To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Expansion (geometry)

From Wikipedia, the free encyclopedia

An example of expanding pentagon into a decagon by moving edges away from the center and inserting new edges in the gaps. The expansion is uniform if all the edges are the same length.
Animation showing an expanded cube (and octahedron)

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 position but reducing their size.

The expansion of a regular convex polytope creates a uniform convex polytope.

For polyhedra, an expanded polyhedron has all the faces of the original polyhedron, all the faces of the dual polyhedron, and new square faces in place of the original edges.

Expansion of regular polytopes

According to Coxeter, this multidimensional term was defined by Alicia Boole Stott[1] for creating new polytopes, specifically starting from regular polytopes to construct new uniform polytopes.

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

It has somewhat different meanings by dimension. In a Wythoff construction, an expansion is generated by reflections from the first and last mirrors. In higher dimensions, lower dimensional expansions can be written with a subscript, so e2 is the same as t0,2 in any dimension.

By dimension:

  • A regular {p} polygon expands into a regular 2n-gon.
  • 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, e{p,q} = e2{p,q} = t0,2{p,q} = rr{p,q}, and has Coxeter 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} 4-polytope (4-polytope) expands into a new 4-polytope with the original {p,q} cells, new cells {r,q} in place of the old vertices, p-gonal prisms in place of the old faces, and r-gonal prisms in place of the old edges.
    • This operation for 4-polytopes is also called runcination, e{p,q,r} = e3{p,q,r} = t0,3{p,q,r}, and has Coxeter diagram .
  • Similarly a regular {p,q,r,s} 5-polytope expands into a new 5-polytope with facets {p,q,r}, {s,r,q}, {p,q}×{ } prisms, {s,r}×{ } prisms, and {p}×{s} duoprisms.
    • This operation is called sterication, e{p,q,r,s} = e4{p,q,r,s} = t0,4{p,q,r,s} = 2r2r{p,q,r,s} and has Coxeter diagram .

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

See also

Notes

  1. ^ Coxeter, Regular Polytopes (1973), p. 123. p.210

References

  • Weisstein, Eric W. "Expansion". MathWorld.
  • Coxeter, H. S. M., Regular Polytopes. 3rd edition, Dover, (1973) ISBN 0-486-61480-8.
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Polyhedron operators
Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations
t0{p,q}
{p,q}
t01{p,q}
t{p,q}
t1{p,q}
r{p,q}
t12{p,q}
2t{p,q}
t2{p,q}
2r{p,q}
t02{p,q}
rr{p,q}
t012{p,q}
tr{p,q}
ht0{p,q}
h{q,p}
ht12{p,q}
s{q,p}
ht012{p,q}
sr{p,q}
This page was last edited on 12 February 2024, at 17:47
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.