Noble Polyhedra.  Fissary examples.

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, 'hull' will then display the figure with a framework convex hull, 'solid' returns to the initial view.

The dual to a noble polyhedron is generally another noble polyhedron.  However in the four cases in the left column below, the polyhedra contain faces which are coplanar but which do not share an edge.  As a result their duals contain distinct vertices which coincide.  Hill calls these 'fissary' polyhedra.  They are shown in the right column below next to their dual partners.

Ih-5-120-300-120-W42968-(gD-19.1)
gD-19.1
Hull: Great Dodecahedron
Gonality / Valence: 5 / 5
V-E-F: 120-300-120
Genus / Wiener Index: 31 / 42968
 Ih-5-60-300-120-W3687-(rD-F)
rD-F
Hull: Rhombicosidodecahedron
Gonality / Valence: 5 / 10
V-E-F: 60-300-120
Genus / Wiener Index: 61 / 3687
Ih-5-120-300-120-W40941-(gD-28.1)
gD-28.1
Hull: Great Dodecahedron
Gonality / Valence: 5 / 5
V-E-F: 120-300-120
Genus / Wiener Index: 31 / 40941
 Ih-5-60-300-120-W3680-(tI-F)
tI-F
Hull: Truncated Icosahedron
Gonality / Valence: 5 / 10
V-E-F: 60-300-120
Genus / Wiener Index: 61 / 3680
I-4-20-120-60-W286-(D-4)
D-4
Hull: Dodecahedron
Gonality / Valence: 4 / 12
V-E-F: 20-120-60
Genus / Wiener Index: 21 / 286
 I-12-60-120-20-W8236-(D-F1)
D-F1
Hull: Dodecahedron
Gonality / Valence: 12 / 4
V-E-F: 60-120-20
Genus / Wiener Index: 21 / 8236
I-3-20-90-60-W302-(D-5)
D-5
Hull: Dodecahedron
Gonality / Valence: 3 / 9
V-E-F: 20-90-60
Genus / Wiener Index: 6 / 302
 I-12-60-90-20-W9258-(D-F2)
D-F2
Hull: Dodecahedron
Gonality / Valence: 12 / 3
V-E-F: 60-90-20
Genus / Wiener Index: 6 / 9258


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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 polyehdra 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

'The terms 'fissary' and 'legit' were coined by Jonathon Bowers (see polytope.net/hedrondude/glossary.htm)