Explanation of Topological Data for Polyhedra

Steven Dutch, Natural and Applied Sciences, University of 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.

Explanation of Data Listing

Since the raw plantri listing doesn't group similar polyhedra, the listing was sorted by number of vertices and face topology. Although this approach groups polyhedra with the same types of faces, the listing still doesn't group similar polyhedra very well, especially for classes with large numbers of polyhedra. The illustrations attempt to do so, and are labeled by their order in the sorted list. The numbering is a label, nothing more. Numbering starts over again for each number of verices, so for completeness, polyhedron 385 with 12 vertices might be best labeled 12-385. Also, zero is a perfectly good number and following standard computer practice, numbering begins at zero.
The example here, 333333455, has 6 triangular faces, one quadrilateral anf two pentagons.
Number of faces (9)
Number of vertices (9)
Number of edges (16, because F + V = E + 2)
List of faces adjoining each face. First face in the list is "a," second is "b," and so on. Face a is adjoined by b, c, d and e, face b by a, e, and f, and so on. Note that faces that meet only at a vertex (like f and a, or g and a) are not listed.
Face Topology
List of face types adjoining each face. Order of faces is identical to the Faces list. Face a is bounded by two triangles (b and c) and two pentagons (d and e).
List of faces adjoining each vertex. Vertex A is the intersection of a, b, f, g and c.
Vertex Topology
List of face types adjoining each vertex. Vertex A is the intersection of four triangles and a quadrilateral.
Face-Vertex Adjacency
A list of faces and vertices adjoining each face. Going around face A we find face b, vertex A, face c, vertex B, then d, C, e, D, and back to b. Going around face b, we find face a, vertex D, then e, E, f, A, and finally a again. (The program was originally intended to truncate the listing without repeating, but it's nice to see the relationship between the closing faces and vertices. It's not a bug, it's a feature.)
Only a few representative examples are shown. Faces a and b meet along edge ab, etc.

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Created 17 June 2014, Last Update 17 January 2014

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