Stellation diagram

In geometry, a stellation diagram or stellation pattern is a two-dimensional diagram in the plane of some face of a polyhedron, showing lines where other face planes intersect with this one. The lines cause 2D space to be divided up into regions. Regions not intersected by any further lines are called elementary regions. Usually unbounded regions are excluded from the diagram, along with any portions of the lines extending to infinity. Each elementary region represents a top face of one cell, and a bottom face of another.

The stellation diagram for the regular dodecahedron with the central pentagon highlighted. This diagram represents the dodecahedron face itself.

A collection of these diagrams, one for each face type, can be used to represent any stellation of the polyhedron, by shading the regions which should appear in that stellation.

A stellation diagram exists for every face of a given polyhedron. In face transitive polyhedra, symmetry can be used to require all faces have the same diagram shading. Semiregular polyhedra like the Archimedean solids will have different stellation diagrams for different kinds of faces.

See also

• List of Wenninger polyhedron models
• The fifty nine icosahedra

References

• M Wenninger, Polyhedron models; Cambridge University Press, 1st Edn (1983), Ppbk (2003).
• Coxeter, Harold Scott MacDonald; Du Val, P.; Flather, H. T.; Petrie, J. F. (1999), The fifty-nine icosahedra (3rd ed.), Tarquin, ISBN 978-1-899618-32-3, MR 0676126 (1st Edn University of Toronto (1938))

External links

• Stellation diagram
• Polyhedra Stellations Applet Vladimir Bulatov, 1998
  • http://bulatov.org/polyhedra/stellation/index.html Polyhedra Stellation (VRML)
  • http://bulatov.org/polyhedra/icosahedron/index_vrml.html 59 stellations of icosahedron
• http://www.queenhill.demon.co.uk/polyhedra/FacetingDiagrams/FacetingDiags.htm facetting diagrams
• http://fortran.orpheusweb.co.uk/Poly/Ex/dodstl.htm Stellating the Dodecahedron
• http://www.queenhill.demon.co.uk/polyhedra/icosa/stelfacet/StelFacet.htm Towards stellating the icosahedron and faceting the dodecahedron
• http://www.mathconsult.ch/showroom/icosahedra/index.html 59 stellations of the icosahedron
• http://www.uwgb.edu/dutchs/symmetry/stellate.htm Stellations of Polyhedra
  • http://www.uwgb.edu/dutchs/symmetry/stelicos.htm Coxeter's Classification and Notation
• http://www.georgehart.com/virtual-polyhedra/stellations-icosahedron-index.html