Diagonalisation describes a non-symmetric relationship between some pairs of regular maps.

If a regular map is described by
M:{2q,q}   2F   F   E
(meaning, it is in manifold M, each face has 2q edges, each vertex has q edges, it has V vertices, F faces and E edges), then it may be possible to diagonalise it, by adding a diagonal to each face, to give something described by
M:{q+1,q+1}   2F   2F   E(q+1)/q.
This may or may not be a regular map.

If you have a regular map with twice as many vertices as faces, and want to diagonalise it, draw a major diagonal of each face so as to involve each vertex once. This may prove impossible. If it is possible, the result may or may not be a regular map.

the monodigon S0:{2,1} to give the 2-hosohedron S0:{2,2}
the di-square S0:{4,2} to give the tetrahedron S0:{3,3}
any {6,3} to give a {4,4} which however is not a regular map (exists,
is not regular)
S2:{8,4} fails,
we find that each major diagonal of an octagon joins a vertex to itself
(does not exist)
C5:{8,4} (which is not a regular map) to give C5:{5,5}, which is

For example, we can try:

Other relationships between regular maps
General Index