# Duals

Duality is a symmetric relationship between pairs of regular maps of the same genus. Every regular map has a dual. In some cases its dual is itself; it is then described as "self-dual".

If a regular map is described by

M:{p,q}r   V   F   E
(meaning, it is in manifold M, each face has p edges, each vertex has q edges, each Petrie polygon has r edges, it has V vertices, F faces and E edges), then its dual exists, is a regular map, and is described by
M:{q,p}r   F   V   E.

This relationship is symmetric: if A is the dual of B, then B is the dual of A.

For example, the dual of the cube is the octahedron. A regular map can be self-dual: an example is the tetrahedron.

ARM denotes duality by δ.

If you have a regular map and want to construct its dual,

• replace each face by a vertex
• replace each vertex by a face
• rotate each edge through a right angle

### The name "dual"

Dual polyhedra are so called because they occur in pairs, each having the same relationship to the other. The term extends naturally to regular maps, and to "irregular maps".