Stellation diagram

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....

, a stellation diagram or stellation pattern is a two-dimensional diagram in the plane of some face of a polyhedron

Polyhedron

A polyhedron is often defined as a geometric solid with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is...

, 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 infinite regions are excluded from the diagram, along with any infinite portions of the lines.

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....

, a stellation diagram or stellation pattern is a two-dimensional diagram in the plane of some face of a polyhedron

Polyhedron

A polyhedron is often defined as a geometric solid with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is...

, 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 infinite regions are excluded from the diagram, along with any infinite portions of the lines. Each elementary region represents a top face of one cell

Cell (geometry)

In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object.- In polytopes :A cell is a three-dimensional polyhedron element that is part of the boundary of a higher-dimensional polytope, such as a polychoron or honeycomb .For example, a cubic honeycomb is made...

, and a bottom face of another.

A collection of these diagrams, one for each face type, can be used to represent any stellation

Stellation

Stellation is a process of constructing new polygons , new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again...

 of the polyhedron, by shading the regions which should appear in that stellation.

A stellation diagram exists for every face of a give polyhedron. In face transitive polyhedra, symmetry can be used to require all faces have the same diagram shading. Semiregular polyhedra like the Archimedean solid

Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices. They are distinct from the Platonic solids, which are composed of only one type of polygon meeting in identical vertices, and from...

s will have different stellation diagrams for different kinds of faces.

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