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In geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are edge-transitive and hence a step closer to regular polyhedra than the semiregular which are merely vertex-transitive.
Contents
- Wythoff construction
- The convex quasiregular polyhedra
- Quasiregular tilings
- Nonconvex examples
- Quasiregular duals
- Quasiregular polytopes and honeycombs
- References
There are only two convex quasiregular polyhedra, the cuboctahedron and the icosidodecahedron. Their names, given by Kepler, come from recognizing their faces contain all the faces of the dual-pair cube and octahedron, in the first, and 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
More generally, a quasiregular figure can have a vertex configuration (p.q)r, representing r (2 or more) instances 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.
Some regular polyhedra and tilings (those with an even number of faces at each vertex) can also be considered quasiregular by differentiating between faces of the same number of sides, but representing them differently, like having different colors, but no surface features defining their orientation. A regular figure with Schläfli symbol {p,q} can be quasiregular, with vertex configuration (p.p)q/2, if q is even.
The octahedron can be considered quasiregular as a tetratetrahedron (2 sets of 4 triangles of the tetrahedron), (3a.3b)2, alternating two colors of triangular faces. Similarly the square tiling (4a.4b)2 can be considered quasiregular, colored as a checkerboard. Also the triangular tiling can have alternately colored triangle faces, (3a.3b)3.
Wythoff construction
Coxeter defines a quasiregular polyhedron as one having a Wythoff symbol in the form p | q r, and it is regular if q=2 or q=r.
The Coxeter-Dynkin diagram is another symbolic representation that shows the quasiregular relation between the two dual-regular forms:
The convex quasiregular polyhedra
There are two convex quasiregular polyhedra:
- The cuboctahedron
{ 3 4 } , vertex configuration (3.4)2, Coxeter-Dynkin diagram - The icosidodecahedron
{ 3 5 } , vertex configuration (3.5)2, Coxeter-Dynkin diagram
In addition, the octahedron, which is also regular,
Each of these forms the common core of a dual pair of regular polyhedra. The names of two of these give clues to the associated dual pair, respectively the cube + octahedron and the icosahedron + dodecahedron. The octahedron is the core of a dual pair of tetrahedra (an arrangement known as the stella octangula), and when derived in this way is sometimes called the tetratetrahedron.
Each of these quasiregular polyhedra can be constructed by a rectification operation on either regular parent, truncating the edges fully, until the original edges are reduced to a point.
Quasiregular tilings
This sequence continues as the trihexagonal tiling, vertex figure (3.6)2 - a quasiregular tiling based on the triangular tiling and hexagonal tiling.
The checkerboard pattern is a quasiregular coloring of the square tiling, vertex figure (4.4)2:
The triangular tiling can also be considered quasiregular, with three sets of alternating triangles at each vertex, (3.3)3:
In the hyperbolic plane, this sequence continues further, for example the triheptagonal tiling, vertex figure (3.7)2 - a quasiregular tiling based on the order-7 triangular tiling and heptagonal tiling.
Nonconvex examples
Coxeter, H.S.M. et al. (1954) also classify certain star polyhedra having the same characteristics as being quasiregular:
Two are based on dual pairs of regular Kepler–Poinsot solids, in the same way as for the convex examples.
The great icosidodecahedron
Lastly there are three ditrigonal forms, whose vertex figures contain three alternations of the two face types:
Quasiregular duals
Some authorities argue that, since the duals of the quasiregular solids share the same symmetries, these duals should be called quasiregular too. But not everybody uses this terminology. These duals are transitive on their edges and faces (but not on their vertices); they are the edge-transitive Catalan solids. The convex ones are, in corresponding order as above:
- The rhombic dodecahedron, with two types of alternating vertices, 8 with three rhombic faces, and 6 with four rhombic faces.
- The rhombic triacontahedron, with two types of alternating vertices, 20 with three rhombic faces, and 12 with five rhombic faces.
In addition, by duality with the octahedron, the cube, which is usually regular, can be made quasiregular if alternate vertices are given different colors.
Their face configuration are of the form V3.n.3.n, and Coxeter-Dynkin diagram
These three quasiregular duals are also characterised by having rhombic faces.
This rhombic-faced pattern continues as V(3.6)2, the rhombille tiling.
Quasiregular polytopes and honeycombs
In Euclidean 4-space, the regular 16-cell can also be seen as quasiregular as an alternated tesseract, h{4,3,3}, Coxeter diagrams: = , composed of alternating tetrahedron and tetrahedron cells. Its vertex figure is the quasiregular tetratetrahedron (an octahedron with tetrahedral symmetry), .
The only quasiregular honeycomb in Euclidean 3-space is the alternated cubic honeycomb, h{4,3,4}, Coxeter diagrams: = , composed of alternating tetrahedral and octahedral cells. Its vertex figure is the quasiregular cuboctahedron, .
In hyperbolic 3-space, one quasiregular honeycomb is the alternated order-5 cubic honeycomb, h{4,3,5}, Coxeter diagrams: = , composed of alternating tetrahedral and icosahedral cells. Its vertex figure is the quasiregular icosidodecahedron, . A related paracompact alternated order-6 cubic honeycomb, h{4,3,6} has alternating tetrahedral and hexagonal tiling cells with vertex figure is a quasiregular trihexagonal tiling, .
Regular polychora honeycombs of the form {p,3,4} or can have their symmetry cut in half as into quasiregular form , creating alternately colored {p,3} cells. These cases include the Euclidean cubic honeycomb {4,3,4} with cubic cells, and compact hyperbolic {5,3,4} with dodecahedral cells, and paracompact {6,3,4} with infinite hexagonal tiling cells. They have four cells around each edge, alternating in 2 colors. Their vertex figures are quasiregular tetratetrahedra, = .
Similarly regular hyperbolic honeycombs of the form {p,3,6} or can have their symmetry cut in half as into quasiregular form , creating alternately colored {p,3} cells. They have six cells around each edge, alternating in 2 colors. Their vertex figures are quasiregular triangular tilings, .