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
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
- 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
Return to Symmetry Index
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Created 17 June 2014, Last Update 17 January 2014
Not an official UW Green Bay site