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.
|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|
|dodecahedron||five tetrahedra||five cubes||ten tetrahedra|
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.
- 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