Hole

An octahedron with one of its holes highlighted in red.

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

Definition

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 $\langle \rho _{0},\rho _{1},\rho _{2}\rangle$ is a polygon:

$\langle \rho _{0},\rho _{1}\rho _{2}\rho _{1}\rangle$

k -holes

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 $\langle \rho _{0},\rho _{1},\rho _{2}\rangle$ is a polygon:

$\left\langle \rho _{0},\rho _{1}(\rho _{2}\rho _{1})^{k-1}\right\rangle$

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

Deep holes

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

For a regular polytope P  of rank n  with distinguished generators $\langle \rho _{0},\rho _{1},\rho _{2},\dots ,\rho _{n-1},\rho _{n}\rangle$ the deep hole of P  is

$\langle \rho _{0},\rho _{1}\rho _{2}\dots \rho _{n-1}\rho _{n}\rho _{n-1}\dots \rho _{2}\rho _{1}\rangle$

Schläfli symbol

The pentagonal duocomb is given the extended Schläfli symbol {4,4∣5} indicating it is a quotient of {4,4} with pentagonal holes.

See also: Schläfli symbol § k-holes

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 $\{4,6\mid 4,8\}$ 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.

See also

• Zigzag walk