Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof Wisconsin - Green Bay
The polyhedra on these pages were enumerated using plantri.exe, a graph enumeration program by Gunnar Brinkmann of the University of Ghent and Brendan McKay of the Australian National University, and then drawn using software written by me. Plantri is intended for graph theory use, and while polyhedra and graph theory overlap considerably, the two are not identical. The example shown here illustrates some of the problems from using plantri, plus explains the format used in the data tabulation.
Index = 170; Topology = 333333455; NFace = 9; NVert = 9; NEdge = 16
Faces:
bcde aef agd achie adifb beg cfh dgi dhe
Face Topology: 3355 453 435 43335
45333 353 333 533 535
Vertices: abfgc acd ade aeb bef cghd dhi die eihgf
Vertex Topology: 43333 435 455 453 353 3335 533 535 53333
Face-Vertex Adjacency:
bAcBdCeDb aDeEfAa aAgFdBa aBcFhGiHeCa aCdHiIfEbDa bEeIgAb cAfIhFc dFgIiGd dGhIeHd
Edges: ab ac ad ae be bf cg cd dh di de ei ef fg gh hi
Faces are designated by lower case letters, edges (only a few shown) by the letters of the faces intersecting along that edge, and vertices in capital letters. Labels are in order of the listing and have no other significance. This is a Schlegel net, a representation of a polyhedron flattened into a plane. Imagine the polyhedron resting on one face (referred to from now on as the base) and flattened. The base face can either be pictured as beneath the net, or alternatively, as the infinite "face" exterior to the net. In the example here, face d is the base.
Note first of all that face a is not one of the pentagonal faces. For visualizing
polyhedra and their relationships, it's desirable to use the largest face as
the base (with a few exceptions), and whichever face is selected as the base,
use that face consistently as the base for all polyhedra of the same type. Plantri
lists results in a way that makes sense for graph problems (and enumerating
all 2606 enneahedra would be all but impossible without it) but doesn't necessarily
group similar polyhedra together or even use the same face for a base.
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Created 17 June 2014, Last Update 17 January 2014