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Facet (mathematics)
From Wikipedia, the free encyclopedia
(Redirected from Facet (geometry))
- For other uses, see facet (disambiguation).
A facet of a simplicial complex is a maximal simplex.
In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:
- A facet of a geometric polyhedron is traditionally any polygon whose corners are vertices of the polyhedron. By extension to higher dimensions, it is any j-tope (j-dimensional polytope) whose vertices are shared by some n-tope (n-dimensional polytope where 0<j<n). To facet a polytope is to find and join such facets to form a new polytope - this process is called facetting or faceting and is the reciprocal process to stellation.
- A facet of an n-polytope is, more recently, an (n-1)-dimensional face or (n-1)-face.
- For example:
- The facets of a polygon are edges. (1-faces)
- The facets of a polyhedron are faces. (2-faces)
- The facets of a polychoron (4-polytope) are cells. (3-faces)
- The facets of a polyteron (5-polytope) are hypercells. (4-faces)
- Exactly two facets meet at any ridge in a polytope. By extension, facet or j-facet is sometimes used to mean any j-dimensional element of a polytope.
- For example:
- Eric W. Weisstein, Facet at MathWorld.
- Olshevsky, George, Facet at Glossary for Hyperspace.