Noble Polyhedra: Icosahedral 6 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.

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

A tight 5-colouring is not possible for these polyhedra due to topological odd-cycle parity collisions.  When a maximal set of vertex-disjoint faces possesses 5-fold dihedral symmetry (D5), the faces partition into orbits of 10, requiring a 6-colour arrangement.

 
I-5-60-150-60-W8013-(rD-7.1)_I_6col_D5
rD-7.1
Hull: Rhombicosidodecahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 8013
 I-5-60-150-60-W5244-(rD-7.2)_I_6col_D5
rD-7.2
Hull: Rhombicosidodecahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 5244
I-5-60-150-60-W7444-(sD-14.1)_I_6col_D5
sD-14.1
Hull: Snub Dodecahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 7444
 I-5-60-150-60-W5000-(sD-15.1)_I_6col_D5
sD-15.1
Hull: Snub Dodecahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 5000
I-5-60-150-60-W4858-(sD-23.1)_I_6col_D5
sD-23.1
Hull: Snub Dodecahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 4858
 I-5-60-150-60-W6831-(sD-4.1)_I_6col_D5
sD-4.1
Hull: Snub Dodecahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 6831
I-5-60-150-60-W8750-(sD-6.1)_I_6col_D5
sD-6.1
Hull: Snub Dodecahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 8750
 I-5-60-150-60-W4731-(sD-8.1)_I_6col_D5
sD-8.1
Hull: Snub Dodecahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 4731
I-5-60-150-60-W7751-(tD-4.1)_I_6col_D5
tD-4.1
Hull: Truncated Dodecahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 7751
 I-5-60-150-60-W6489-(tI-6.1)_I_6col_D5
tI-6.1
Hull: Truncated Icosahedron
Gonality / Valence: 5 / 5
V-E-F: 60-150-60
Genus / Wiener Index: 16 / 6489

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