Editing Regular polytope (section)

===Duality of the regular polytopes===

The [[Dual polytope|dual]] of a regular polytope is also a regular polytope. The Schläfli symbol for the dual polytope is just the original symbol written backwards: {3, 3} is self-dual, {3, 4} is dual to {4, 3}, {4, 3, 3} to {3, 3, 4} and so on.

The [[vertex figure]] of a regular polytope is the dual of the dual polytope's [[Facet (geometry)|facet]]. For example, the vertex figure of {3, 3, 4} is {3, 4}, the dual of which is {4, 3} &mdash; a [[Cell (geometry)|cell]] of {4, 3, 3}.

The [[hypercube|measure]] and [[cross polytope]]s in any dimension are dual to each other.

If the Schläfli symbol is [[palindromic]] (i.e. reads the same forwards and backwards), then the polytope is self-dual. The self-dual regular polytopes are:
* All [[regular polygon]]s - {a}.
* All regular ''n''-[[simplex]]es - {3,3,...,3}.
* The regular [[24-cell]] - {3,4,3} - in 4&nbsp;dimensions.
* The [[great 120-cell]] - {5,5/2,5} - and [[grand stellated 120-cell]] - {5/2,5,5/2} - in 4&nbsp;dimensions.
* All regular ''n''-dimensional [[Hypercubic honeycomb|hypercubic]] [[Honeycomb (geometry)|honeycombs]] - {4,3,...,3,4}. These may be treated as [[#Apeirotopes — infinite polytopes|infinite polytope]]s.
* Hyperbolic tilings and honeycombs (tilings {p,p} with p>4 in 2 dimensions; [[Order-4 square tiling honeycomb|{4,4,4}]], [[order-5 dodecahedral honeycomb|{5,3,5}]], [[Icosahedral honeycomb|{3,5,3}]], [[order-6 hexagonal tiling honeycomb|{6,3,6}]], and [[Hexagonal tiling honeycomb|{3,6,3}]] in 3 dimensions; [[order-5 120-cell honeycomb|{5,3,3,5}]] in 4 dimensions; and [[16-cell honeycomb honeycomb|{3,3,4,3,3}]] in 5 dimensions).