# Triambulation

Triambulation denotes a non-symmetric relationship which is seen between some pairs of regular maps. Some regular maps can be "triambulated" to create larger regular maps in the same genus.

To the right is a diagram of a hexagonal face, in black, subdivided in red into four triangular faces. If we have a regular map with hexagonal faces, and we divide each face into four triangles in this way, we may obtain a new regular map.

This can only work if the original map has an even number of hexagons meeting at each vertex, so that at each vertex we can arrange alternate hexagons to have new edges incident to that vertex. Even then, the structure generated is not always a regular map.

Triambulation converts a regular map {6,2n} with 3p vertices, np faces, and 3np edges to a regular map {3,4n} with 3p vertices, 4np faces, and 6np edges, in the same genus.

We can also reverse the process. If we have a regular map with a multiple of four triangular faces meeting at each vertex, we can fuse the triangles together in fours, forming a structure with a quarter as many hexagonal faces meeeting at each vertex. This may be a regular map.

Some examples are
the di-hexagon to the octahedron
S2:{6,4} to S2:{3,8}
S3:{6,4} to the dual Dyck map
S4:{6,6}3,2 to S4:{3,12}