Trivial group

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The trivial group also known as the zero group, is the simplest possible group. It contains 1 element, its identity. Polytopes whose symmetry group are isomorphic to the trivial group are called asymmetrical.

Symmetries isomorphic to the trivial group[edit | edit source]

Notable asymmetric polytopes[edit | edit source]

Most classes of polytopes that are objects of study (regular, uniform, noble, etc.) are symmetric. However, interesting questions arise from asymmetric polytopes as well, such as asking what the smallest asymmetric polytopes are in a given category using some measure such as facet count.

Asymmetric abstract polytopes[edit | edit source]

The flag graph of the non-trivial asymmetric abstract polytope with the fewest flags.

The nullitope and point are trivial examples of asymmetric abstract polytopes. However non-trivial abstract polytopes are somewhat rare, since abstract polytopes have at least as much symmetry as their realizations. No abstract polygons are asymmetric (in fact all abstract polygons are regular).

By facet count, the smallest non-trivial asymmetric abstract polytope has 3 facets. It is impossible for an abstract polytope with 2 facets to be asymmetric since its faces must be identical and thus symmetric.

By flag count, the smallest known non-trivial asymmetric abstract polytope has 24 flags. It is is a tetrahedron of rank 3 with 1 digon, 2 triangles, and 1 quadrilateral as faces. The triangles share an edge.

Asymmetric acrohedra[edit | edit source]

Asymmetrical polyhedra with all regular faces are believed to have a minimum possible face count of 9. An example is a square pyramid blended with a pentagonal pyramid, angled so that the apex of each pyramid is adjacent to the base of the other.

All Johnson solids are symmetrical, but there are asymmetrical Blind polytopes, all of which are special cuts of the hexacosichoron.