Stella octangula as a faceting of the cube

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, faceting (also spelled facetting) is the process of removing parts of a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all o ...

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

, 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 In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specif ...

''. For every stellation of some

convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the w ...

, there exists a dual faceting of the

dual polytope In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other ...

Faceted polygons

For example, a regular

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be sim ...

has one symmetry faceting, the

pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...

, and the regular

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

has two symmetric facetings, one as a polygon, and one as a compound of two triangles.

Faceted polyhedra

The

regular icosahedron In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. It has five equilateral triangular faces meeting at each vertex. It ...

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

dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

can be faceted into one regular Kepler–Poinsot polyhedron, three

uniform star polyhedra In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figure ...

, and three regular polyhedral compound. The uniform stars and

compound of five cubes The compound of five cubes is one of the five regular polyhedral compounds. It was first described by Edmund Hess in 1876. It is one of five regular compounds, and dual to the compound of five octahedra. It can be seen as a faceting of a regul ...

are constructed by face diagonals. The excavated dodecahedron is a facetting with star hexagon faces.

History

Faceting has not been studied as extensively as

. * In 1568 Wenzel Jamnitzer published his book ''

Perspectiva Corporum Regularium (from Latin: ''Perspective of the Regular Solids'') is a book of perspective drawings of polyhedra by German Renaissance goldsmith Wenzel Jamnitzer, with engravings by Jost Amman, published in 1568. Despite its Latin title, is written mainly ...

'', showing many stellations and facetings of polyhedra.''Mathematical Treasure: Wenzel Jamnitzer's Platonic Solids''
by Frank J. Swetz (2013): "In this study of the five Platonic solids, Jamnitzer truncated, stellated, and faceted the regular solids .. * In 1619,

Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...

described a regular compound of two

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...

which fits inside a cube, and which he called the Stella octangula. * In 1858,

Bertrand Bertrand may refer to: Places * Bertrand, Missouri, US * Bertrand, Nebraska, US * Bertrand, New Brunswick, Canada * Bertrand Township, Michigan, US * Bertrand, Michigan * Bertrand, Virginia, US * Bertrand Creek, state of Washington * Saint-Ber ...

derived the regular

star polyhedra In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two general kinds of star polyhedron: *Polyhedra which self-intersect in a repetitive way. *Concave ...

( Kepler–Poinsot polyhedra) by faceting the regular

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...

and

. * In 1974, Bridge enumerated the more straightforward facetings of the regular polyhedra, including those of the

. * 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 In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the oth ...

'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

External links

* *{{GlossaryForHyperspace , anchor= Faceting , title= Faceting Polyhedra Polygons Polytopes