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 Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

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 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 aroun ...

, 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 can be faceted into one regular

Kepler–Poinsot polyhedron In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic faces or vertex figure ...

, 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.

History

Faceting has not been studied as extensively as

. * In 1568

Wenzel Jamnitzer Wenzel Jamnitzer (sometimes Jamitzer, or Wenzel ''Gemniczer'') (1507/1508 – 19 December 1585) was a Northern Mannerism, Northern Mannerist goldsmith, artist, and printmaker in etching, who worked in Nuremberg. He was the best known Germa ...

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 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 polyto ...

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

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 other. ...

'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