Faceting

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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 amay 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[edit]

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.

Convex Regular Quasiregular Regular compound
Regular pentagon
{5}
Regular hexagon
{6}
Pentagram
{5/2}
Star hexagon Hexagram
{6/2}
Regular polygon 5.svg Regular polygon 6.svg Regular star polygon 5-2.svg Regular polygon truncation 3 2.svg Regular star figure 2(3,1).svg

Faceted polyhedra[edit]

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
Icosahedron.png Great dodecahedron.png Small stellated dodecahedron.png Great icosahedron.png

The regular dodecahedron can be faceted into one regular Kepler–Poinsot polyhedra, 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
Dodecahedron.png Great stellated dodecahedron.png Small ditrigonal icosidodecahedron.png Ditrigonal dodecadodecahedron.png Great ditrigonal icosidodecahedron.png Excavated dodecahedron highlighted.png
Convex Regular compounds
dodecahedron five tetrahedra five cubes ten tetrahedra
Dodecahedron.png Compound of five tetrahedra.png Compound of five cubes.png Compound of ten tetrahedra.png

History[edit]

Faceting has not been studied as extensively as stellation.

References[edit]

  • 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.

External links[edit]