Quasiregular polyhedron

Last updated • 6 min readFrom Wikipedia, The Free Encyclopedia
Quasiregular figures
Right triangle domains (p q 2), CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png = r{p,q}
r{4,3} r{5,3} r{6,3} r{7,3}...r{,3}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t1.svg
(3.4)2
Uniform polyhedron-53-t1.svg
(3.5)2
Uniform tiling 63-t1.svg
(3.6)2
Triheptagonal tiling.svg
(3.7)2
H2 tiling 23i-2.png
(3.)2
Isosceles triangle domains (p p 3), CDel branch 10ru.pngCDel split2-pp.pngCDel node.png = CDel node h.pngCDel 6.pngCDel node.pngCDel p.pngCDel node.png = h{6,p}
h{6,4} h{6,5}h{6,6}h{6,7}...h{6,}
CDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png = CDel branch 10ru.pngCDel split2-44.pngCDel node.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png = CDel branch 10ru.pngCDel split2-55.pngCDel node.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png = CDel branch 10ru.pngCDel split2-66.pngCDel node.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 7.pngCDel node.png = CDel branch 10ru.pngCDel split2-77.pngCDel node.pngCDel node h.pngCDel 6.pngCDel node.pngCDel infin.pngCDel node.png = CDel branch 10ru.pngCDel split2-ii.pngCDel node.png
H2 tiling 344-4.png
(4.3)4
H2 tiling 355-4.png
(5.3)5
H2 tiling 366-4.png
(6.3)6
H2 tiling 377-4.png
(7.3)7
H2 tiling 3ii-4.png
(.3)
Isosceles triangle domains (p p 4), CDel label4.pngCDel branch 10ru.pngCDel split2-pp.pngCDel node.png = CDel node h.pngCDel 8.pngCDel node.pngCDel p.pngCDel node.png = h{8,p}
h{8,3} h{8,5}h{8,6}h{8,7}...h{8,}
CDel node h.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png =CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel node h.pngCDel 8.pngCDel node.pngCDel 5.pngCDel node.png =CDel label4.pngCDel branch 10ru.pngCDel split2-55.pngCDel node.pngCDel node h.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.png =CDel label4.pngCDel branch 10ru.pngCDel split2-66.pngCDel node.pngCDel node h.pngCDel 8.pngCDel node.pngCDel 7.pngCDel node.png =CDel label4.pngCDel branch 10ru.pngCDel split2-77.pngCDel node.pngCDel node h.pngCDel 8.pngCDel node.pngCDel infin.pngCDel node.png =CDel label4.pngCDel branch 10ru.pngCDel split2-ii.pngCDel node.png
H2 tiling 334-1.png
(4.3)3
H2 tiling 455-1.png
(4.5)5
H2 tiling 466-1.png
(4.6)6
H2 tiling 477-1.png
(4.7)7
H2 tiling 4ii-1.png
(4.)
Scalene triangle domain (5 4 3), CDel branch.pngCDel split2-45.pngCDel node.png
CDel branch 01rd.pngCDel split2-45.pngCDel node.pngCDel branch.pngCDel split2-45.pngCDel node 1.pngCDel branch 10ru.pngCDel split2-45.pngCDel node.png
H2 tiling 345-1.png
(3.5)4
H2 tiling 345-2.png
(4.5)3
H2 tiling 345-4.png
(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), CDel node 1.pngCDel split1-pp.pngCDel nodes.png = CDel node 1.pngCDel p.pngCDel node.pngCDel 4.pngCDel node h0.png = r{p,p} = {p,4}12
{3,4}12
r{3,3}
{4,4}12
r{4,4}
{5,4}12
r{5,5}
{6,4}12
r{6,6}...
{,4}12
r{,}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel split1.pngCDel nodes.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel split1-44.pngCDel nodes.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel split1-55.pngCDel nodes.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel split1-66.pngCDel nodes.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel split1-ii.pngCDel nodes.png
Uniform polyhedron-33-t1.png
(3.3)2
Uniform tiling 44-t1.svg
(4.4)2
H2 tiling 255-2.png
(5.5)2
H2 tiling 266-2.png
(6.6)2
H2 tiling 2ii-2.png
(.)2
Isosceles triangle domains (p p 3), CDel node 1.pngCDel split1-pp.pngCDel branch.png = CDel node 1.pngCDel p.pngCDel node.pngCDel 6.pngCDel node h0.png = {p,6}12
{3,6}12 {4,6}12 {5,6}12 {6,6}12... {,6}12
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel split1.pngCDel branch.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel split1-44.pngCDel branch.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel split1-55.pngCDel branch.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel split1-66.pngCDel branch.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel split1-ii.pngCDel branch.png
Uniform tiling 333-t1.svg
(3.3)3
H2 tiling 344-2.png
(4.4)3
H2 tiling 355-2.png
(5.5)3
H2 tiling 366-2.png
(6.6)3
H2 tiling 3ii-2.png
(.)3
Isosceles triangle domains (p p 4), CDel node 1.pngCDel split1-pp.pngCDel branch.pngCDel label4.png = CDel node 1.pngCDel p.pngCDel node.pngCDel 8.pngCDel node h0.png = {p,8}12
{3,8}12 {4,8}12 {5,8}12 {6,8}12... {,8}12
CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node h0.png =CDel node 1.pngCDel split1.pngCDel branch.pngCDel label4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node h0.png =CDel node 1.pngCDel split1-44.pngCDel branch.pngCDel label4.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 8.pngCDel node h0.png =CDel node 1.pngCDel split1-55.pngCDel branch.pngCDel label4.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 8.pngCDel node h0.png =CDel node 1.pngCDel split1-66.pngCDel branch.pngCDel label4.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 8.pngCDel node h0.png =CDel node 1.pngCDel split1-ii.pngCDel branch.pngCDel label4.png
H2 tiling 334-4.png
(3.3)4
H2 tiling 444-2.png
(4.4)4
H2 tiling 455-2.png
(5.5)4
H2 tiling 466-2.png
(6.6)4
H2 tiling 4ii-2.png (.)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.

Contents

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 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 = (3a.3b)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 = (4a.4b)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 = (3a.3b)3, alternating two colors of triangular faces.

Wythoff construction

Wythoffian construction diagram.svg
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 Wythoff construction-pqr.png
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
{p,q}CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngq | 2 p
{q,p}CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngp | 2 q
r{p,q}CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png or CDel node 1.pngCDel split1-pq.pngCDel nodes.png2 | p q

The convex quasiregular polyhedra

There are two uniform convex quasiregular polyhedra:

  1. The cuboctahedron , vertex configuration (3.4)2, Coxeter-Dynkin diagram CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
  2. The icosidodecahedron , vertex configuration (3.5)2, Coxeter-Dynkin diagramCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png

In addition, the octahedron, which is also regular, , 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 diagramCDel 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 cube octahedron, and icosahedron dodecahedron. The octahedron is the common core of a dual pair of tetrahedra (a compound known as the stella octangula); when derived in this way, the octahedron is sometimes called the tetratetrahedron, as tetrahedron tetrahedron.

RegularDual regularQuasiregular common core 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
r{3,3}
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.svg
Cube
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
3 | 2 4
Uniform polyhedron-43-t2.svg
Octahedron
{3,4}
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
4 | 2 3
Uniform polyhedron-43-t1.svg
Cuboctahedron
r{3,4}
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.svg
Dodecahedron
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
3 | 2 5
Uniform polyhedron-53-t2.svg
Icosahedron
{3,5}
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
5 | 2 3
Uniform polyhedron-53-t1.svg
Icosidodecahedron
r{3,5}
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 vertices fully, until each original edge is reduced to its midpoint.

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.

RegularDual regularQuasiregular combination Vertex figure
Uniform tiling 63-t0.svg
Hexagonal tiling
{6,3}
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
6 | 2 3
Uniform tiling 63-t2.svg
Triangular tiling
{3,6}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
3 | 2 6
Uniform tiling 63-t1.svg
Trihexagonal tiling
r{6,3}
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
2 | 3 6
Trihexagonal tiling vertfig.png
(3.6)2

The checkerboard pattern is a quasiregular coloring of the square tiling, vertex figure (4.4)2:

RegularDual regularQuasiregular combination Vertex figure
Uniform tiling 44-t0.svg
{4,4}
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
4 | 2 4
Uniform tiling 44-t2.svg
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
4 | 2 4
Uniform tiling 44-t1.svg
r{4,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
2 | 4 4
Square tiling vertfig.png
(4.4)2

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

Uniform tiling 333-t1.svg
h{6,3}
3 | 3 3
CDel branch 10ru.pngCDel split2.pngCDel node.png = CDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png

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.

RegularDual regularQuasiregular combination Vertex figure
Heptagonal tiling.svg
Heptagonal tiling
{7,3}
CDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png
7 | 2 3
Order-7 triangular tiling.svg
Triangular tiling
{3,7}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
3 | 2 7
Triheptagonal tiling.svg
Triheptagonal tiling
r{3,7}
CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png
2 | 3 7
Triheptagonal tiling vertfig.png
(3.7)2

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 , and the dodecadodecahedron :

RegularDual regularQuasiregular common core 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
r{3,5/2}
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
r{5,5/2}
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

Nine more are the hemipolyhedra, which are faceted forms of the aforementioned quasiregular polyhedra derived from rectification of regular polyhedra. These include equatorial faces passing through the centre of the polyhedra:

Quasiregular (rectified) Rectified tetrahedron.png
Tetratetrahedron
Cuboctahedron.png
Cuboctahedron
Icosidodecahedron.png
Icosidodecahedron
Great icosidodecahedron.png
Great icosidodecahedron
Dodecadodecahedron.png
Dodecadodecahedron
Quasiregular (hemipolyhedra) Tetrahemihexahedron.png
Tetrahemihexahedron
3/2 3 | 2
Octahemioctahedron.png
Octahemioctahedron
3/2 3 | 3
Small icosihemidodecahedron.png
Small icosihemidodecahedron
3/2 3 | 5
Great icosihemidodecahedron.png
Great icosihemidodecahedron
3/2 3 |5/3
Small dodecahemicosahedron.png
Small dodecahemicosahedron
5/35/2| 3
Vertex figure Tetrahemihexahedron vertfig.png
3.4.3/2.4
Octahemioctahedron vertfig.png
3.6.3/2.6
Small icosihemidodecahedron vertfig.png

3.10.3/2.10
Great icosihemidodecahedron vertfig.png
3.10/3.3/2.10/3
Small dodecahemicosahedron vertfig.png
5/2.6.5/3.6
Quasiregular (hemipolyhedra)  Cubohemioctahedron.png
Cubohemioctahedron
4/3 4 | 3
Small dodecahemidodecahedron.png
Small dodecahemidodecahedron
5/4 5 | 5
Great dodecahemidodecahedron.png
Great dodecahemidodecahedron
5/35/2|5/3
Great dodecahemicosahedron.png
Great dodecahemicosahedron
5/4 5 | 3
Vertex figure  Cubohemioctahedron vertfig.png
4.6.4/3.6
Small dodecahemidodecahedron vertfig.png
5.10.5/4.10
Great dodecahemidodecahedron vertfig.png
5/2.10/3.5/3.10/3
Great dodecahemicosahedron vertfig.png
5.6.5/4.6

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

ImageFaceted form
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

In the Euclidean plane, the sequence of hemipolyhedra continues with the following four star tilings, where apeirogons appear as the aforementioned equatorial polygons:

Original
rectified
tiling
Edge
diagram
SolidVertex
Config
WythoffSymmetry group
Uniform tiling 44-t1.svg
Square
tiling
4.oo.4-3.oo tiling frame.png Star tiling sha.gif 4..4/3.
4..-4.
4/3 4 |p4m
Uniform tiling 333-t1.svg
Triangular
tiling
3.oo.3.oo.3oo tiling-frame.png Star tiling ditatha.gif (3..3..3.)/23/2 | 3 p6m
Uniform tiling 63-t1.svg
Trihexagonal
tiling
6.oo.6-5.oo tiling-frame.png Star tiling hoha.gif 6..6/5.
6..-6.
6/5 6 |
Star tiling tha.gif .3..3/2
.3..-3
3/2 3 |

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:

  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 configurations 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.svg 7-3 rhombille tiling.svg H2-8-3-rhombic.svg
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

In higher dimensions, Coxeter defined a quasiregular polytope or honeycomb to have regular facets and quasiregular vertex figures. It follows that all vertex figures are congruent and that there are two kinds of facets, which alternate. [2]

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}
SpaceFiniteAffineCompactParacompact
Schläfli
symbol
h{4,3,3} h{4,3,4} h{4,3,5} h{4,3,6} h{4,4,3} h{4,4,4}
Coxeter
diagram
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel nodes 10ru.pngCDel split2-44.pngCDel node.pngCDel 3.pngCDel node.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel nodes 10ru.pngCDel split2-44.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1-43.pngCDel nodes.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1-53.pngCDel nodes.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1-63.pngCDel nodes.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel split1-43.pngCDel nodes.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel split1-44.pngCDel nodes.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.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.svg
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-53-t1.svg
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t1.svg
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t1.svg
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 44-t1.svg
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png

Regular polychora or 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 tetratetrahedron, , same as regular octahedron Uniform polyhedron-33-t1.png
Common vertex figure is the quasiregular tetratetrahedron, CDel node 1.pngCDel split1.pngCDel nodes.png, same as regular octahedron
Regular and Quasiregular honeycombs: {p,3,4} and {p,31,1}
SpaceEuclidean 4-spaceEuclidean 3-spaceHyperbolic 3-space
Name{3,3,4}
{3,31,1} =
{4,3,4}
{4,31,1} =
{5,3,4}
{5,31,1} =
{6,3,4}
{6,31,1} =
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.pngCDel 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.pngCDel 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.pngCDel 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, = Uniform tiling 333-t1.svg
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]}
FormParacompactNoncompact
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 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 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 Hyperbolic honeycomb 7-3-6 poincare.png Hyperbolic honeycomb 8-3-6 poincare.png Hyperbolic honeycomb i-3-6 poincare.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.svg
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Heptagonal tiling.svg
{7,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
H2-8-3-dual.svg
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
H2-I-3-dual.svg
{,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png

See also

Notes

  1. Coxeter, H.S.M., Longuet-Higgins, M.S. and Miller, J.C.P. Uniform Polyhedra, Philosophical Transactions of the Royal Society of London246 A (1954), pp. 401–450. (Section 7, The regular and quasiregular polyhedra p | q r)
  2. 1 2 Coxeter, Regular Polytopes, 4.7 Other honeycombs. p.69, p.88

Related Research Articles

<span class="mw-page-title-main">Cuboctahedron</span> Polyhedron with 8 triangular faces and 6 square faces

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron.

<span class="mw-page-title-main">Icosidodecahedron</span> Archimedean solid with 32 faces

In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such, it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

<span class="mw-page-title-main">Truncated cuboctahedron</span> Archimedean solid in geometry

In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.

<span class="mw-page-title-main">Rhombic dodecahedron</span> Catalan solid with 12 faces

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

In geometry, the term semiregular polyhedron is used variously by different authors.

<span class="mw-page-title-main">Rectification (geometry)</span> Operation in Euclidean geometry

In Euclidean geometry, rectification, also known as critical truncation or complete-truncation, is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.

<span class="mw-page-title-main">Uniform polyhedron</span> Isogonal polyhedron with regular faces

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular, quasi-regular, or semi-regular. The faces and vertices don't need to be convex, so many of the uniform polyhedra are also star polyhedra.

<span class="mw-page-title-main">Hexagonal prism</span> Prism with a 6-sided base

In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices.

<span class="mw-page-title-main">Cubic honeycomb</span> Only regular space-filling tessellation of the cube

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.

<span class="mw-page-title-main">Tetrahedral-octahedral honeycomb</span> Quasiregular space-filling tesselation

The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

<span class="mw-page-title-main">Bitruncated cubic honeycomb</span>

The bitruncated cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedra. It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.

<span class="mw-page-title-main">Truncation (geometry)</span> Operation that cuts polytope vertices, creating a new facet in place of each vertex

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

<span class="mw-page-title-main">Alternation (geometry)</span> Removal of alternate vertices

In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

<span class="mw-page-title-main">Conway polyhedron notation</span> Method of describing higher-order polyhedra

In geometry and topology, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.

<span class="mw-page-title-main">Honeycomb (geometry)</span> Tiling of euclidean or hyperbolic space of three or more dimensions

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.

In geometry, a polytope or a tiling is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.

<span class="mw-page-title-main">Uniform polytope</span> Isogonal polytope with uniform facets

In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the same must also be true within each lower-dimensional face of the polytope. In two dimensions a stronger definition is used: only the regular polygons are considered as uniform, disallowing polygons that alternate between two different lengths of edges.

In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

<span class="mw-page-title-main">Snub (geometry)</span> Geometric operation applied to a polyhedron

In geometry, a snub is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube and snub dodecahedron.

References