These pages describe "regular maps", objects lying somewhere between
**graphs**, and
**polyhedra**.
This page aims to show the relationship between graphs, regular maps, and polyhedra.

A graph is a collection of vertices and lines, joined up in a specified way, but not "in" any kind of space. A regular map is a graph that has been constrained to fit in a 2-manifold, which it partitions into faces. A polyhedron is a regular map whose 2-manifold is the 2-sphere, which has been embedded in Euclidean 3-space, so as to impose a nice metric on it; the polyhedron can then be regarded as a 3-dimensional solid object.

These differences are described further in the table below.

Graphs | Regular Maps | Polyhedra (and Polytopes) | |
---|---|---|---|

components | Vertices and edges | Vertices, edges, and faces | Vertices, edges, and faces (and n-faces) |

embedding | Not embedded | Embedded in a 2-manifold | Embedded in Euclidean 3-space (n-space) |

each one is defined by | An incidence table of vertices and edges | Incidence tables of vertices, edges and faces; and an indication of how these can be threaded through the tunnels of the manifold. | A set of vertices in Euclidean 3-space (n-space).Simpler polyhedra (polytopes) can be defined as the convex hull of such a set of vertices. Star polyhedra (polytopes) require something more, which may be incidence tables. |

metric | No metric | It is sometimes possible to impose a metric | The Euclidean metric of the embedding space |

how to draw them on paper or computer screen | How you like, with the arcs crossing as you wish | The arcs can't cross. The diagram will include "sewing instructions" appropriate to the manifold | Typically by a projection of an opaque space-filling object |

a typical regular specimen |

The "Polyhedra" column also mentions polytopes, in brackets. Could the middle column similarly mention 3-manifolds, and higher manifolds?

It could, and I am sure the non-sphere analogues of polytopes are interesting. But I suspect that listing polytope-analogues for 3-manifolds other than the 3-sphere would be very difficult. Even listing 3-manifolds is difficult.

Some regular maps

Copyright N.S.Wedd 2009