Noble Polyhedra: Octahedral 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.

Structural Excess Minimum Injective Face-Colourings of Noble Polyhedra with 'O' (Chiral Octahedral) Symmetry

A tight 5-colouring is not possible for these 5-valent polyhedra due to topological parity constraints.  When a maximal set of vertex-disjoint faces possesses 4-fold cyclic symmetry (C4), the 24 faces partition into orbits of 4, requiring a 6-colour arrangement.

 
O-5-24-60-24-W806-(sC-2.1)_O_6col_C4
sC-2.1
Hull: Snub Cube
Gonality / Valence: 5 / 5
V-E-F: 24-60-24
Genus / Wiener Index: 7 / 806
 O-5-24-60-24-W578-(sC-4.1)_O_6col_C4
sC-4.1
Hull: Snub Cube
Gonality / Valence: 5 / 5
V-E-F: 24-60-24
Genus / Wiener Index: 7 / 578

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