Weakly regular polytope

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An object is weakly regular if its symmetry acts transitively on elements of each rank. For example a polyhedron is weakly regular if its symmetry is isohedral, isotoxal, and isogonal. All regular polytopes are weakly regular, but there are non-regular polytopes that are weakly regular.

Weakly regular polyhedra[edit | edit source]

A weakly regular compound of five tetrahedra.

A weakly regular polyhedron is isotoxal, isogonal, and isotopic. Under classical definitions of polyhedra (finite, planar, not a compound) the weakly regular polyhedra are exactly the nine regular polyhedra, i.e. the Platonic solids and the Kepler-Poinsot solids.

Relaxing these requirements allows polyhedra that are weakly regular but not regular:

  • The regular compound polyhedra, despite their name, are only weakly regular, except for the stella octangula which is truly regular. No other compound polyhedra are weakly regular.
  • The rhombic tiling is a weakly regular, but not regular, Euclidean tiling.
  • The quotient of the square tiling by (2,1) and (-1,2) is a weakly regular, but not regular, finite abstract polyhedron. It can also be embedded as a skew polyhedron in 4-dimensional space.