Editing Dual polyhedron (section)

==Dual polytopes and tessellations==
Duality can be generalized to ''n''-dimensional space and '''dual [[polytope]]s;''' in two dimensions these are called [[dual polygon]]s.

The vertices of one polytope correspond to the (''n'' &minus; 1)-dimensional elements, or facets, of the other, and the ''j'' points that define a (''j'' &minus; 1)-dimensional element will correspond to ''j'' hyperplanes that intersect to give a (''n'' &minus; ''j'')-dimensional element. The dual of an ''n''-dimensional tessellation or [[Honeycomb (geometry)|honeycomb]] can be defined similarly.

In general, the facets of a polytope's dual will be the topological duals of the polytope's vertex figures. For the polar reciprocals of the [[regular polytope|regular]] and [[Uniform polytope|uniform]] polytopes, the dual facets will be polar reciprocals of the original's vertex figure. For example, in four dimensions, the vertex figure of the [[600-cell]] is the [[Regular icosahedron|icosahedron]];  the dual of the 600-cell is the [[120-cell]], whose facets are [[dodecahedron|dodecahedra]], which are the dual of the icosahedron.

===Self-dual polytopes and tessellations===
[[File:Kah 4 4.png|thumb|The [[square tiling]], {4,4}, is self-dual, as shown by these red and blue tilings]]
[[File:Infinite-order apeirogonal tiling and dual.png|thumb|The [[Infinite-order apeirogonal tiling]], {&infin;,&infin;} in red, and its dual position in blue]]
The primary class of self-dual polytopes are [[regular polytope]]s with [[palindromic]] [[Schläfli symbol]]s. All regular polygons, {a} are self-dual, [[polyhedron|polyhedra]] of the form {a,a}, [[4-polytope]]s of the form {a,b,a}, [[5-polytope]]s of the form {a,b,b,a}, etc.

The self-dual regular polytopes are:

* All [[regular polygon]]s, {a}.
* Regular [[tetrahedron]]: {3,3}
* In general, all regular ''n''-[[simplex]]es, {3,3,...,3}
* The regular [[24-cell]] in 4&nbsp;dimensions, {3,4,3}.
* The [[great 120-cell]] {5,5/2,5} and the [[grand stellated 120-cell]] {5/2,5,5/2}
The self-dual (infinite) regular Euclidean [[Honeycomb (geometry)|honeycombs]] are:
* [[Apeirogon]]: {&infin;}
* [[Square tiling]]: {4,4}
* [[Cubic honeycomb]]: {4,3,4}
* In general, all regular ''n''-dimensional Euclidean [[hypercubic honeycomb]]s: {4,3,...,3,4}.
The self-dual (infinite) regular [[Coxeter diagram#Hyperbolic Coxeter groups|hyperbolic]] honeycombs are:
* Compact hyperbolic tilings: [[Order-5 pentagonal tiling|{5,5}]], [[Order-6 hexagonal tiling|{6,6}]], ... {p,p}.
* Paracompact hyperbolic tiling: [[Infinite-order apeirogonal tiling|{&infin;,&infin;}]]
* Compact hyperbolic honeycombs: [[Icosahedral honeycomb|{3,5,3}]], [[Order-5 dodecahedral honeycomb|{5,3,5}]], and [[Order-5 120-cell honeycomb|{5,3,3,5}]]
* Paracompact hyperbolic honeycombs: [[Triangular tiling honeycomb|{3,6,3}]], [[Order-6 hexagonal tiling honeycomb|{6,3,6}]], [[Order-4 square tiling honeycomb|{4,4,4}]], and [[16-cell honeycomb honeycomb|{3,3,4,3,3}]]