Compact symmetry

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The order-5 cubic honeycomb is a hyperbolic honeycomb with a compact symmetry group.

A hyperbolic symmetry group is compact if its fundamental domain is finite.

A polytope may be called compact if its symmetry group is compact. A compact polytope never has ideal or ultra-ideal vertices.

Paracompact symmetry[edit | edit source]

The square tiling honeycomb is a paracompact hyperbolic honeycomb

A symmetry group is paracompact if its fundamental domain has finite area. Some authors may additionally require it to not be compact.

In 2 dimensions, the paracompact Coxeter groups are precisely those Coxeter groups with an ∞ in their diagram.

Hypercompact symmetry[edit | edit source]

The heptagonal tiling honeycomb is a hypercompact hyperbolic honeycomb

A symmetry group is noncompact if it is not compact or paracompact. Thus the fundamental domain of a noncompact symmetry group has infinite area and noncompact polytopes have ultra-ideal vertices.

Some authors may call this condition hypercompactness in which case the term "noncompact" simply refers to any symmetry group which is not compact.