 antipodes

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

A "blade" comprises a vertex, an edge incident with that vertex, and a face incident
with both. It is the same as a flag but restricted to
twodimensional things.
 C&D

In 2001, Conder and Dobcsányi listed "all regular maps of small genus"
C96, and assigned them unique identifiers. These
identifiers start with R for refexible orientable regular maps, C
for chiral orientable regular maps, and N for nonorientable regular maps.
The letter is followed by an integer denoting the genus, then a dot, then an
arbitrarilyassigned integer, and finally, in the case of dual pairs, a prime
to indicate the member of the pair with larger faces. Thus what I have called
S^{3}:{7,3} has the C&D identifier
R3.1'.
These identifiers have the huge advantage that they are unique. Other naming
systems for regular maps lack uniqueness, see Schläfli
symbol below.
 cantankerous

"Cantankerous" was coined by Stephen Wilson in W89
to designate a certain class of nonorientable regular map.
 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 selfdual 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 nonchiral 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.
 dart

A "dart" comprises a vertex and an edge incident with that vertex. A dart
comprises two blades.

degenerate

A term applied to things 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 S^{0}:{4,3} is the octahedron S^{0}:{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 doubleEulerian 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. In twodimensional structures, flags are
also known as blades.
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 S^{n}, with the sphere being S^{0}, the torus
S^{1}, etc.; and that of a nonorientable surface by
C^{n}, with the projective plane being C^{1}, the
Klein bottle C^{2}, 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.
 halfedge

The term "halfedge" 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 halfedges per edge.
A regular map is edgetransitive if any edge can be mapped to any other edge.
It is halfedgetransitive 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 (nfold) 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 secondsharpest 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.
 multiplicity

The vertexmultiplicity of a regular map is the number of edges connecting
those pairs of vertices that are connected by at least one edge.
The facemultiplicity of a regular map is the number of edges shared by
those pairs of faces that share at least one edge.

pathological

A term applied to structures 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. Also, the Petrie dual of the dual of the Petrie dual is the dual
of the Petrie dual of the dual.

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 structure in 3space, 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 a 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 continuously 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
nonintegers. {5/2,5} is the small stellated dodecahedron, with five
pentagrams meeting at each vertex, and {3,5/2} is the great icosahedron,
with "twoandahalf" 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
600cell 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 genus0 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.
 {4,64} 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 twonumber subscripts specifies one
of these. Unfortunately I use a notation different from that used by
ARM, although we both use bracketed twonumber
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. {8,84}_{2}
and {8,84}_{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 S^{2}:{8,8} and
S^{3}:{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 2spaces,
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 facetransitive if, for any two faces a,
b there is some operation of its symmetry group which maps a
to b. Likewise for vertextransitive, edgetransitive,
halfedgetransitive, flagtransitive, etc.