Noble Polyhedra: Icosahedral 5 colour arrangements

Notes:  In all cases one representative set of faces has been highlighted.  This face remains solid through the various displays in the VRML window.  Pressing the 'frame' hud button will display a 'framework' version of the figure, 'hull' will then display the figure with a framework convex hull, 'solid' returns to the initial view.

Tight Minimum Injective Face-Colourings of Noble Polyhedra with 'I' (Chiral Icosahedral) Symmetry

A tight minimum injective face-colouring of a polyhedron is a face colouring where each colour is present exactly once at each vertex.  These are also 'perfect' colourings.

For an icosahedral figure to admit a 5 colouring the orbit of the vertex disjoint face sets must be tetrahedral.

 
I-5-60-150-60-W7083-(rD-5.4)_I_5col_T
rD-5.4
Hull: Rhombicosidodecahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 7083
 I-5-60-150-60-W6991-(rD-5.5)_I_5col_T
rD-5.5
Hull: Rhombicosidodecahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 6991
I-5-60-150-60-W6761-(rD-5.6)_I_5col_T
rD-5.6
Hull: Rhombicosidodecahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 6761
 I-5-60-150-60-W6156-(sD-12.1)_I_5col_T
sD-12.1
Hull: Snub Dodecahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 6156
I-5-60-150-60-W8280-(tI-5.3)_I_5col_T
tI-5.3
Hull: Truncated Icosahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 8280
 I-5-60-150-60-W8182-(tI-5.4)_I_5col_T
tI-5.4
Hull: Truncated Icosahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 8182
I-5-60-150-60-W8202-(tI-5.5)_I_5col_T
tI-5.5
Hull: Truncated Icosahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 8202


Notes: The face colouring is also chiral, but the chirality of the face colouring is independent of the chirality of the polyhedron.  There could therefore be considered to be 4 'colour isomers' of each polyhedron.  Note that there are only 2 derangements of the colours, one of each chirality.  Swapping any two colours merely moves the colour arrangement from one chirality to the other.

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