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 c ...

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

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 ...

, showing lines where other face planes

intersect Intersection or intersect may refer to: * Intersection in mathematics, including: ** Intersection (set theory), the set of elements common to some collection of sets ** Intersection (geometry) ** Intersection theory * Intersection (road), a pl ...

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 Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...

. Each elementary region represents a top face of one cell, and a bottom face of another. A collection of these diagrams, one for each face type, can be used to represent any

stellation In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific el ...

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 In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...

polyhedra, symmetry can be used to require all faces have the same diagram shading. Semiregular polyhedra like the

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...

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

References

* M Wenninger, ''Polyhedron models''; Cambridge University Press, 1st Edn (1983), Ppbk (2003). * (1st Edn University of Toronto (1938))

External links

Stellation diagramPolyhedra 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 {{geometry-stub

See also

References

External links