Facetting operation

Not to be confused with Faceting.

The great dodecahedron (right) is the second-order facetting, $\varphi _{2}$, of the icosahedron.

The facetting operation $\varphi _{k}$ is an operation which replaces the faces of a polyhedron with its k -holes. If a polyhedron has a triangular vertex figure, then its (second-order) facetting will be itself.

Definition

For a regular polyhedron P  with distinguished generators $\langle \rho _{0},\rho _{1},\rho _{2}\rangle$ the k -order facetting, $\varphi _{k}$, of P  is given by the generators:

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

The first-order facetting of a polyhedron is itself.

Examples

• The second-order facetting of the icosahedron is the great dodecahedron.
• The second-order facetting of the halved mucube is the tetrahelical triangular tiling.
• The second-order facetting of the tetrahedron is the tetrahedron.
• The second-order facetting of the triangular tiling is the hexagonal tiling.
• The second-order facetting of the mucube is the cube.
• The first-order facetting of the icosahedron is the icosahedron.