From Surf Wiki (app.surf) — the open knowledge base

Faceting

Removing parts of a polytope without creating new vertices

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	Pentagram{5/2}	Star hexagon	Compound2{3}	Decagram{10/3}	Compound2{5}	Compound2{5/2}	Star decagon
[[File:Regular polygon 5.svg	100px]]	[[File:Regular polygon truncation 3 1.svg	80px]]	[[File:Regular polygon truncation 5 1.svg	80px]]
[[File:Regular_star_polygon_5-2.svg	80px]]	[[File:Regular polygon truncation 3 2.svg	80px]]	[[File:Regular star figure 2(3,1).svg	100px]]	[[File:Regular star polygon 10-3.svg	80px]]	[[File:Regular star figure 2(5,1).svg	80px]]	[[File:Regular star figure 2(5,2).svg	80px]]	[[File:Regular polygon truncation 5 2.svg	80px]]	[[File:Regular polygon truncation 5 3.svg	80px]]	[[File:Regular star truncation 5-3 2.svg	80px]]	[[File:Regular star truncation 5-3 3.svg	80px]]

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	150px]]	[[File:Great dodecahedron.png	150px]]	[[File:Small stellated dodecahedron.png	150px]]	[[File:Great icosahedron.png	150px]]

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	100px]]	[[File:Great stellated dodecahedron.png	100px]]	[[Image:Small ditrigonal icosidodecahedron.png	100px]]	[[Image:Ditrigonal dodecadodecahedron.png	100px]]	[[Image:Great ditrigonal icosidodecahedron.png	100px]]	[[File:Excavated dodecahedron highlighted.png	100px]]

Convex	Regular compounds	dodecahedron	five tetrahedra	five cubes	ten tetrahedra
[[File:Dodecahedron.png	100px]]	[[File:Compound of five tetrahedra.png	100px]]	[[File:Compound of five cubes.png	100px]]	[[File:Compound of ten tetrahedra.png	100px]]

History

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

Notes

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

References

[https://www.maa.org/press/periodicals/convergence/mathematical-treasure-wenzel-jamnitzers-platonic-solids ''Mathematical Treasure: Wenzel Jamnitzer's Platonic Solids''] {{Webarchive. link. (2021-10-05 by Frank J. Swetz (2013): "In this study of the five Platonic solids, Jamnitzer truncated, stellated, and faceted the regular solids [...]")

Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

polyhedra polygons polytopes

Want to explore this topic further?

Ask Mako anything about Faceting — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report