In geometry, facetting (also spelled 'faceting') is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices. Table of Geometry, from the 1728 Cyclopaedia. ... Look up polygon in Wiktionary, the free dictionary. ... A polyhedron is a geometric shape which in mathematics is defined by three related meanings. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...

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. Look up Dual in Wiktionary, the free dictionary A dual is a pair or a grouping of two. ... Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ... Look up convex in Wiktionary, the free dictionary. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the others. ...

History

Facetting has not been studied as extensively as stellation. Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ...

In 1858, Bertrand derived the regular star polyhedra (Kepler-Poinsot polyhedra) by facetting the regular convex icosahedron and dodecahedron. Joseph Louis François Bertrand (March 11, 1822 - April 5, 1900, born and died in Paris) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, and thermodynamics. ... In geometry, the term star polyhedron does not seem to have been propely defined, even though it is in common use. ... A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids). ... In ordinary English, regular is an adjective or noun used to mean in accordance with the usual customs, conventions, or rules, or frequent, periodic, or symmetric. ... Look up convex in Wiktionary, the free dictionary. ... An icosahedron noun (plural: -drons, -dra ) is a polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ...

In 1974, Bridge enumerated the more straightforward facettings of the regular polyhedra, including those of the dodecahedron. In ordinary English, regular is an adjective or noun used to mean in accordance with the usual customs, conventions, or rules, or frequent, periodic, or symmetric. ... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ...

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. Look up Dual in Wiktionary, the free dictionary A dual is a pair or a grouping of two. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...

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.

Joseph Louis François Bertrand (March 11, 1822 - April 5, 1900, born and died in Paris) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, and thermodynamics. ...

External links

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