Quasiregular polyhedron

Quasiregular figures (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.∞)2 $\begin{Bmatrix} 3 \\ 4 \end{Bmatrix}$ $\begin{Bmatrix} 3 \\ 5 \end{Bmatrix}$ $\begin{Bmatrix} 3 \\ 6 \end{Bmatrix}$ $\begin{Bmatrix} 3 \\ 7 \end{Bmatrix}$ $\begin{Bmatrix} 3 \\ \infin \end{Bmatrix}$ cuboctahedron icosidodecahedron trihexagonal tiling triheptagonal tiling tricircular tiling A quasiregular polyhedron or tiling has exactly two kinds of regular face, which alternate around each vertex.

Regular and quasiregular figures {3,4} {4,4} {3,6} {5,4} $\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}$ $\begin{Bmatrix} 4 \\ 4 \end{Bmatrix}$ $\begin{Bmatrix} 5 \\ 4 \end{Bmatrix}$ (3.3)2 (4.4)2 (3.3)3 (5.4)2 octahedron square tiling triangular tiling tetrapentagonal tiling 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 semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are edge-transitive and hence step closer to regularity than the semiregular which are merely vertex-transitive.

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 $\begin{Bmatrix} p \\ q \end{Bmatrix}$ to represent their containing the faces of both the regular {p,q} and 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) 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), (3_a.3_b)², alternating two colors of triangular faces. Similarly the square tiling (4_a.4_b)² can be considered quasiregular, colored as a checkerboard. Also the triangular tiling can have alternately colored triangle faces, (3_a.3_b)³.

Wythoff construction

Regular (p | 2 q) and quasiregular polyhedra (2 | p q) are created from a Wythoff construction with the generator point at one of 3 corners of the fundamental domain. This defines a single edge within the fundamental domain.

Quasiregular polyhedra are generated from all 3 corners of the fundamental domain for Schwarz triangles that have no right angles:
q | 2 p, p | 2 q, 2 | p q

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:

Schläfli symbol	Coxeter-Dynkin diagram	Wythoff symbol
$\begin{Bmatrix} p , q \end{Bmatrix}$		q | 2 p
$\begin{Bmatrix} q , p \end{Bmatrix}$		p | 2 q
$\begin{Bmatrix} p \\ q \end{Bmatrix}$		2 | p q

The convex quasiregular polyhedra

There are two convex quasiregular polyhedra:

1. The cuboctahedron $\begin{Bmatrix} 3 \\ 4 \end{Bmatrix}$, vertex configuration 3.4.3.4, Coxeter-Dynkin diagram
2. The icosidodecahedron $\begin{Bmatrix} 3 \\ 5 \end{Bmatrix}$, vertex configuration 3.5.3.5, Coxeter-Dynkin diagram

In addition, the octahedron, which is also regular, $\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}$, vertex configuration 3.3.3.3, can be considered quasiregular if alternate faces are given different colors. In this form it is sometimes known as the tetratetrahedron. The remaining convex regular polyhedra have an odd number of faces at each vertex so cannot be colored in a way that preserves edge transitivity. It has Coxeter-Dynkin diagram

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.

Regular	Dual regular	Quasiregular	Vertex figure
Tetrahedron {3,3} 3 | 2 3	Tetrahedron {3,3} 3 | 2 3	Tetratetrahedron (Octahedron) 2 | 3 3	3.3.3.3
Cube {4,3} 3 | 2 4	Octahedron {3,4} 4 | 2 3	Cuboctahedron 2 | 3 4	3.4.3.4
Dodecahedron {5,3} 3 | 2 5	Icosahedron {3,5} 5 | 2 3	Icosidodecahedron 2 | 3 5	3.5.3.5

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 3.6.3.6 - a quasiregular tiling based on the triangular tiling and hexagonal tiling.

Regular	Dual regular	Quasiregular	Vertex figure
Hexagonal tiling {6,3} 6 | 2 3	Triangular tiling {3,6} 3 | 2 6	Trihexagonal tiling 2 | 3 6	3.6.3.6

The checkerboard pattern is a quasiregular coloring of the square tiling, vertex figure 4.4.4.4:

Regular	Dual regular	Quasiregular	Vertex figure
{4,4} 4 | 2 4	{4,4} 4 | 2 4	2 | 4 4	4.4.4.4

The triangular tiling can also be considered quasiregular, with three sets of alternating triangles at each vertex, (3.3)³:

3 | 3 3

In the hyperbolic plane, this sequence continues further, for example the triheptagonal tiling, vertex figure 3.7.3.7 - a quasiregular tiling based on the order-7 triangular tiling and heptagonal tiling.

Regular	Dual regular	Quasiregular	Vertex figure
Heptagonal tiling {7,3} 7 | 2 3	Triangular tiling {3,7} 3 | 2 7	Triheptagonal tiling 2 | 3 7	3.7.3.7

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 $\begin{Bmatrix} 3 \\ 5/2 \end{Bmatrix}$ and the dodecadodecahedron $\begin{Bmatrix} 5 \\ 5/2 \end{Bmatrix}$:

Regular	Dual regular	Quasiregular	Vertex figure
great stellated dodecahedron {5/2,3} 3 | 2 5/2	great icosahedron {3,5/2} 5/2 | 2 3	Great icosidodecahedron   2 | 3 5/2	3.5/2.3.5/2
Small stellated dodecahedron {5/2,5} 5 | 2 5/2	Great dodecahedron {5,5/2} 5/2 | 2 5	Dodecadodecahedron   2 | 5 5/2	5.5/2.5.5/2

Lastly there are three ditrigonal forms, whose vertex figures contain three alternatations of the two face types:

Polyhedron	Vertex figure
Ditrigonal dodecadodecahedron 3 | 5/3 5	(5.5/3)3
Small ditrigonal icosidodecahedron 3 | 5/2 3	(3.5/2)3
Great ditrigonal icosidodecahedron 3/2 | 3 5	((3.5)3)/2

Quasiregular duals

Some authorities argue that, since the duals of the quasiregular solids share the same symmetries, these duals must be quasiregular too. But not everybody accepts this view. 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:

1. The rhombic dodecahedron, with two types of alternating vertices, 8 with three rhombic faces, and 6 with four rhombic faces.
2. 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:

Cube V3.3.3.3

rhombic dodecahedron V3.4.3.4

rhombic triacontahedron V3.5.3.5

These three quasiregular duals are also characterised by having rhombic faces.

This rhombic-faced pattern continues as V3.6.3.6, the rhombille tiling.

See also

• Rectification (geometry)

References

• Cromwell, P. Polyhedra, Cambridge University Press (1977).

External links

• Weisstein, Eric W., "Quasiregular polyhedron" from MathWorld.
• Weisstein, Eric W., "Uniform polyhedron" from MathWorld. Quasi-regular polyhedra: (p.q)^r
• George Hart, Quasiregular polyhedra