Regular Noble Polyhedra: Platonic and Kepler-Poinsot

Notes:  In all cases one representative face 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, for the Kepler-Poinsot polyhedra 'hull' will then display the figure with a framework convex hull, 'solid' returns to the initial view.

Platonic Polyhedra

Td-3-4-6-4-W9-(T-1)
T-1
Hull: Tetrahedron
Gonality / Valence: 3 / 3
V-E-F: 4-6-4
Genus / Wiener Index: 0 / 9
Oh-3-6-12-8-W25-(O-1)
O-1
Hull: Octahedron
Gonality / Valence: 3 / 4
V-E-F: 6-12-8
Genus / Wiener Index: 0 / 25
 Oh-4-8-12-6-W55-(C-1)
C-1
Hull: Cube
Gonality / Valence: 4 / 3
V-E-F: 8-12-6
Genus / Wiener Index: 0 / 55
I-3-12-30-20-W113-(I-1)
I-1
Hull: Icosahedron
Gonality / Valence: 3 / 5
V-E-F: 12-30-20
Genus / Wiener Index: 0 / 113
 I-5-20-30-12-W356-(D-1)
D-1
Hull: Dodecahedron
Gonality / Valence: 5 / 3
V-E-F: 20-30-12
Genus / Wiener Index: 0 / 356


Kepler-Poinsot Polyhedra

I-3-12-30-20-W183-(I-2)
I-2
Hull: Icosahedron
Gonality / Valence: 3 / 5
V-E-F: 12-30-20
Genus / Wiener Index: 0 / 183
 I-5-20-30-12-W934-(D-6)
D-6
Hull: Dodecahedron
Gonality / Valence: 5 / 3
V-E-F: 20-30-12
Genus / Wiener Index: 0 / 934
I-5-12-30-12-W183-(I-4)
I-4
Hull: Icosahedron
Gonality / Valence: 5 / 5
V-E-F: 12-30-12
Genus / Wiener Index: 4 / 183
 I-5-12-30-12-W113-(I-3)
I-3
Hull: Icosahedron
Gonality / Valence: 5 / 5
V-E-F: 12-30-12
Genus / Wiener Index: 4 / 113



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Footnote.

The Wiener Index is more specifically the Geometric Wiener Index, which is the sum of the shortest paths (along the edges) between any two vertices.  I find it a useful metric as apart from some duplication in the Kepler-Poinsot cases it gives a unique value for every noble polyhedron even when truncated to an integer.  In all cases the polyhedra are scaled to a radius of 1 for this calculation. The topological Wiener Index (which is the sum of the number of edges traversed) is described in wikipedia and in Wolfram Mathworld.  The 'geometric' variant used here is an extension of this using the sum of actual edge lengths.  See Mohar and Pisanski: "How to Compute the Wiener index of a Graph" https://users.fmf.uni-lj.si/mohar/Reprints/1988/BM88_JMC2_Pisanki_WienerIndex.pdf