahedronGraph(), DodecahedronGraph(), IcosahedronGraph()]); # Robert Israel, Aug 24 2014

Corrected value of a(4) due to discovery of a new coloring for the dodecahedron.

Corrected value of a(5) due to discovery of a new coloring for the icosahedron.

A317978 The number of ways to paint the cells of the six convex regular 4-polytopes using exactly n colors where n is the number of cells of each 4-polytope.

Original entry on oeis.org

2, 210, 108972864000, 1077167364120207360000

Offset: 1

Frank M Jackson, Aug 12 2018

Let G, the group of rotations in 4 dimensional space, act on the set of n! paintings of each convex regular 4-polytopes having n cells. 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 A273509/2. So by Burnside's Lemma a(n)=n!/|G|. a(5) = 120!/7200 and a(6) = 600!/72000 and they are too large to display.
See A198861 for the Platonic solids which are the analogs of the regular polyhedra in three dimensions.
a(6) = 17577...66368*10^146 has 1405 digits. - Georg Fischer, Jun 16 2025

			The second of these six 4-polytopes (in sequence of cell count) is the 4-cube (with 8 cells). It has |G| = 192 rotations with n = 8. Hence a(2) = 8!/192 = 210.

Georg Fischer, Table of n, a(n) for n = 1..5
Wikipedia, Regular 4-polytope

Cf. A063924, A098427, A198861, A273509.

{5!/60, 8!/192, 16!/192, 24!/576, 120!/7200, 600!/7200};

{5!/60, 8!/192, 16!/192, 24!/576, 120!/7200, 600!/7200}

a(n) = 2*A063924(n)! / A273509(n). [Corrected by Georg Fischer, Jun 16 2025]

A317251 a(n) is the number of ways to paint the 2^n cells of dimension n-1 that bound a regular convex n-orthoplex polytope using exactly 2^n colors where n is the dimension of Euclidean space.

Original entry on oeis.org

2, 6, 1680, 108972864000, 137047310902965380295426048000000, 5507245320567889066989296412116383715402149139520190633628554443368693760000000000000

Offset: 1

Frank M Jackson, Aug 13 2018

nonn

Let G, the group of rotations in n-dimensional Euclidean space, act on the set of (2^n)! paintings of an n-orthoplex bound by 2^n cells of dimension n-1. There are (2^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 2^(n-1)*n! = A002866(n). So by Burnside's Lemma a(n) = (2^n)!/|G|.

See A198861(3) for the number of ways to paint the octahedron a(3) in the Platonic solids and A317978(3) for the 4-orthoplex a(4) in the regular convex 4-polytopes.

Frank M Jackson, Table of n, a(n) for n = 1..8
Wikipedia, Cross-polytope

Cf. A000723, A002866, A097801, A198861, A317978.

Mathematica
```
a[n_]:=(2^n)!/(2^(n-1)*n!); Array[a,10]
```

a(n) = (2^n)!/(2^(n-1)*n!) = (2^n)!/A002866(n).

a(n) = 2 * A000723(n). - Alois P. Heinz, Aug 15 2018

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A244951 Minimum number of colors needed to color the faces of the Platonic solids such that no two faces meeting at a common edge share the same color (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

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A317978 The number of ways to paint the cells of the six convex regular 4-polytopes using exactly n colors where n is the number of cells of each 4-polytope.

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A317251 a(n) is the number of ways to paint the 2^n cells of dimension n-1 that bound a regular convex n-orthoplex polytope using exactly 2^n colors where n is the dimension of Euclidean space.

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