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Quasiregular polyhedron

  (Redirected from Quasiregular polytope)
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.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 infin.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t1.png
(3.4)2
Uniform polyhedron-53-t1.png
(3.5)2
Uniform polyhedron-63-t1.png
(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.png CDel node h.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png = CDel branch 10ru.pngCDel split2-55.pngCDel node.png CDel node h.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png = CDel branch 10ru.pngCDel split2-66.pngCDel node.png CDel node h.pngCDel 6.pngCDel node.pngCDel 7.pngCDel node.png = CDel branch 10ru.pngCDel split2-77.pngCDel node.png CDel 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.png CDel node h.pngCDel 8.pngCDel node.pngCDel 5.pngCDel node.png =CDel label4.pngCDel branch 10ru.pngCDel split2-55.pngCDel node.png CDel node h.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.png =CDel label4.pngCDel branch 10ru.pngCDel split2-66.pngCDel node.png CDel node h.pngCDel 8.pngCDel node.pngCDel 7.pngCDel node.png =CDel label4.pngCDel branch 10ru.pngCDel split2-77.pngCDel node.png CDel 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.png CDel branch.pngCDel split2-45.pngCDel node 1.png CDel 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.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel split1-44.pngCDel nodes.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel split1-55.pngCDel nodes.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel split1-66.pngCDel nodes.png CDel 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.png
(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.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel split1-44.pngCDel branch.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel split1-55.pngCDel branch.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel split1-66.pngCDel branch.png CDel 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.png
(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.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node h0.png =CDel node 1.pngCDel split1-44.pngCDel branch.pngCDel label4.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 8.pngCDel node h0.png =CDel node 1.pngCDel split1-55.pngCDel branch.pngCDel label4.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 8.pngCDel node h0.png =CDel node 1.pngCDel split1-66.pngCDel branch.pngCDel label4.png CDel 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 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. The dual figures is also sometimes considered quasiregular, except that they are edge-transitive, face-transitive, and alternate between two regular vertex figures.

There are only two regular 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 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 uniform 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.

Contents

Wythoff constructionEdit

 
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
  {p,q}       q | 2 p
  {q,p}       p | 2 q
  r{p,q}       or     2 | p q

The convex quasiregular polyhedraEdit

There are two uniform convex quasiregular polyhedra:

  1. The cuboctahedron  , vertex configuration (3.4)2, Coxeter-Dynkin diagram      
  2. The icosidodecahedron  , vertex configuration (3.5)2, Coxeter-Dynkin diagram      

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 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
r{3,3}
     
2 | 3 3
 
3.3.3.3
 
Cube
{4,3}
     
3 | 2 4
 
Octahedron
{3,4}
     
4 | 2 3
 
Cuboctahedron
r{3,4}
     
2 | 3 4
 
3.4.3.4
 
Dodecahedron
{5,3}
     
3 | 2 5
 
Icosahedron
{3,5}
     
5 | 2 3
 
Icosidodecahedron
r{3,4}
     
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 tilingsEdit

This sequence continues as the trihexagonal tiling, vertex figure (3.6)2 - 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
r{6,3}
     
2 | 3 6
 
(3.6)2

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

Regular Dual regular Quasiregular Vertex figure
 
{4,4}
     
4 | 2 4
 
{4,4}
     
4 | 2 4
 
r{4,4}
     
2 | 4 4
 
(4.4)2

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

 
h{6,3}
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.

Regular Dual regular Quasiregular Vertex figure
 
Heptagonal tiling
{7,3}
     
7 | 2 3
 
Triangular tiling
{3,7}
     
3 | 2 7
 
Triheptagonal tiling
r{3,7}
     
2 | 3 7
 
(3.7)2

Nonconvex examplesEdit

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  :

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
r{3,5/2}
       
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
r{5,5/2}
       
2 | 5 5/2
 
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)  
Tetratetrahedron
 
Cuboctahedron
 
Icosidodecahedron
 
Great icosidodecahedron
 
Dodecadodecahedron
Quasiregular (hemipolyhedra)  
Tetrahemihexahedron
3/2 3 | 2
 
Octahemioctahedron
3/2 3 | 3
 
Small icosihemidodecahedron
3/2 3 | 5
 
Great icosihemidodecahedron
3/2 3 | 5/3
 
Small dodecahemicosahedron
5/3 5/2 | 3
Vertex figure  
3.4.3/2.4
 
3.6.3/2.6
 

3.10.3/2.10
 
3.10/3.3/2.10/3
 
5/2.6.5/3.6
Quasiregular (hemipolyhedra)    
Cubohemioctahedron
4/3 4 | 3
 
Small dodecahemidodecahedron
5/4 5 | 5
 
Great dodecahemidodecahedron
5/3 5/2 | 5/3
 
Great dodecahemicosahedron
5/4 5 | 3
Vertex figure    
4.6.4/3.6
 
5.10.5/4.10
 
5/2.10/3.5/3.10/3
 
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:

Image Polyhedron name
Wythoff symbol
Coxeter diagram
Vertex figure
  Ditrigonal dodecadodecahedron
3 | 5/3 5
  or      
 
(5.5/3)3
  Small ditrigonal icosidodecahedron
3 | 5/2 3
  or      
 
(3.5/2)3
  Great ditrigonal icosidodecahedron
3/2 | 3 5
  or      
 
((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
Solid Vertex
Config
Wythoff Symmetry
 
Square
tiling
    4.∞.4/3.∞
4.∞.-4.∞
4/3 4 | ∞ p4m
 
Triangular
tiling
    (3.∞.3.∞.3.∞)/2 3/2 | 3 ∞ p6m
 
Trihexagonal
tiling
    6.∞.6/5.∞
6.∞.-6.∞
6/5 6 | ∞
  ∞.3.∞.3/2
∞.3.∞.-3
3/2 3 | ∞

Quasiregular dualsEdit

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 configuration are of the form V3.n.3.n, and Coxeter-Dynkin diagram      

           
Cube
V(3.3)2
     
Rhombic dodecahedron
V(3.4)2
     
Rhombic triacontahedron
V(3.5)2
     
Rhombille tiling
V(3.6)2
     
V(3.7)2
     
V(3.8)2
     

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 honeycombsEdit

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:         =      , 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,      .[2]

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 or 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,       =    .

 
Common vertex figure is the quasiregular tetratetrahedron,    , same as regular octahedron

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,    .

 
The common vertex figure is a quasiregular triangular tiling,       =    
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]}
       
     
       
     
       
     
       
     
       
     
       
     
       
     
       
     
Image              
Cells  
{3,3}
     
 
{4,3}
     
 
{5,3}
     
 
{6,3}
     
 
{7,3}
     
 
{8,3}
     
 
{∞,3}
     

See alsoEdit

NotesEdit

  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. ^ a b Coxeter, Regular Polytopes, 4.7 Other honeycombs. p.69, p.88

ReferencesEdit

  • Cromwell, P. Polyhedra, Cambridge University Press (1977).
  • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 2.3 Quasi-Regular Polyhedra. (p. 17), Quasi-regular honeycombs p.69

External linksEdit