## English

### Verb

faceting- present participle of facet

In geometry, facetting (also
spelled 'faceting') is the process of removing parts of a polygon, polyhedron or polytope, without creating any
new vertices.

Facetting is the reciprocal or dual process to stellation. For every
stellation of some convex
polytope, there exists a dual facetting of the dual
polytope.

Facetting has not been studied as extensively as
stellation.

In 1619, Kepler
described a regular
compound
of two tetrahedra
which fits inside a cube, and which he called the Stella
octangula. This seems to be the first known example of
facetting.

In 1858,
Bertrand derived the regular star
polyhedra (Kepler-Poinsot
polyhedra) by facetting the regular convex icosahedron and dodecahedron.

In 1974, Bridge
enumerated the more straightforward facettings of the regular polyhedra, including
those of the dodecahedron.

In 2006, Inchbald
described the basic theory of facetting diagrams for polyhedra. For
some vertex, the diagram shows all the possible edges and facets
(new faces) which may be used to form facettings of the original
hull. It is dual to the
dual
polyhedron's stellation diagram, which shows all the possible
edges and vertices for some face plane of the original core.

- Bertrand, J. Note sur la théorie des polyèdres réguliers, Comptes rendus des séances de l'Académie des Sciences, 46 (1858), pp. 79-82.
- Bridge, N.J. Facetting the dodecahedron, Acta crystallographica A30 (1974), pp. 548-552.
- Inchbald, G. Facetting diagrams, The mathematical gazette, 90 (2006), pp. 253-261.

faceting in French: Facettage

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