Faceting

Stella octangula as a faceting of the cube

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

New edges of a faceted polyhedron may be created along face diagonals or internal space diagonals. A faceted polyhedron will have two faces on each edge and creates new polyhedra or compounds of polyhedra.

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

Faceted polygons

For example, a regular pentagon has one symmetry faceting, the pentagram, and the regular hexagon has two symmetric facetings, one as a polygon, and one as a compound of two triangles.

Convex		Regular	Quasiregular	Regular compound
Regular pentagon {5}	Regular hexagon {6}	Pentagram {5/2}	Star hexagon	Hexagram {6/2}

Faceted polyhedra

The regular icosahedron can be faceted into three regular Kepler–Poinsot polyhedra: small stellated dodecahedron, great dodecahedron, and great icosahedron. They all have 30 edges.

Convex	Regular stars
icosahedron	great dodecahedron	small stellated dodecahedron	great icosahedron

The regular dodecahedron can be faceted into one regular Kepler–Poinsot polyhedron, three uniform star polyhedra, and three regular polyhedral compound. The uniform stars and compound of five cubes are constructed by face diagonals. The excavated dodecahedron is a facetting with star hexagon faces.

Convex	Regular star	Uniform stars			Vertex-transitive
dodecahedron	great stellated dodecahedron	Small ditrigonal icosi-dodecahedron	Ditrigonal dodeca-dodecahedron	Great ditrigonal icosi-dodecahedron	Excavated dodecahedron

Convex	Regular compounds
dodecahedron	five tetrahedra	five cubes	ten tetrahedra

History

Faceting 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 faceting.
• In 1858, Bertrand derived the regular star polyhedra (Kepler–Poinsot polyhedra) by faceting the regular convex icosahedron and dodecahedron.
• In 1974, Bridge enumerated the more straightforward facetings of the regular polyhedra, including those of the dodecahedron.
• In 2006, Inchbald described the basic theory of faceting diagrams for polyhedra. For a given vertex, the diagram shows all the possible edges and facets (new faces) which may be used to form facetings 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.

References

• 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.
• Alan Holden, Shapes, Space, and Symmetry. New York: Dover, 1991. p.94

External links

• Weisstein, Eric W., "Faceting", MathWorld.
• Olshevsky, George, Faceting at Glossary for Hyperspace.