Expansion (geometry)

Expansion (geometry)

[
dodecahedron creates a rhombicosidodecahedron and a reverse expansion creates an icosahedron.] In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements (vertices, edges, etc). (Equivalently this operation can be imagined by keeping facets in the same location, but reducing their size.)

According to Coxeter, this multidimensional term was defined by Alicia Boole Stott [Coxeter, H. S. M., "Regular Polytopes". 3rd edition, Dover, 1973, p. 123. ISBN 0-486-61480-8. p.210] for creating new polytopes, specifically starting from regular polytopes constructs new uniform polytopes.

The "expansion" operation is symmetric with respects to a regular polytope and its dual. The resulting figure contains the facets of both the regular and regular dual, along with various prismatic facets filling the gaps created between intermediate dimensional elements.

It has somewhat different meanings by dimension, and correspond to reflections from the first and last mirrors in a Wythoff construction.

By dimension:
* A regular {p} polygon "expands" into a regular 2n-gon.
** The operation is identical to truncation for polygons, t0,1{p} and has Coxeter-Dynkin diagram .
* A regular {p,q} polyhedron (3-polytope) "expands" into a polyhedron with vertex figure "p.4.q.4".
**This operation for polyhedra is also called cantellation, t0,2{p,q} and has Coxeter-Dynkin diagram .
**:
**: For example, a rhombicuboctahedron can be called an "expanded cube", "expanded octahedron", as well as a "cantellated cube" or "cantellated octahedron".
* A regular {p,q,r} polychoron (4-polytope) "expands" into a new polychoron with the original {p,q} cells, new cells {q,r} in place of he old vertices and q-gonal prisms in place of the old edges.
** This operation for polychora is also called runcination, t0,3{p,q,r} and has Coxeter-Dynkin diagram .
* Similar a regular {p,q,r,s} polyteron (5-polytope) "expands" into a new polyteron with facets {p,q,r}, {q,r,s}, {q,r} hyperprisms, {p} duoprisms, {q} duoprisms.
** This operation is called sterication, t0,4{p,q,r,s} and has Coxeter-Dynkin diagram .

The general operator for "expansion" for an n-polytope is t0,n-1{p,q,r,...}. New simplex facets are added at each vertex, and new prismatic polytopes are added at each divided edge, face, ... ridge, etc.

References

*


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Geometry-Shader — Eine Geometry Shader Hardware ist ein relativ neuartiger Baustein in der 3D Computergrafik. Als Shader in der klassischen Grafikpipeline wird der Geometry Shader nach dem Vertex Shader aufgerufen. Er kann neue primitive Geometrien aus bereits… …   Deutsch Wikipedia

  • History of geometry — Geometry (Greek γεωμετρία ; geo = earth, metria = measure) 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. Classic geometry… …   Wikipedia

  • Metric expansion of space — Physical cosmology Universe · Big Bang …   Wikipedia

  • Differential geometry of surfaces — Carl Friedrich Gauss in 1828 In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives:… …   Wikipedia

  • Algebraic geometry and analytic geometry — In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally… …   Wikipedia

  • Odd greedy expansion — In number theory, the odd greedy expansion problem concerns a method for forming Egyptian fractions in which all denominators are odd. If a rational number x/y is a sum of odd unit fractions, then y must be odd. Conversely, it is known that… …   Wikipedia

  • Bay Area Rapid Transit expansion — Throughout the history of the Bay Area Rapid Transit, better known as BART, there have been plans to extend service to other areas. Contents 1 Proposals 1.1 Warm Springs extension 1.2 San Jose extension …   Wikipedia

  • Cantellation (geometry) — In geometry, a cantellation is an operation in any dimension that cuts a regular polytope edges and vertices, creating a new facet in place of each edge and vertex. The operation also applies to regular tilings and honeycombs.It is represented by …   Wikipedia

  • Cupola (geometry) — For other uses, see cupola (disambiguation). Pentagonal cupola (example) Type Set of cupolas Faces n triangles, n squares 1 n agon, 1 2n agon Edges …   Wikipedia

  • List of formulas in Riemannian geometry — This is a list of formulas encountered in Riemannian geometry.Christoffel symbols, covariant derivativeIn a smooth coordinate chart, the Christoffel symbols are given by::Gamma {ij}^m=frac12 g^{km} left( frac{partial}{partial x^i} g {kj}… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”