# Hole

The **holes** of a polytope are a particular type of pseudo-face.

## Definition[edit | edit source]

A hole is a path along the edges of a map such that it always takes the second rightmost (or alternatively always takes the second leftmost) edge.

In terms of distinguished generators, the hole of a regular polyhedron P with distinguished generators is a polygon:

## k -holes[edit | edit source]

k -holes generalize the notion further.

A k -hole is a path along the edges of a map such that it always takes the k th rightmost (or alternative always takes the k th leftmost) edge.

In terms of distinguished generators, the k -hole of a regular polyhedron P with distinguished generators is a polygon:

A 1-hole is therefore a face of the polyhedron with a 2-hole being the normal hole.

## Deep holes[edit | edit source]

This section needs expansion. You can help by adding to it. |

Deep holes are a generalization of holes to higher rank polytopes.

For a regular polytope P of rank n with distinguished generators the deep hole of P is

## Schläfli symbol[edit | edit source]

A common extension to Schläfli symbols allows the symbol to indicate the size of its k -holes when its different from that of the universal cover. This is done with a | character so {p,q∣h} indicates a polytope with Schläfli type {p,q} and holes of size h . Higher order holes can be added following the 2-hole. For example is a quotient of {4,6} with square 2-holes and octagonal 3-holes. Since 1-holes are just faces, which are already indicated the first number after the | is the 2-hole instead.