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