Quasiregular polyhedron

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Quasiregular figures
(3.3)2 (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.8)2 (3.∞)2
\begin{Bmatrix} 3 \\ 3 \end{Bmatrix} \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 \\ 8 \end{Bmatrix} \begin{Bmatrix} 3 \\ \infin \end{Bmatrix}
r{3,3} r{3,4} r{3,5} r{3,6} r{3,7} r{3,8} r{3,∞}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-33-t1.png Uniform polyhedron-43-t1.png Uniform polyhedron-53-t1.png Uniform polyhedron-63-t1.png Uniform tiling 73-t1.png Uniform tiling 83-t1.png Uniform tiling infin32-t1.png
A quasiregular polyhedron or tiling has exactly two kinds of regular face, which alternate around each vertex. Their vertex figures are rectangles.

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.

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} or r{p,q} 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.

Regular and quasiregular figures
Right triangles (p p 2)
{3,4}
r{3,3}
{4,4}
r{4,4}
{5,4}
r{5,5}
{6,4}
r{6,6}
{7,4}
r{7,7}
{8,4}
r{8,8}
{∞,4}
r{∞,∞}
\begin{Bmatrix} 3 \\ 3 \end{Bmatrix} \begin{Bmatrix} 4 \\ 4 \end{Bmatrix} \begin{Bmatrix} 5 \\ 5 \end{Bmatrix} \begin{Bmatrix} 6 \\ 6 \end{Bmatrix} \begin{Bmatrix} 7 \\ 7 \end{Bmatrix} \begin{Bmatrix} 8 \\ 8 \end{Bmatrix} \begin{Bmatrix} \infin \\ \infin \end{Bmatrix}
(3.3)2 (4.4)2 (5.5)2 (6.6)2 (7.7)2 (8.8)2 (∞.∞)2
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.png
Uniform polyhedron-33-t1.png Uniform tiling 44-t1.png
square tiling
H2 tiling 255-2.png
order-4 pentagonal tiling
H2 tiling 266-2.png
order-4 hexagonal tiling
H2 tiling 277-2.png
order-4 heptagonal tiling
H2 tiling 288-2.png
order-4 octagonal tiling
H2 tiling 2ii-2.png
Order-4 apeirogonal tiling
General triangles (p p 3)
{3,6} {4,6} {5,6} {6,6} {7,6} {8,6} {∞,6}
(3.3)3 (4.4)3 (5.5)3 (6.6)3 (7.7)3 (8.8)3 (∞.∞)3
CDel branch.pngCDel split2.pngCDel node 1.png CDel branch.pngCDel split2-44.pngCDel node 1.png CDel branch.pngCDel split2-55.pngCDel node 1.png CDel branch.pngCDel split2-ii.pngCDel node 1.png
Uniform tiling 333-t1.png H2 tiling 344-2.png H2 tiling 355-2.png H2 tiling 366-2.png H2 tiling 377-2.png H2 tiling 388-2.png H2 tiling 3ii-2.png
General triangles (p p 4)
{3,8} {4,8} {5,8} {6,8} {7,8} {8,8} {∞,8}
(3.3)4 (4.4)4 (5.5)4 (6.6)4 (7.7)4 (8.8)4 (∞.∞)4
CDel label4.pngCDel branch.pngCDel split2.pngCDel node 1.png CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node 1.png CDel label4.pngCDel branch.pngCDel split2-55.pngCDel node 1.png CDel label4.pngCDel branch.pngCDel split2-ii.pngCDel node 1.png
H2 tiling 334-4.png H2 tiling 444-2.png H2 tiling 455-2.png H2 tiling 466-2.png H2 tiling 477-2.png H2 tiling 488-2.png H2 tiling 4ii-2.png
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).

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[edit]

Wythoffian construction diagram.png
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.[1]

The Coxeter-Dynkin diagram is another symbolic representation that shows the quasiregular relation between the two dual-regular forms:

Schläfli symbol Coxeter diagram Wythoff symbol
\begin{Bmatrix} p , q \end{Bmatrix} {p,q} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png q | 2 p
\begin{Bmatrix} q , p \end{Bmatrix} {q,p} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png p | 2 q
\begin{Bmatrix} p \\ q \end{Bmatrix} r{p,q} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png 2 | p q

The convex quasiregular polyhedra[edit]

Further information: Rectification (geometry)

There are two convex quasiregular polyhedra:

  1. The cuboctahedron \begin{Bmatrix} 3 \\ 4 \end{Bmatrix}, vertex configuration (3.4)2, Coxeter-Dynkin diagram CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
  2. The icosidodecahedron \begin{Bmatrix} 3 \\ 5 \end{Bmatrix}, vertex configuration (3.5)2, Coxeter-Dynkin diagram CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png

In addition, the octahedron, which is also regular, \begin{Bmatrix} 3 \\ 3 \end{Bmatrix}, vertex configuration (3.3)2, 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 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png

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
Uniform polyhedron-33-t0.png
Tetrahedron
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
3 | 2 3
Uniform polyhedron-33-t2.png
Tetrahedron
{3,3}
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
3 | 2 3
Uniform polyhedron-33-t1.png
Tetratetrahedron
(Octahedron)

CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
2 | 3 3
Tetratetrahedron vertfig.png
3.3.3.3
Uniform polyhedron-43-t0.png
Cube
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
3 | 2 4
Uniform polyhedron-43-t2.png
Octahedron
{3,4}
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
4 | 2 3
Uniform polyhedron-43-t1.png
Cuboctahedron
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
2 | 3 4
Cuboctahedron vertfig.png
3.4.3.4
Uniform polyhedron-53-t0.png
Dodecahedron
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
3 | 2 5
Uniform polyhedron-53-t2.png
Icosahedron
{3,5}
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
5 | 2 3
Uniform polyhedron-53-t1.png
Icosidodecahedron
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
2 | 3 5
Icosidodecahedron vertfig.png
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[edit]

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
Uniform tiling 63-t0.png
Hexagonal tiling
{6,3}
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
6 | 2 3
Uniform tiling 63-t2.png
Triangular tiling
{3,6}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
3 | 2 6
Uniform tiling 63-t1.png
Trihexagonal tiling
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
2 | 3 6
Trihexagonal tiling vertfig.png
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
Uniform tiling 44-t0.png
{4,4}
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
4 | 2 4
Uniform tiling 44-t2.png
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
4 | 2 4
Uniform tiling 44-t1.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
2 | 4 4
Square tiling vertfig.png
4.4.4.4

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

Uniform tiling 333-t1.png
3 | 3 3
CDel node 1.pngCDel split1.pngCDel branch.png

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
Uniform tiling 73-t0.png
Heptagonal tiling
{7,3}
CDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png
7 | 2 3
Uniform tiling 73-t2.png
Triangular tiling
{3,7}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
3 | 2 7
Uniform tiling 73-t1.png
Triheptagonal tiling
CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png
2 | 3 7
Triheptagonal tiling vertfig.png
3.7.3.7

Nonconvex examples[edit]

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.png
great stellated dodecahedron
{5/2,3}
CDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 3.pngCDel node.png
3 | 2 5/2
Great icosahedron.png
great icosahedron
{3,5/2}
CDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 3.pngCDel node 1.png
5/2 | 2 3
Great icosidodecahedron.png
Great icosidodecahedron
 
CDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.pngCDel 3.pngCDel node.png
2 | 3 5/2
Great icosidodecahedron vertfig.png
3.5/2.3.5/2
Small stellated dodecahedron.png
Small stellated dodecahedron
{5/2,5}
CDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 5.pngCDel node.png
5 | 2 5/2
Great dodecahedron.png
Great dodecahedron
{5,5/2}
CDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 5.pngCDel node 1.png
5/2 | 2 5
Dodecadodecahedron.png
Dodecadodecahedron
 
CDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.pngCDel 5.pngCDel node.png
2 | 5 5/2
Dodecadodecahedron vertfig.png
5.5/2.5.5/2

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

Image Polyhedron name
Wythoff symbol
Coxeter diagram
Vertex figure
Ditrigonal dodecadodecahedron.png Ditrigonal dodecadodecahedron
3 | 5/3 5
Ditrigonal dodecadodecahedron cd.png or CDel node.pngCDel 5.pngCDel node h3.pngCDel 5-2.pngCDel node.png
Ditrigonal dodecadodecahedron vertfig.png
(5.5/3)3
Small ditrigonal icosidodecahedron.png Small ditrigonal icosidodecahedron
3 | 5/2 3
Small ditrigonal icosidodecahedron cd.png or CDel node h3.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Small ditrigonal icosidodecahedron vertfig.png
(3.5/2)3
Great ditrigonal icosidodecahedron.png Great ditrigonal icosidodecahedron
3/2 | 3 5
Great ditrigonal icosidodecahedron cd.png or CDel node h3.pngCDel 5-2.pngCDel node.pngCDel 3.pngCDel node.png
Great ditrigonal icosidodecahedron vertfig.png
((3.5)3)/2

Quasiregular duals[edit]

Some authorities argue that, since the duals of the quasiregular solids share the same symmetries, these duals must be quasiregular too. But not everybody believes this to be true. 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, and Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node f1.pngCDel n.pngCDel node.png

Hexahedron.svg Rhombicdodecahedron.jpg Rhombictriacontahedron.svg Rhombic star tiling.png Order73 qreg rhombic til.png Uniform dual tiling 433-t01-yellow.png
Cube
V(3.3)2
CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png
Rhombic dodecahedron
V(3.4)2
CDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png
Rhombic triacontahedron
V(3.5)2
CDel node.pngCDel 3.pngCDel node f1.pngCDel 5.pngCDel node.png
Rhombille tiling
V(3.6)2
CDel node.pngCDel 3.pngCDel node f1.pngCDel 6.pngCDel node.png
V(3.7)2
CDel node.pngCDel 3.pngCDel node f1.pngCDel 7.pngCDel node.png
V(3.8)2
CDel node.pngCDel 3.pngCDel node f1.pngCDel 8.pngCDel node.png

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[edit]

In Euclidean 4-space, the regular 16-cell can also be seen as quasiregular as an alternated tesseract, h{4,3,3}, Coxeter diagrams: CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png, composed of alternating tetrahedron and tetrahedron cells. Its vertex figure is the quasiregular tetratetrahedron (an octahedron with tetrahedral symmetry), CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png.

The only quasiregular honeycomb in Euclidean 3-space is the alternated cubic honeycomb, h{4,3,4}, Coxeter diagrams: CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png, composed of alternating tetrahedral and octahedral cells. Its vertex figure is the quasiregular cuboctahedron, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png.[2]

In hyperbolic 3-space, one quasiregular honeycomb is the alternated order-5 cubic honeycomb, h{4,3,5}, Coxeter diagrams: CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png, composed of alternating tetrahedral and icosahedral cells. Its vertex figure is the quasiregular icosidodecahedron, CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png. 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, CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png.

Quasiregular polychora and honeycombs: h{4,p,q}
Space Finite Affine Compact Paracompact
Name h{4,3,3} h{4,3,4} h{4,3,5} h{4,3,6} h{4,4,3} h{4,4,4}
\left\{3,{3\atop3}\right\} \left\{3,{3\atop4}\right\} \left\{3,{3\atop5}\right\} \left\{3,{3\atop6}\right\} \left\{4,{3\atop4}\right\} \left\{4,{4\atop4}\right\}
Coxeter
diagram
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node.png CDel nodes 10ru.pngCDel split2-44.pngCDel node.pngCDel 3.pngCDel node.png CDel nodes 10ru.pngCDel split2-44.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png CDel nodes.pngCDel split2-43.pngCDel node.pngCDel 3.pngCDel node 1.png CDel nodes.pngCDel split2-53.pngCDel node.pngCDel 3.pngCDel node 1.png CDel nodes.pngCDel split2-63.pngCDel node.pngCDel 3.pngCDel node 1.png CDel nodes.pngCDel split2-43.pngCDel node.pngCDel 4.pngCDel node 1.png CDel nodes.pngCDel split2-44.pngCDel node.pngCDel 4.pngCDel node 1.png
Image 16-cell nets.png Tetrahedral-octahedral honeycomb.png Alternated order 5 cubic honeycomb.png H3 444 FC boundary.png
Vertex
figure

r{p,3}
Uniform polyhedron-33-t1.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t1.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-53-t1.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-63-t1.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t1.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Square tiling uniform coloring 7.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png

Regular polychora honeycombs of the form {p,3,4} or CDel node 1.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png can have their symmetry cut in half as CDel node 1.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png into quasiregular form CDel node 1.pngCDel p.pngCDel node.pngCDel split1.pngCDel nodes.png, 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, CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel split1.pngCDel nodes.png.

Common vertex figure is the quasiregular tetratetrahedra, CDel node 1.pngCDel split1.pngCDel nodes.png, same as regular octahedron
Regular and Quasiregular honeycombs: {p,3,4} and {p,31,1}
Space Euclidean 4-space Euclidean 3-space Hyperbolic 3-space
Name {3,3,4}
{3,31,1} = \left\{3,{3\atop3}\right\}
{4,3,4}
{4,31,1} = \left\{4,{3\atop3}\right\}
{5,3,4}
{5,31,1} = \left\{5,{3\atop3}\right\}
{6,3,4}
{6,31,1} = \left\{6,{3\atop3}\right\}
Coxeter
diagram
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel nodes.png
Image 16-cell nets.png Bicolor cubic honeycomb.png H3 534 CC center.png H3 634 FC boundary.png
Cells
{p,3}
Uniform polyhedron-33-t0.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t0.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-53-t0.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-63-t0.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png

Similarly regular hyperbolic honeycombs of the form {p,3,6} or CDel node 1.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png can have their symmetry cut in half as CDel node 1.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.png into quasiregular form CDel node 1.pngCDel p.pngCDel node.pngCDel split1.pngCDel branch.png, creating alternately colored {p,3} cells. They have six cells around each edge, alternating in 2 colors. Their vertex figures are quasiregular triangular tilings, CDel node 1.pngCDel split1.pngCDel branch.png.

The common vertex figure is a quasiregular triangular tiling, CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel split1.pngCDel branch.png
Hyperbolic uniform honeycombs: {p,3,6} and {p,3[3]}
Form Paracompact Noncompact
Name {3,3,6}
{3,3[3]}
{4,3,6}
{4,3[3]}
{5,3,6}
{5,3[3]}
{6,3,6}
{6,3[3]}
{7,3,6}
{7,3[3]}
{8,3,6}
{8,3[3]}
... {∞,3,6}
{∞,3[3]}
CDel node 1.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel p.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel ultra.pngCDel node.pngCDel split1.pngCDel branch.pngCDel uaub.pngCDel nodes 11.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel splitplit1u.pngCDel branch4u 11.pngCDel uabc.pngCDel branch4u.pngCDel splitplit2u.pngCDel node.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel split1.pngCDel branch.png
Image H3 336 CC center.png H3 436 CC center.png H3 536 CC center.png H3 636 FC boundary.png
Cells Tetrahedron.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Hexahedron.png
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Dodecahedron.png
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 63-t0.png
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 237-1.png
{7,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 238-1.png
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 23i-1.png
{∞,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png

See also[edit]

Notes[edit]

  1. ^ Coxeter, H.S.M., Longuet-Higgins, M.S. and Miller, J.C.P. Uniform Polyhedra, Philosophical Transactions of the Royal Society of London 246 A (1954), pp. 401–450. (Section 7, The regular and quasiregular polyhedra p | q r)
  2. ^ Coxeter, Regular Polytopes, 4.7 Other honeycombs. p.69, p.88

References[edit]

External links[edit]