Facetting

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Stella octangula as a facetting of the cube

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.

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

icosahedron	small stellated dodecahedron	great icosahedron

[edit] History

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.

[edit] 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.

[edit] External links

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