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 ...

, 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 to ...

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 on ...

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''. For every stellation of some convex polytope, 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 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 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 figures. ...

, 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 stellation. * In 1568

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

published his book '' Perspectiva Corporum Regularium'', 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 described a regular compound of two tetrahedra which fits inside a cube, and which he called the

Stella octangula The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depict ...

. * In 1858, Bertrand derived the regular star polyhedra ( 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 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'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