Faceting

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

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.

For example the stellated dodecahedron and great icosahedron are two facetings of the icosahedron:

icosahedron	small stellated dodecahedron	great icosahedron

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.

External links

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