Faceting

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Short description: Removing parts of a polytope without creating new vertices


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 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
Regular polygon 5.svg Regular polygon truncation 3 1.svg Regular polygon truncation 5 1.svg
Pentagram
{5/2}
Star hexagon Compound
2{3}
Decagram
{10/3}
Compound
2{5}
Compound
2{5/2}
Star decagon
Regular star polygon 5-2.svg Regular polygon truncation 3 2.svg Regular star figure 2(3,1).svg Regular star polygon 10-3.svg Regular star figure 2(5,1).svg Regular star figure 2(5,2).svg Regular polygon truncation 5 2.svg Regular polygon truncation 5 3.svg Regular star truncation 5-3 2.svg Regular star truncation 5-3 3.svg

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

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
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

Facetings of icosahedron (giving the shape of a great dodecahedron) and pentakis dodecahedron in Jamnitzer's book

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.[1]
  • 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

  1. 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 [...]"

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