Weakly regular polytope
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 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.
- 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.
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