Quasiregular polyhedron

Quasiregular figures Right triangle domains (p q 2), = r{p,q} r{4,3} r{5,3} r{6,3} r{7,3}... r{∞,3} (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.∞)2 Isosceles triangle domains (p p 3), = = h{6,p} h{6,4} h{6,5} h{6,6} h{6,7}... h{6,∞} = = = = = (4.3)4 (5.3)5 (6.3)6 (7.3)7 (∞.3)∞ Isosceles triangle domains (p p 4), = = h{8,p} h{8,3} h{8,5} h{8,6} h{8,7}... h{8,∞} = = = = = (4.3)3 (4.5)5 (4.6)6 (4.7)7 (4.∞)∞ Scalene triangle domain (5 4 3), (3.5)4 (4.5)3 (3.4)5 A quasiregular polyhedron or tiling has exactly two kinds of regular face, which alternate around each vertex. Their vertex figures are isogonal polygons.

Regular and quasiregular figures Right triangle domains (p p 2), = = r{p,p} = {p,4}1⁄2 {3,4}1⁄2 r{3,3} {4,4}1⁄2 r{4,4} {5,4}1⁄2 r{5,5} {6,4}1⁄2 r{6,6}... {∞,4}1⁄2 r{∞,∞} = = = = = (3.3)2 (4.4)2 (5.5)2 (6.6)2 (∞.∞)2 Isosceles triangle domains (p p 3), = = {p,6}1⁄2 {3,6}1⁄2 {4,6}1⁄2 {5,6}1⁄2 {6,6}1⁄2... {∞,6}1⁄2 = = = = = (3.3)3 (4.4)3 (5.5)3 (6.6)3 (∞.∞)3 Isosceles triangle domains (p p 4), = = {p,8}1⁄2 {3,8}1⁄2 {4,8}1⁄2 {5,8}1⁄2 {6,8}1⁄2... {∞,8}1⁄2 = = = = = (3.3)4 (4.4)4 (5.5)4 (6.6)4 (∞.∞)4 A regular polyhedron or tiling can be considered quasiregular if it has an even number of faces around each vertex (and thus can have alternately colored faces).

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

Their dual figures are face-transitive and edge-transitive; they have exactly two kinds of regular vertex figures, which alternate around each face. They are sometimes also considered quasiregular.

There are only two convex quasiregular polyhedra: the cuboctahedron and the icosidodecahedron. Their names, given by Kepler, come from recognizing that their faces are all the faces (turned differently) of the dual-pair cube and octahedron, in the first case, and of the dual-pair icosahedron and dodecahedron, in the second case.

These forms representing a pair of a regular figure and its dual can be given a vertical Schläfli symbol ${\displaystyle {\begin{Bmatrix}p\\q\end{Bmatrix}}}$ or r{p,q}, to represent that their faces are all the faces (turned differently) of both the regular {p,q} and the dual regular {q,p}. A quasiregular polyhedron with this symbol will have a vertex configuration p.q.p.q (or (p.q)²).

More generally, a quasiregular figure can have a vertex configuration (p.q)^r, representing r (2 or more) sequences of the faces around the vertex.

Tilings of the plane can also be quasiregular, specifically the trihexagonal tiling, with vertex configuration (3.6)². Other quasiregular tilings exist on the hyperbolic plane, like the triheptagonal tiling, (3.7)². Or more generally: (p.q)², with 1/p + 1/q < 1/2.

Regular polyhedra and tilings with an even number of faces at each vertex can also be considered quasiregular by differentiating between faces of the same order, by representing them differently, like coloring them alternately (without defining any surface orientation). A regular figure with Schläfli symbol {p,q} can be considered quasiregular, with vertex configuration (p.p)^q/2, if q is even.

Examples:

The regular octahedron, with Schläfli symbol {3,4} and 4 being even, can be considered quasiregular as a tetratetrahedron (2 sets of 4 triangles of the tetrahedron), with vertex configuration (3.3)^4/2 = (3_a.3_b)², alternating two colors of triangular faces.

The square tiling, with vertex configuration 4⁴ and 4 being even, can be considered quasiregular, with vertex configuration (4.4)^4/2 = (4_a.4_b)², colored as a checkerboard.

The triangular tiling, with vertex configuration 3⁶ and 6 being even, can be considered quasiregular, with vertex configuration (3.3)^6/2 = (3_a.3_b)³, alternating two colors of triangular faces.