# Glossary

antipodes
In these pages, the words "antipodes" and "antipodal" are used in a non-standard 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.
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 two-dimensional 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 non-orientable regular maps.

The letter is followed by an integer denoting the genus, then a dot, then an arbitrarily-assigned 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 S3:{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 non-negative 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 non-orientable regular map.

A regular map is cantankerous iff its vertices have an even number of edges and its highest-order holes have the same set of edges as the highest-order Petris polygons but traverse each of them 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 non-orientable 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 non-chiral object's full symmetry group is generally twice the size of its rotational symmetry group. But as objects embedded in non-orientable 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 n-fold 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 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. 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 double-Eulerian circuit is one which traverses every edge of a graph once in each direction, ending on the vertex where it started.

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 two-dimensional 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 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 cannot exceed 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. 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.

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 nth-order hole, or an n-hole, a 1st-order hole is what is usually called a face, a 2nd-order hole is what is usually called a hole, a 3rd-order hole is found by taking the third-sharpest 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 vertex-transitive. g03
isohedral
A map is isohedral if it is face-transitive. g03
isotoxal
A map is isotoxal if it is edge-transitive. 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 two-sided face, also known as a digon.
multiplicity
The vertex-multiplicity of a regular map is the number of edges connecting those pairs of vertices that are connected by at least one edge. The face-multiplicity 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 vertex-transitive and face-transitive but not necessarily edge-transitive. 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 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.

polyhedral map
A polyhedral mapB97 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 self-dual.
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 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. This is described more fully here under rectification.

regular
A "regular map" is an embedding of a graph in a compact 2-manifold such that
• the 2-manifold is partitioned into faces,
• each face has the topology of a disc,
• its rotational symmetry group is dart-transitive: for any two darts, there is a rotation of the whole thing that takes one dart to the other.
Grünbaum defines regular as meaning flag-transitive. 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 non-orientable 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 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. 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. {8,8|4}2 and {8,8|4}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 S2:{8,8} and S3:{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 four-valent vertices. The half-shuriken 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 2-spaces, 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 half-edge 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 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.

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.