Glossary

antipodes

In these pages, the words "antipodes" and "antipodal" are used in a non-standard sense, as defined in the page Antipodes.

cantellation

Cantellation is the process which takes a regular polyhedron and shaves down the vertices so as to form new faces. It is described in the Wikipedia article cantellation.

It is of interest to us because the uniform cantellation of a self-dual regular map yields another regular map. Each vertex becomes a face, each face remains a face, and each edge becomes a vertex. If the original was {p,p} with q vertices, q faces and pq/2 edges, then the new regular map is {p,4} with pq/2 vertices, 2q faces, and pq edges. It is still in the same manifold.

chiral

An object is chiral if it has no mirror symmetry.

A chiral regular map in the torus is shown to the right.

A chiral object's full symmetry group is the same as its rotational symmetry group; a non-chiral object's full symmetry group is twice the size of its rotational symmetry group.

chromatic number

The chromatic number of a regular map or other graph is the number of colours needed to colour its vertices so that no two vertices with a common edge are the same colour.

chromatic index

The chromatic index of a regular map or other graph is the number of colours needed to colour its edges so that no two edges meeting at a vertex are the same colour.

degenerate

A term applied to things which one does not like, maybe such as the regular map shown to the right. Cf. pathological.

diameter

The diameter of a regular map or other graph is the greatest number of edges it can be necessary to traverse to reach one vertex from another.

dual

The dual of a polyhedron or other regular map can be formed from it by replacing each face by a vertex, replacing each vertex by a face, and rotating each edge through a right angle about its centre while keeping it in the plane of the manifold. Thus the dual of the cube S⁰:{4,3} is the octahedron S⁰:{3,4}.

Duality is a symmetric relation: if A is the dual of B then B is the dual of A, hence the name.

Eulerian

An Eulerian path is one which traverses every edge of a graph exactly once, as in the bridges of Königsberg.

An Eulerian circuit is one which traverses every edge of a graph exactly once, ending on the vertex where it started.

A double-Eulerian circuit is one which traverses every edge of a graph exactly once in each direction, ending on the vertex where it started.

See also Hamiltonian.

flag

"Flag" is a general concept used in the study of polytopes. For a polyhedron, it comprises a vertex, an edge bounded by that vertex, and a face bounded by that edge. In general for a polytope, it goes on to comprise a polyhedron bounded by that face, a polytope bounded by that polyhedron, etc. The concept extends naturally to regular maps.

A polyhedron or other regular map has four flags for each edge.

genus

The genus of an orientable manifold is the number of "handles" you need to stitch onto a sphere to make it. For instance, the sphere has genus 0 and the torus has genus 1.

In these pages, the genus of an orientable surface may be designated by Sⁿ, with the sphere being S⁰, the torus S¹, etc.; and that of a non-orientable surface by Cⁿ, with the projective plane being C¹, the Klein bottle C², etc.

The genus of a group is the least genus of any manifold on which its Cayley diagram can be drawn without the arcs crossing.

girth

The girth of a graph is the number of edges in the smallest cycle. For a regular map it may be the number of edges of each face.

half-edge

The term "half-edge" is used here for a pair of adjacent flags. These flags may share a vertex and an edge, or share an edge and a face, either way there are two half-edges per edge.

A regular map is edge-transitive if any edge can be mapped to any other edge. It is half-edge-transitive if any edge can be mapped to any other edge with the edge either way round.

Hamiltonian

A Hamiltonian path is one which visits every vertex of a graph exactly once, using no edge more than once.

A Hamiltonian circuit is one which visits every vertex of a graph exactly once, using no edge more than once, and ending on the vertex where it started.

A double (n-fold) Hamiltonian circuit is one which visits every vertex of a graph exactly twice (n times), using no edge more than once, and ending on the vertex where it started.

See also Eulerian.

hole

A hole is a polygon found in a regular map by travelling along its edges, taking the second-sharpest left at each vertex. This is only of interest if the regular map has more than three edges meeting at each vertex.

An octahedron with a hole highlighted in red is shown to the right.

See also Petrie polygon.

pathological

A term applied to structures which one strongly dislikes, maybe including the regular map shown to the right. Cf. degenerate.

Petrie dual

The Petrie dual of a polyhedron or other regular map is the regular map whose vertices and edges correspond to the vertices and edges of the original, and whose faces correspond to the Petrie polygons of the original. It is sometimes shortened to the portmanteau word "Petrial".

Petrie duality is a symmetric relation: if A is the Petrie dual of B then B is the Petrie dual of A.

Petrie polygon

A Petrie polygon is a polygon found in a polyhedron or other regular map by travelling along its edges, turning sharp left and sharp right at alternate vertices.

A cube with a Petrie polygon highlighted in red is shown to the right.

If you have embedded the polygon in 3-space, you will find that its Petrie polygons are skew.

The concept of holes and Petrie polygons can be generalised, as described in holes and Petrie polygons.

pyritification

Pyritification is the process that converts a regular map into a larger regular map by dividing up each of its faces in the same way. It is explained in the page pyritification

quality

The quality of a regular map is an arbitrary measure of its aesthetic value. It has no mathematical significance or justification. It is currently defined as follows: if the Schläfli symbol of the polyhedron is {G,H}, and it has E edges, then its quality is E/max(G,H). Thus the quality of a dodecahedron, with Schläfli symbol {5,3} and 30 edges, is 6.

In every genus where I have identified the regular maps, the highest quality is a multiple of 3. I expect there is a simple reason for this.

rotation

Rotation is used in these pages for the operation of moving a regular map while keeping it embedded in its manifold.

Schläfli symbol

A simple Schläfli symbol has the form {G,H}. The first number specifies the number of edges per face, the second number specifies the number of faces meeting at each vertex. Thus the Schläfli symbol for the cube is {4,3}.

A Schläfli symbol can specify a stellated polyhedron, by using non-integers. {5/2,5} is the small stellated dodecahedron, with five pentagrams meeting at each vertex, and {3,5/2} is the great icosahedron, with "two-and-a-half" triangles meeting at each vertex, i.e. its vertex figures are pentagrammal. Stellated polyhedra are not considered in these pages.

This can be extended to polytopes. The Schläfli symbol for the 600-cell is {3,3,5}. The 3,3 specifies tetrahedron; the 5 specifies that 3 of these meet at each edge. But in these pages we are only concerned with polyhedra, having two main numbers in the Schläfli symbol.

If we are only concerned with genus-0 regular maps (regular polyhedra), a simple Schläfli symbol of the form {G,H} is sufficient to specify a polyhedron. But if we look more broadly, more numbers may be used, to disambiguate. Here are some examples.

• {4,6|4} specifies a polyhedron with six squares meeting at each vertex. The 4 after the | specifies that its holes have four edges. This is an infinite polyhedron of infinite genus: it can be seen in the first picture in the Wikipedia article Regular skew polyhedron.
• {7,3}₈ specifies a regular map with three heptagons meeting at each vertex. The single subscript 8 specifies that its Petrie polygons have eight edges. We have a picture of {7,3}₈, it has genus 3.
• {6,3}_(2,0) specifies a regular map with three hexagons meeting at each vertex. There are infinitely many regular maps designated by {6,3}, all of genus 1. The bracketed two-number subscripts specifies one of these. Unfortunately I use a notation different from that used by ARM, although we both use bracketed two-number subscripts: the difference is explained in the page Regular Maps on the Torus.

However, these various enhancements to Schläfli symbols are not enough to make them unambiguous. {8,8|4}₂ and {8,8|4}₂ are different regular maps, of genus 2 and 3 respectively, the latter being the double cover of the former.

Therefore in these pages I disambiguate Schläfli symbols with a prefix to indicate the genus where necessary. E.g. I will designate those two polyhedra as S²:{8,8} and S³:{8,8}. Where I omit the prefix it should be clear from the context.

skew polygon

A skew polygon is a polygon whose vertices are not coplanar. This concept is only meaningful when the structure has been embedded in a space of more than two dimensions. As these pages are concerned only with polyhedra in 2-spaces, and not their embedding in higher spaces, they do not use the concept.

transitive

A group which permutes a set is said to be transitive on that set if, for any two members a, b of the set there is some operation of the group which maps a to b.

Thus a map is said to be face-transitive if, for any two faces a, b there is some operation of its symmetry group which maps a to b. Likewise for vertex-transitive, edge-transitive, half-edge-transitive, flag-transitive, etc.