Faceting

For other uses, see Facet (disambiguation).

File:CubeAndStel.svg
Stella octangula as a faceting of the cube

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

Faceted polygons

For example, a regular pentagon has one symmetry faceting, the pentagram, and the regular hexagon has two symmetric facetings, one as a polygon, and one as a compound of two triangles.

Pentagon	Hexagon		Decagon
File:Regular polygon 5.svg	File:Regular polygon truncation 3 1.svg		File:Regular polygon truncation 5 1.svg
Pentagram {5/2}	Star hexagon	Compound 2{3}	Decagram {10/3}	Compound 2{5}	Compound 2{5/2}	Star decagon
File:Regular star polygon 5-2.svg	File:Regular polygon truncation 3 2.svg	File:Regular star figure 2(3,1).svg	File:Regular star polygon 10-3.svg	File:Regular star figure 2(5,1).svg	File:Regular star figure 2(5,2).svg	File:Regular polygon truncation 5 2.svg	File:Regular polygon truncation 5 3.svg	File:Regular star truncation 5-3 2.svg	File:Regular star truncation 5-3 3.svg

Faceted polyhedra

The regular icosahedron can be faceted into three regular Kepler–Poinsot polyhedra: small stellated dodecahedron, great dodecahedron, and great icosahedron. They all have 30 edges.

Convex	Regular stars
icosahedron	great dodecahedron	small stellated dodecahedron	great icosahedron
File:Icosahedron.png	File:Great dodecahedron.png	File:Small stellated dodecahedron.png	File:Great icosahedron.png

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
File:Dodecahedron.png	File:Great stellated dodecahedron.png	File:Small ditrigonal icosidodecahedron.png	File:Ditrigonal dodecadodecahedron.png	File:Great ditrigonal icosidodecahedron.png	File:Excavated dodecahedron highlighted.png

Convex	Regular compounds
dodecahedron	five tetrahedra	five cubes	ten tetrahedra
File:Dodecahedron.png	File:Compound of five tetrahedra.png	File:Compound of five cubes.png	File:Compound of ten tetrahedra.png

History

File:Perspectiva Corporum Regularium 22c.jpg

File:Perspectiva Corporum Regularium 28a.jpg

Facetings of icosahedron (giving the shape of a great dodecahedron) and pentakis dodecahedron in Jamnitzer's book

Faceting has not been studied as extensively as stellation.

• In 1568 Wenzel Jamnitzer published his book Perspectiva Corporum Regularium, showing many stellations and facetings of polyhedra.
• In 1619, Kepler described a regular compound of two tetrahedra which fits inside a cube, and which he called the Stella octangula.
• 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

Bibliography

• 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

External links

• Weisstein, Eric W. "Faceting". MathWorld.
• Olshevsky, George. "Faceting". Glossary for Hyperspace. Archived from the original on 4 February 2007.