 antipodes

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

A bipartite graph is one whose chromatic number is 2.
Its vertices can be partitioned into two sets such that every edge connects
one vertex from each set.
 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.
In these pages, I have extended these identifiers to regular maps of nonnegative
Euler characteristic; so, for example, the cube has the identifies R0.2′.
 cantankerous

"Cantankerous" was coined by Stephen Wilson in W89
to designate a certain class of nonorientable regular map.
A regular map is cantankerous iff it is in a nonorientable surface, its vertices
have an even number of edges v, and the orderv/2 hole (and hence the
orderv/2 Petrie polygon) has an even number of distinct edges, each visited
twice.

canvas

In these pages, a canvas is a "blank" diagram on which regular maps, of the
genus of that canvas, may be portrayed.
To the right is an example: a canvas for nonorientable genus 4. The small round
pink things are crosscaps.

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 generally twice the size
of its rotational symmetry group. But as objects embedded in nonorientable
surfaces can be reflected by translating them, their full symmetry groups are
the same as their rotational symmetry groups.
 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.
 cover

A double cover of a regular map is another regular map having twice as many
vertices, faces, and edges. An nfold cover of a regular map is another
regular map having n times as many vertices, faces, and edges. The
covering map will be in a surface with n time the Euler
characteristic of the surface of the map which it covers. This is
explained in the page double cover.

crosscap

Crosscaps in these pages are shown as on the right.
 dart

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

Diagonalisation is a process which can be applied to any regular map that has
twice as many vertices as faces. It may yield another regular map, it may yield
something that is not quite a regular map, or it may prove impossible. The
process is described in the page diagonalisation.
 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.
 digon

A digon is a face having two vertices and two edges.
Regular maps with digonal faces only exist in the surfaces of positive
Euler characteristic: the sphere and the projective plane.
 double

A double Petrie polygon, hole, or other circuit is one which traverses a set
of edges twice each before returning to its starting point with the same phase
it started in.
 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. See dual for more details.
 Euler characteristic

The Euler characteristic of a regular (or irregular) map is its number of faces plus
number of vertices minus number of edges. It is the same for all regular maps in a
particular surface (and for the irregular maps, so long as the faces all have the
topology of a disk).
See Wikipedia for
more information.
 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 once in each direction, ending on the vertex where it started.
See also Hamiltonian.
 face adjacency graph

A graph showing, for a regular map or similar structure, which of its faces
adjoin which. It is the underlying graph of the dual.
 face multiplicity

In any one regular map, the number of edges that can be shared by any pair
of faces can take at most two values, 0 and mF. mF is known as the "face
multiplicity". See also vertex multiplicity.
 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.
 full symmetry group

The set of all the rotations and reflections which can be applied
to an object so as to leave its appearance unchanged forms a group.
This is called its full symmetry group. See also "rotational
symmetry group".
If the object is a regular map, then its full symmetry group has four
times as many elements as the object has edges.
 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
nonorientable 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 cannot exceed 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.
In these pages, we regard an octahedron as having six distinct holes.
Some writers identify pairs of holes that comprise identical sets of
edges, and therefore consider that an octahedron has only three holes.
See also Petrie polygon.
When we write of an nthorder hole, or an nhole, a 1storder
hole is what is usually called a face, a 2ndorder hole is what is usually
called a hole, a 3rdorder hole is found by taking the thirdsharpest left
at each vertex, etc.
 hosohedron

A hosohedron is any regular map, in the sphere, with exactly two vertices.
 isogonal

A map is isogonal if it is vertextransitive.
g03
 isohedral

A map is isohedral if it is facetransitive.
g03
 isotoxal

A map is isotoxal if it is edgetransitive.
g03
 lucanicohedron

A lucanicohedron is a quasiregular map, derived from a
hosohedron. The name derives from a fancied resemblance to a string, or
loop, of sausages: The Greek for sausage is
λουκάνικο.
 lune

A lune is a twosided face, also known as a digon.
 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.
 noble

A map is noble if it is vertextransitive and facetransitive
but not necessarily edgetransitive. g03
 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. See
Petrie dual for more details.

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.

polyhedral map

A polyhedral map^{B97}
is such that the intersection of two distinct faces is one of
 empty
 one vertex
 one edge
Thus for example
{4,4}_{(2,1)},
shown to the right, is not a polyhedral map: the intersection of two
of its distinct faces is an edge and a vertex.
In these pages, regular maps which are known to be polyhedral maps are
indicated by , and those which are known
not to be polyhedral maps are indicated by ,
 portray

These pages portray regular maps as sets of faces, vertices and edges
in diagrams which generally also have pink sewing
instructions, showing how the diagram is to be assembled into the
required manifold. Such a diagram, before it has had a regular map
portrayed on it, is described as a canvas.
 presentation

A presentation of a group is a way of specifying it by means of generators
and relations. You can learn more from the Wikipedia article on
group
presentation. The presentations given in these page for full symmetry
groups are triangle
groups, taken from the work of Professor Marston Conder
c09.
 pyritify

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.
 quasiregular

In these pages, a quasiregular map is like a regular map except that it
is not quite regular, having faces of two shapes. Quasiregular maps are
analogous to
quasiregular
polyhedra. One can obtained by rectification of any
regular map which is not selfdual.
 rectify

rectification 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
rectification.
It is of interest to us because the uniform rectification 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. This is described more fully here under
rectification.
 regular

A "regular map" is an embedding of a graph in a compact 2manifold such
that
 the 2manifold is partitioned into faces,
 each face has the topology of a disc,
 its rotational symmetry group is darttransitive:
for any two darts, there is a rotation
of the whole thing that takes one dart to the other.
Grünbaum defines regular as meaning flagtransitive. g03
 replete

A regular map is said to be replete if there is a rotation that fixes
one face but not all faces, and a rotation that fixes one vertex but
not all vertices.
 rotation

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

The set of all the rotations which can be applied to an object so
as to leave it appearance unchanged forms a group. This is called
its rotational symmetry group. See also "full
symmetry group".
If the object is a regular map embedded in an orientable manifold,
then its rotational symmetry group has twice times as many elements
as the object has edges. If the object embedded in a nonorientable
manifold, then its rotational symmetry group is the same as its full
symmetry group.
 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. Where ARM writes {6,3}_{(s,0)} we write
{6,3}_{(s,s)}, and where ARM writes {6,3}_{(s,s)} we
write {6,3}_{(0, 2s)}.
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 a 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.
 Schlegel diagram

A Schlegel diagram portrays a regular map or other structure in a sphere,
on a finite flat diagram. It is a projection from the whole sphere to one
of the faces of the structure, using a point a short distance outside that
face.
In these pages, almost all the diagrams showing regular
maps on the sphere are Schlegel diagrams.
 shuriken

The full shuriken of a regular map is another map created by replacing all
its vertices by crosscaps and all its edges by fourvalent vertices.
The halfshuriken of a regular map is another map created by replacing
alternate vertices by crosscaps. This is explained more fully in the page
on shurikens.
 side

A polyhedron does not have things called "sides", it has vertices, edges,
and faces. It is best to avoid using the term "side" in the context of a
polyhedron or other regular map, as some people use it to mean "edge" and
others "face". However, I use the term "side" for the "cut edges" of the
canvases on which regular maps can be portrayed.
 singular

A regular map is said to be singular if no pair of vertices has more than
one common edge, and no pair of faces has more than one common edge.
 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 with their embedding in higher spaces, they do not use the concept.
 split

Splitting is a process that converts a regular map into a larger
regular map by replacing each vertex by a pair of vertices. It is
explained in the page splitting.
 stellate

Some regular maps can be stellated, by a process analogous to the stellation
of polyhedra. The process is described, and examples listed, at
stellation.
 symmetric graph

A symmetric graph is one which is halfedge transitive. There is more
information in the Wikipedia article
"Symmetric graph".
 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.
 trivial

A regular map is said to be trivial if its faces have 2 edges, or
if its vertices have 2 edges, or if its Petrie
polygons have 2 edges.
 underlying graph

A regular map is an embedding of a symmetric graph in a surface.
The embedded graph is the "underlying graph" of the regular map.
 vertex multiplicity

In any regular map, the number of edges that can connect any pair of
vertices can take at most two values, 0 and mV. mV is known as the
"vertex multiplicity". See also face multiplicity.