# Facetting operation

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The **facetting operation** 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[edit | edit source]

For a regular polyhedron P with distinguished generators the k -order facetting, , of P is given by the generators:

The first-order facetting of a polyhedron is itself.

## Examples[edit | edit source]

- 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.

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