Diagonalisation describes a nonsymmetric relationship between some pairs of regular maps.
If a regular map is described byIf 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 2hosohedron S0:{2,2}  
the disquare 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: