Noble Polyhedra: Octahedral 8 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 'O' (Chiral Octahedral) Symmetry

A 5-colour or 6-colour arrangement is not possible for these 5-valent polyhedra due to topological and adjacency constraints.  When a maximal set of vertex-disjoint faces possesses 3-fold cyclic symmetry (C3), the 24 faces partition into orbits of 3, requiring an 8-colour arrangement.

 
O-5-24-60-24-W683-(rC-1.1)_O_8col_C3
rC-1.1
Hull: Rhombicuboctahedron
Gonality / Valence: 5 / 5
V-E-F: 24-60-24
Genus / Wiener Index: 7 / 683
 O-5-24-60-24-W693-(sC-1.1)_O_8col_C3
sC-1.1
Hull: Snub Cube
Gonality / Valence: 5 / 5
V-E-F: 24-60-24
Genus / Wiener Index: 7 / 693
O-5-24-60-24-W832-(sC-3.1)_O_8col_C3
sC-3.1
Hull: Snub Cube
Gonality / Valence: 5 / 5
V-E-F: 24-60-24
Genus / Wiener Index: 7 / 832
 O-5-24-60-24-W540-(sC-5.1)_O_8col_C3
sC-5.1
Hull: Snub Cube
Gonality / Valence: 5 / 5
V-E-F: 24-60-24
Genus / Wiener Index: 7 / 540
O-5-24-60-24-W532-(sC-6.1)_O_8col_C3
sC-6.1
Hull: Snub Cube
Gonality / Valence: 5 / 5
V-E-F: 24-60-24
Genus / Wiener Index: 7 / 532
 O-5-24-60-24-W1019-(sC-6.2)_O_8col_C3
sC-6.2
Hull: Snub Cube
Gonality / Valence: 5 / 5
V-E-F: 24-60-24
Genus / Wiener Index: 7 / 1019
O-5-24-60-24-W901-(tC-1.1)_O_8col_C3
tC-1.1
Hull: Truncated Cube
Gonality / Valence: 5 / 5
V-E-F: 24-60-24
Genus / Wiener Index: 7 / 901

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