This is the most widespread of constructions, and applies only to regular
figures.

Notion

A regular polyhderon is made up from regular polygons of one kind, with
a constant number at a corner. A regular polychora is done in the same manner,
with polyhedra of one kind, the same at each edge.

Th. Gosset uses the vertex figure, building them up, eg "pentagon-edged
tetrahedron"

Notation

The notation for the polygons is simply the number of sides it has, eg
3 for a triangle.

For a polyhedron the notation is to use two numbers like {p,q}.
This symbol can be read as p-gons, q at a corner. So, the
dodecahedron is {5,3}, this means pentagons, three at a corner.

For the polychora, the notation uses three numbers, eg {p,q,r}.
This is read as {p,q}, r at an edge.

Comments

There are not a lot of regular figures past two dimensions. In three
dimensions, there are just five (the Platonic figures), and in four, there
are just six. Thereafter, there are just three.

The uniform figures are made by applying the Stott-style variations
to the regular figures.