Editing Polytope compound (section)

==Compounds of tilings==
There are eighteen two-parameter families of regular compound tessellations of the Euclidean plane. In the hyperbolic plane, five one-parameter families and seventeen isolated cases are known, but the completeness of this listing has not been enumerated.

The Euclidean and hyperbolic compound families 2 {''p'',''p''} (4 ≤ ''p'' ≤ ∞, ''p'' an integer) are analogous to the spherical [[stella octangula]], 2 {3,3}.
{| class="wikitable"
|+ A few examples of Euclidean and hyperbolic regular compounds
!Self-dual
!colspan=2|Duals
!Self-dual
|-
!2 [[square tiling|{4,4}]]
!2 [[hexagonal tiling|{6,3}]]
!2 [[triangular tiling|{3,6}]]
!2 [[infinite-order apeirogonal tiling|{∞,∞}]]
|- align=center
|[[File:Kah 4 4.png|160px]] 
|[[File:Compound 2 hexagonal tilings.svg|160px]]
|[[File:Compound 2 triangular tilings.svg|160px]]
| [[File:Infinite-order apeirogonal tiling and dual.png|160px]]

|-
!
!3 {6,3}
!3 {3,6}
!3 [[infinite-order apeirogonal tiling|{∞,∞}]]
|-
|
|[[File:Compound 3 hexagonal tilings.svg|160px]]
|[[File:Compound 3 triangular tilings.svg|160px]]
|[[File:Iii symmetry 000.png|160px]]
|}

A known family of regular Euclidean compound honeycombs in any number of dimensions is an infinite family of compounds of [[hypercubic honeycomb]]s, all sharing vertices and faces with another hypercubic honeycomb. This compound can have any number of hypercubic honeycombs.

There are also ''dual-regular'' tiling compounds. A simple example is the E<sup>2</sup> compound of a [[hexagonal tiling]] and its dual [[triangular tiling]], which shares its edges with the [[deltoidal trihexagonal tiling]]. The Euclidean compounds of two hypercubic honeycombs are both regular and dual-regular.