In
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 th ...
, 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
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 amo ...
. Each elementary region represents a top face of one
cell
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery ...
, 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 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.
See also
*
List of Wenninger polyhedron models
This is an indexed list of the uniform and stellated polyhedra from the book ''Polyhedron Models'', by Magnus Wenninger.
The book was written as a guide book to building polyhedra as physical models. It includes templates of face elements for cons ...
*
The fifty nine icosahedra
''The Fifty-Nine Icosahedra'' is a book written and illustrated by H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic icosahedron, according to a set of rules put for ...
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
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