Coxeter–Dynkin diagram

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See also: Dynkin diagram
Coxeter–Dynkin diagrams for the fundamental finite Coxeter groups
Coxeter–Dynkin diagrams for the fundamental affine Coxeter groups

In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge). An unlabeled branch implicitly represents order-3.

Each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams.

Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed, while Coxeter diagrams are undirected; secondly, Dynkin diagrams must satisfy an additional (crystallographic) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6. See Dynkin diagrams for details. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras.[1]

Description[edit]

Branches of a Coxeter–Dynkin diagram are labeled with a rational number p, representing a dihedral angle of 180°/p. When p = 2 the angle is 90° and the mirrors have no interaction, so the branch can be omitted from the diagram. If a branch is unlabeled, it is assumed to have p = 3, representing an angle of 60°. Two parallel mirrors have a branch marked with "∞". In principle, n mirrors can be represented by a complete graph in which all n(n − 1) / 2 branches are drawn. In practice, nearly all interesting configurations of mirrors include a number of right angles, so the corresponding branches are omitted.

Diagrams can be labeled by their graph structure. The first studied forms by Ludwig Schläfli are the orthoschemes as linear and generate regular polytopes and regular honeycombs. Plagioschemes are simplices represented by branching graphs, and cycloschemes are simplices represented by cyclic graphs.

Schläfli matrix[edit]

Every Coxeter diagram has a corresponding Schläfli matrix with matrix elements ai,j = aj,i = −2cos (π / p) where p is the branch order between the pairs of mirrors. As a matrix of cosines, it is also called a Gramian matrix after Jørgen Pedersen Gram. All Coxeter group Schläfli matrices are symmetric because their root vectors are normalized. It is related closely to the Cartan matrix, used in the similar but directed graph Dynkin diagrams in the limited cases of p = 2,3,4, and 6, which are NOT symmetric in general.

The determinant of the Schläfli matrix, called the Schläflian, and its sign determines whether the group is finite (positive), affine (zero), indefinite (negative). This rule is called Schläfli's Criterion.[2]

The eigenvalues of the Schläfli matrix determines whether a Coxeter group is of finite type (all positive), affine type (all non-zero, at least one is zero), or indefinite type (otherwise). The indefinite type is sometimes further subdivided, e.g. into hyperbolic and other Coxeter groups. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups. We use the following definition: A Coxeter group with connected diagram is hyperbolic if it is neither of finite nor affine type, but every proper connected subdiagram is of finite or affine type. A hyperbolic Coxeter group is compact if all subgroups are finite (i.e. have positive determinants), and paracompact if all its subgroups are finite or affine (i.e. have nonnegative determinants).

Finite and affine groups are also called elliptical and parabolic respectively. Hyperbolic groups are also called Lannér and F. Lannér who enumerated the compact hyperbolic groups in 1950,[3] and Koszul (or quasi-Lannér) for the paracompact groups.

Rank 2 Coxeter groups[edit]

For rank 2, the type of a Coxeter group is fully determined by the determinant of the Schläfli matrix, as it is simply the product of the eigenvalues: Finite type (positive determinant), affine type (zero determinant) or hyperbolic (negative determinant). Coxeter uses an equivalent bracket notation which lists sequences of branch orders as a substitute for the node-branch graphic diagrams.

Type Finite Affine Hyperbolic
Geometry Dihedral symmetry domains 1.png Dihedral symmetry domains 2.png Dihedral symmetry domains 3.png Dihedral symmetry domains 4.png ... Dihedral symmetry domains infinity.png Horocycle mirrors.png Dihedral symmetry ultra.png
Coxeter CDel node c1.png
[ ]
CDel node c1.pngCDel 2.pngCDel node c3.png
[2]
CDel node c1.pngCDel 3.pngCDel node c1.png
[3]
CDel node c1.pngCDel 4.pngCDel node c3.png
[4]
CDel node.pngCDel p.pngCDel node.png
[p]
CDel node c1.pngCDel infin.pngCDel node c3.png
[∞]
CDel node c2.pngCDel infin.pngCDel node c3.png
[∞]
CDel node c2.pngCDel ultra.pngCDel node c3.png
[iπ/λ]
Order 2 4 6 8 2p
Mirror lines are colored to correspond to Coxeter diagram nodes.
Fundamental domains are alternately colored.
Rank 2 Coxeter group diagrams
Order
p
Group Coxeter diagram Schläfli matrix
\left [\begin{matrix}2&a_{12}\\a_{21}&2\end{matrix}\right ] Determinant
(4-a21*a12)
Finite (Determinant>0)
2 I2(2) = A1xA1 CDel node.pngCDel 2.pngCDel node.png [2] \left [\begin{smallmatrix}2&0\\0&2\end{smallmatrix}\right ] 4
3 I2(3) = A2 CDel node.pngCDel 3.pngCDel node.png [3] \left [\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}\right ] 3
4 I2(4) = BC2 CDel node.pngCDel 4.pngCDel node.png [4] \left [\begin{smallmatrix}2&-\sqrt{2}\\-\sqrt{2}&2\end{smallmatrix}\right ] 2
5 I2(5) = H2 CDel node.pngCDel 5.pngCDel node.png [5] \left [\begin{smallmatrix}2&-\phi\\-\phi&2\end{smallmatrix}\right ] 4\sin^2(\pi/5)
=(5-\sqrt{5})/2

~1.38196601125

6 I2(6) = G2 CDel node.pngCDel 6.pngCDel node.png [6] \left [\begin{smallmatrix}2&-\sqrt{3}\\-\sqrt{3}&2\end{smallmatrix}\right ] 1
8 I2(8) CDel node.pngCDel 8.pngCDel node.png [8] \left [\begin{smallmatrix}2&-2\cos(\pi/8)\\-2\cos(\pi/8)&2\end{smallmatrix}\right ] 2-\sqrt{2}

~0.58578643763

10 I2(10) CDel node.pngCDel 10.pngCDel node.png [10] \left [\begin{smallmatrix}2&-2\cos(\pi/10)\\-2\cos(\pi/10)&2\end{smallmatrix}\right ] 4\sin^2(\pi/10)
=(3-\sqrt{5})/2

~0.38196601125

12 I2(12) CDel node.pngCDel 12.pngCDel node.png [12] \left [\begin{smallmatrix}2&-2\cos(\pi/12)\\-2\cos(\pi/12)&2\end{smallmatrix}\right ] 2-\sqrt{3}

~0.26794919243

p I2(p) CDel node.pngCDel p.pngCDel node.png [p] \left [\begin{smallmatrix}2&-2\cos(\pi/p)\\-2\cos(\pi/p)&2\end{smallmatrix}\right ] 4\sin^2(\pi/p)
Affine (Determinant=0)
I2(∞) = {\tilde{I}}_1 = {\tilde{A}}_1 CDel node.pngCDel infin.pngCDel node.png [∞] \left [\begin{smallmatrix}2&-2\\-2&2\end{smallmatrix}\right ] 0
Hyperbolic (Determinant<=0)
CDel node.pngCDel infin.pngCDel node.png [∞] \left [\begin{smallmatrix}2&-2\\-2&2\end{smallmatrix}\right ] 0
CDel node.pngCDel ultra.pngCDel node.png [iπ/λ] \left [\begin{smallmatrix}2&-2cosh(2\lambda)\\-2cosh(2\lambda)&2\end{smallmatrix}\right ] 4(1-\cosh^2(2\lambda)) ≤0

Geometric visualizations[edit]

The Coxeter–Dynkin diagram can be seen as a graphic description of the fundamental domain of mirrors. A mirror represents a hyperplane within a given dimensional spherical or Euclidean or hyperbolic space. (In 2D spaces, a mirror is a line, and in 3D a mirror is a plane).

These visualizations show the fundamental domains for 2D and 3D Euclidean groups, and 2D spherical groups. For each the Coxeter diagram can be deduced by identifying the hyperplane mirrors and labelling their connectivity, ignoring 90-degree dihedral angles (order 2).

Coxeter-dynkin plane groups.png
Coxeter groups in the euclidean plane with equivalent diagrams. Reflections are labeled as graph nodes R1, R2, etc. and are colored by their reflection order. Reflections at 90 degrees are inactive and therefore suppressed from the diagram. Parallel mirrors are connected by an ∞ labeled branch. The prismatic group {\tilde{I}}_1x{\tilde{I}}_1 is shown as a doubling of the {\tilde{C}}_2, but can also be created as rectangular domains from doubling the {\tilde{G}}_2 triangles. The {\tilde{A}}_2 is a doubling of the {\tilde{G}}_2 triangle.
Hyperbolic kaleidoscopes.png
Many Coxeter groups in the hyperbolic plane can be extended from the Euclidean cases as a series of hyperbolic solutions.
Coxeter-Dynkin 3-space groups.png
Coxeter groups in 3-space with diagrams. Mirrors (triangle faces) are labeled by opposite vertex 0..3. Branches are colored by their reflection order.
{\tilde{C}}_3 fills 1/48 of the cube. {\tilde{B}}_3 fills 1/24 of the cube. {\tilde{A}}_3 fills 1/12 of the cube.
Coxeter-Dynkin sphere groups.png
Coxeter groups in the sphere with equivalent diagrams. One fundamental domain is outlined in yellow. Domain vertices (and graph branches) are colored by their reflection order.

Finite Coxeter groups[edit]

See also polytope families for a table of end-node uniform polytopes associated with these groups.
  • Three different symbols are given for the same groups – as a letter/number, as a bracketed set of numbers, and as the Coxeter diagram.
  • The bifurcated Dn groups is half or alternated version of the regular Cn groups.
  • The bifurcated Dn and En groups are also labeled by a superscript form [3a,b,c] where a,b,c are the numbers of segments in each of the three branches.
Connected finite Dynkin graphs up to (ranks 1 to 9)
Rank Simple Lie groups Exceptional Lie groups  
{A}_{1+} {BC}_{2+} {D}_{2+} {E}_{3-8} {F}_{3-4} {G}_{2} {H}_{2-4} {I}_{2}(p)
1 A1=[]
CDel node.png
       
2 A2=[3]
CDel node.pngCDel 3.pngCDel node.png
BC2=[4]
CDel node.pngCDel 4.pngCDel node.png
D2=A1xA1
CDel nodes.png
  G2=[6]
CDel node.pngCDel 6.pngCDel node.png
H2=[5]
CDel node.pngCDel 5.pngCDel node.png
I2[p]
CDel node.pngCDel p.pngCDel node.png
3 A3=[32]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
BC3=[3,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
D3=A3
CDel nodes.pngCDel split2.pngCDel node.png
E3=A2A1
CDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodeb.png
F3=BC3
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
H3 
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
4 A4=[33]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
BC4=[32,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
D4=[31,1,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
E4=A4
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.png
F4
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
H4 
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5 A5=[34]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
BC5=[33,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
D5=[32,1,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
E5=D5
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png
 
6 A6=[35]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
BC6=[34,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
D6=[33,1,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
E6=[32,2,1]
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
7 A7=[36]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
BC7=[35,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
D7=[34,1,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
E7=[33,2,1]
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
8 A8=[37]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
BC8=[36,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
D8=[35,1,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
E8=[34,2,1]
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
9 A9=[38]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
BC9=[37,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
D9=[36,1,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
 
10+ .. .. .. ..

Application with uniform polytopes[edit]

In constructing uniform polytopes, nodes are marked as active by a ring if a generator point is off the mirror, creating a new edge between a generator point and its mirror image. An unringed node represents an inactive mirror that generates no new points.

Coxeter–Dynkin diagrams can explicitly enumerate nearly all classes of uniform polytope and uniform tessellations. Every uniform polytope with pure reflective symmetry (all but a few special cases have pure reflectional symmetry) can be represented by a Coxeter–Dynkin diagram with permutations of markups. Each uniform polytope can be generated using such mirrors and a single generator point: mirror images create new points as reflections, then polytope edges can be defined between points and a mirror image point. Faces can be constructed by cycles of edges created, etc. To specify the generating vertex, one or more nodes are marked with rings, meaning that the vertex is not on the mirror(s) represented by the ringed node(s). (If two or more mirrors are marked, the vertex is equidistant from them.) A mirror is active (creates reflections) only with respect to points not on it. A diagram needs at least one active node to represent a polytope.

All regular polytopes, represented by Schläfli symbol symbol {p, q, r, ...}, can have their fundamental domains represented by a set of n mirrors with a related Coxeter–Dynkin diagram of a line of nodes and branches labeled by p, q, r, ..., with the first node ringed.

Uniform polytopes with one ring correspond to generator points at the corners of the fundamental domain simplex. Two rings correspond to the edges of simplex and have a degree of freedom, with only the midpoint as the uniform solution for equal edge lengths. In general k-rings generators are on k-faces of the simplex, and if all the nodes are ringed, the generator point is in the interior of the simplex.

A secondary markup conveys a special case nonreflectional symmetry uniform polytopes. These cases exist as alternations of reflective symmetry polytopes. This markup removes the central dot of a ringed node, called a hole (circles with nodes removed), to imply alternate nodes deleted. The resulting polytope will have a subsymmetry of the original Coxeter group. If all the nodes are holes, the figure is considered a snub.

  • A single node represents a single mirror. This is called group A1. If ringed this creates a line segment perpendicular to the mirror, represented as {}.
  • Two unattached nodes represent two perpendicular mirrors. If both nodes are ringed, a rectangle can be created, or a square if the point is at equal distance from both mirrors.
  • Two nodes attached by an order-n branch can create an n-gon if the point is on one mirror, and a 2n-gon if the point is off both mirrors. This forms the I1(n) group.
  • Two parallel mirrors can represent an infinite polygon I1(∞) group, also called Ĩ1.
  • Three mirrors in a triangle form images seen in a traditional kaleidoscope and can be represented by three nodes connected in a triangle. Repeating examples will have branches labeled as (3 3 3), (2 4 4), (2 3 6), although the last two can be drawn as a line (with the 2 branches ignored). These will generate uniform tilings.
  • Three mirrors can generate uniform polyhedra; including rational numbers gives the set of Schwarz triangles.
  • Three mirrors with one perpendicular to the other two can form the uniform prisms.
Wythoffian construction diagram.png
There are 7 reflective uniform constructions within a general triangle, based on 7 topological generator positions within the fundamental domain. Every active mirror generates an edge, with two active mirrors have generators on the domain sides and three active mirrors has the generator in the interior. One or two degrees of freedom can be solved for a unique position for equal edge lengths of the resulting polyhedron or tiling.
Polyhedron truncation example3.png
Example 7 generators on octahedral symmetry, fundamental domain triangle (4 3 2), with 8th snub generation as an alternation

The duals of the uniform polytopes are sometimes marked up with a perpendicular slash replacing ringed nodes, and a slash-hole for hole nodes of the snubs. For example CDel node 1.pngCDel 2.pngCDel node 1.png represents a rectangle (as two active orthogonal mirrors), and CDel node f1.pngCDel 2.pngCDel node f1.png represents its dual polygon, the rhombus.

Example polyhedra and tilings[edit]

For example, the BC3 Coxeter group has a diagram: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png. This is also called octahedral symmetry.

There are 7 convex uniform polyhedra that can be constructed from this symmetry group and 3 from its alternation subsymmetries, each with a uniquely marked up Coxeter–Dynkin diagram. The Wythoff symbol represents a special case of the Coxeter diagram for rank 3 graphs, with all 3 branch orders named, rather than suppressing the order 2 branches. The Wythoff symbol is able to handle the snub form, but not general alternations without all nodes ringed.

Uniform octahedral polyhedra
Symmetry: [4,3], (*432) [4,3]+
(432)
[1+,4,3] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3} t{4,3} r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{3,4}
s{31,1}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png =
CDel nodes 10ru.pngCDel split2.pngCDel node.png or CDel nodes 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel nodes 10ru.pngCDel split2.pngCDel node 1.png or CDel nodes 01rd.pngCDel split2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png =
CDel node h.pngCDel split1.pngCDel nodes hh.png
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg
Uniform polyhedron-33-t02.png
Uniform polyhedron-43-t12.svg
Uniform polyhedron-33-t012.png
Uniform polyhedron-43-t2.svg
Uniform polyhedron-33-t1.png
Uniform polyhedron-43-t02.png
Rhombicuboctahedron uniform edge coloring.png
Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Uniform polyhedron-33-t0.pngUniform polyhedron-33-t2.png Uniform polyhedron-33-t01.pngUniform polyhedron-33-t12.png Uniform polyhedron-43-h01.svg
Uniform polyhedron-33-s012.png
Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Octahedron.svg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.svg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Tetrahedron.svg Triakistetrahedron.jpg Dodecahedron.svg

The same constructions can be made on disjointed (orthogonal) Coxeter groups like the uniform prisms, and can be seen more clearly as tilings of dihedrons and hosohedrons on the sphere, like this [6]×[] or [6,2] family:

Uniform hexagonal dihedral spherical polyhedra
Symmetry: [6,2], (*622) [6,2]+, (622) [1+,6,2], (322) [6,2+], (2*3)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png CDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png
Hexagonal dihedron.png Dodecagonal dihedron.png Hexagonal dihedron.png Spherical hexagonal prism.png Spherical hexagonal hosohedron.png Spherical truncated trigonal prism.png Spherical dodecagonal prism2.png Spherical hexagonal antiprism.png Trigonal dihedron.png Spherical trigonal antiprism.png
{6,2} t{6,2} r{6,2} 2t{6,2}=t{2,6} 2r{6,2}={2,6} rr{6,2} tr{6,2} sr{6,2} h{6,2} s{2,6}
Uniform duals
CDel node f1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 2.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 2.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 2.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 2.pngCDel node f1.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 2x.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png CDel node.pngCDel 6.pngCDel node fh.pngCDel 2x.pngCDel node fh.png
Spherical hexagonal hosohedron.png Spherical dodecagonal hosohedron.png Spherical hexagonal hosohedron.png Spherical hexagonal bipyramid.png Hexagonal dihedron.png Spherical hexagonal bipyramid.png Spherical dodecagonal bipyramid.png Spherical hexagonal trapezohedron.png Spherical trigonal hosohedron.png Spherical trigonal trapezohedron.png
V62 V122 V62 V4.4.6 V26 V4.4.6 V4.4.12 V3.3.3.6 V32 V3.3.3.3

In comparison the [6,3], CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png family produces a parallel set of 7 uniform tilings of the Euclidean plane, and their dual tilings. There are again 3 alternations and some half symmetry version.

Uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+
(632)
[1+,6,3]
(*333)
[6,3+]
(3*3)
{6,3} t{6,3} r{6,3}
r{3[3]}
t{3,6}
t{3[3]}
{3,6}
{3[3]}
rr{6,3}
s2{6,3}
tr{6,3} sr{6,3} h{6,3}
{3[3]}
h2{6,3}
r{3[3]}
s{3,6}
s{3[3]}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel branch 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel branch 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel branch.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png =
CDel branch 10ru.pngCDel split2.pngCDel node.png or CDel branch 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel branch 10ru.pngCDel split2.pngCDel node 1.png or CDel branch 01rd.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
= CDel branch hh.pngCDel split2.pngCDel node h.png
Uniform tiling 63-t0.png Uniform tiling 63-t01.png Uniform tiling 63-t1.png
Uniform tiling 333-t01.png
Uniform tiling 63-t12.png
Uniform tiling 333-t012.png
Uniform tiling 63-t2.png
Uniform tiling 333-t2.png
Uniform tiling 63-t02.png
Rhombitrihexagonal tiling snub edge coloring.png
Uniform tiling 63-t012.png Uniform tiling 63-snub.png Uniform tiling 333-t0.pngUniform tiling 333-t1.png Uniform tiling 333-t02.pngUniform tiling 333-t12.png Uniform tiling 63-h12.png
Uniform tiling 333-snub.png
Uniform duals
V63 V3.122 V(3.6)2 V63 V36 V3.4.12.4 V.4.6.12 V34.6 V36 V(3.6)2 V36
CDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Uniform tiling 63-t2.png Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg Rhombic star tiling.png Uniform tiling 63-t2.png Uniform tiling 63-t0.png Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg Uniform tiling 63-t0.png Rhombic star tiling.png Uniform tiling 63-t0.png

In the hyperbolic plane [7,3], CDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png family produces a parallel set of uniform tilings of the Euclidean plane, and their dual tilings. There is only 1 alternation (snub) since all branch orders are odd. Many other hyperbolic families of uniform tilings can be seen at uniform tilings in hyperbolic plane.

Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732) [7,3]+, (732)
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 7.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 73-t0.png Uniform tiling 73-t01.png Uniform tiling 73-t1.png Uniform tiling 73-t12.png Uniform tiling 73-t2.png Uniform tiling 73-t02.png Uniform tiling 73-t012.png Uniform tiling 73-snub.png
{7,3} t{7,3} r{7,3} 2t{7,3}=t{3,7} 2r{7,3}={3,7} rr{7,3} tr{7,3} sr{7,3}
Uniform duals
CDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Uniform tiling 73-t2.png Ord7 triakis triang til.png Order73 qreg rhombic til.png Order3 heptakis heptagonal til.png Uniform tiling 73-t0.png Deltoidal triheptagonal til.png Order-3 heptakis heptagonal tiling.png Ord7 3 floret penta til.png
V73 V3.14.14 V3.7.3.7 V6.6.7 V37 V3.4.7.4 V4.6.14 V3.3.3.3.7

Affine Coxeter groups[edit]

Families of convex uniform Euclidean tessellations are defined by the affine Coxeter groups. These groups are identical to the finite groups with the inclusion of one added node. In letter names they are given the same letter with a "~" above the letter. The index refers to the finite group, so the rank is the index plus 1. (Ernst Witt symbols for the affine groups are given as also)

  1. {\tilde{A}}_{n-1}: diagrams of this type are cycles. (Also Pn)
  2. {\tilde{C}}_{n-1} is associated with the hypercube regular tessellation {4, 3, ...., 4} family. (Also Rn)
  3. {\tilde{B}}_{n-1} related to C by one removed mirror. (Also Sn)
  4. {\tilde{D}}_{n-1} related to C by two removed mirrors. (Also Qn)
  5. {\tilde{E}}_6, {\tilde{E}}_7, {\tilde{E}}_8. (Also T7, T8, T9)
  6. {\tilde{F}}_4 forms the {3,4,3,3} regular tessellation. (Also U5)
  7. {\tilde{G}}_2 forms 30-60-90 triangle fundamental domains. (Also V3)
  8. {\tilde{I}}_1 is two parallel mirrors. ( = {\tilde{A}}_1 = {\tilde{C}}_1) (Also W2)

Composite groups can also be defined as orthogonal projects. The most common use {\tilde{A}}_1, like {\tilde{A}}_1^2, CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png represents square or rectangular checker board domains in the Euclidean plane. And {\tilde{A}}_1 {\tilde{G}}_2 CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel split1.pngCDel branch.png represents triangular prism fundamental domains in Euclidean 3-space.

Affine Coxeter graphs up to (2 to 10 nodes)
Rank {\tilde{A}}_{1+} (P2+) {\tilde{B}}_{3+} (S4+) {\tilde{C}}_{1+} (R2+) {\tilde{D}}_{4+} (Q5+) {\tilde{E}}_n (Tn+1) / {\tilde{F}}_4 (U5) / {\tilde{G}}_2 (V3)
2 {\tilde{A}}_1=[∞]
CDel node.pngCDel infin.pngCDel node.png
  {\tilde{C}}_1=[∞]
CDel node.pngCDel infin.pngCDel node.png
   
3 {\tilde{A}}_{2}=[3[3]]
* CDel branch.pngCDel split2.pngCDel node.png
{\tilde{C}}_{2}=[4,4]
* CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{G}}_{2}=[6,3]
* CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
4 {\tilde{A}}_{3}=[3[4]]
* CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png
{\tilde{B}}_{3}=[4,31,1]
* CDel nodes.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{C}}_{3}=[4,3,4]
* CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{D}}_{3}=[31,1,3−1,31,1]
CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png = {\tilde{A}}_{3}
5 {\tilde{A}}_{4}=[3[5]]
* CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
{\tilde{B}}_{4}=[4,3,31,1]
* CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{C}}_{4}=[4,32,4]
* CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{D}}_{4}=[31,1,1,1]
* CDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png
{\tilde{F}}_{4}=[3,4,3,3]
* CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6 {\tilde{A}}_{5}=[3[6]]
* CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
{\tilde{B}}_{5}=[4,32,31,1]
* CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{C}}_{5}=[4,33,4]
* CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{D}}_{5}=[31,1,3,31,1]
* CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
 
7 {\tilde{A}}_{6}=[3[7]]
* CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
{\tilde{B}}_{6}=[4,33,31,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{C}}_{6}=[4,34,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{D}}_{6}=[31,1,32,31,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
{\tilde{E}}_{6}=[32,2,2]
CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8 {\tilde{A}}_{7}=[3[8]]
* CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
{\tilde{B}}_{7}=[4,34,31,1]
* CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{C}}_{7}=[4,35,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{D}}_{7}=[31,1,33,31,1]
* CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
{\tilde{E}}_{7}=[33,3,1]
* CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
9 {\tilde{A}}_{8}=[3[9]]
* CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
{\tilde{B}}_{8}=[4,35,31,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{C}}_{8}=[4,36,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{D}}_{8}=[31,1,34,31,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
{\tilde{E}}_{8}=[35,2,1]
* CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
10 {\tilde{A}}_{9}=[3[10]]
* CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
{\tilde{B}}_{9}=[4,36,31,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{C}}_{9}=[4,37,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{D}}_{9}=[31,1,35,31,1]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
11 ... ... ... ...

Hyperbolic Coxeter groups[edit]

There are many infinite hyperbolic Coxeter groups. Hyperbolic groups are categorized as compact or not, with compact groups having bounded fundamental domains. Compact simplex hyperbolic groups (Lannér simplices) exist as rank 3 to 5. Paracompact simplex groups (Koszul simplices) exist up to rank 10. Hypercompact (Vinberg polytopes) groups have been explored but not been fully determined. In 2006, Allcock proved that there are infinitely many compact Vinberg polytopes for dimension up to 6, and infinitely many finite-volume Viberg polytopes for dimension up to 19,[4] so a complete enumeration is not possible. All of these fundamental reflective domains, both simplices and nonsimplices, are often called Coxeter polytopes or sometimes less accurately Coxeter polyhedra.

Hyperbolic groups in H2[edit]

Poincaré disk model of fundamental domain triangles
Example right triangles [p,q]
H2checkers 237.png
[3,7]
H2checkers 238.png
[3,8]
Hyperbolic domains 932.png
[3,9]
H2checkers 23i.png
[3,∞]
H2checkers 245.png
[4,5]
H2checkers 246.png
[4,6]
H2checkers 247.png
[4,7]
H2checkers 248.png
[4,8]
H2checkers 24i.png
[∞,4]
H2checkers 255.png
[5,5]
H2checkers 256.png
[5,6]
H2checkers 257.png
[5,7]
H2checkers 266.png
[6,6]
H2checkers 2ii.png
[∞,∞]
Example general triangles [(p,q,r)]
H2checkers 334.png
[(3,3,4)]
H2checkers 335.png
[(3,3,5)]
H2checkers 336.png
[(3,3,6)]
H2checkers 337.png
[(3,3,7)]
H2checkers 33i.png
[(3,3,∞)]
H2checkers 344.png
[(3,4,4)]
H2checkers 366.png
[(3,6,6)]
H2checkers 3ii.png
[(3,∞,∞)]
H2checkers 666.png
[(6,6,6)]
H2checkers iii.png
[(∞,∞,∞)]

Two-dimensional hyperbolic triangle groups exists as rank 3 Coxeter diagrams, defined by triangle (p q r) for:

\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1.

There are infinitely many compact triangular hyperbolic Coxeter groups, including linear and triangle graphs. The linear graphs exist for right triangles (with r=2).[5]

Compact hyperbolic Coxeter groups
Linear Cyclic
[p,q], CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png:
2(p+q)<pq

CDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 9.pngCDel node.pngCDel 3.pngCDel node.png
...
CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
...
CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png
CDel node.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png
...

∞ [(p,q,r)], CDel pqr.png: p+q+r>9

CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png

CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png
CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.png
CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.png

CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.pngCDel 4.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.pngCDel 4.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.pngCDel 5.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.png

CDel 3.pngCDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png
...

Paracompact Coxeter groups of rank 3 exist as limits to the compact ones.

Linear graphs Cyclic graphs
  • [p,∞] CDel node.pngCDel p.pngCDel node.pngCDel infin.pngCDel node.png
  • [∞,∞] CDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png
  • [(p,q,∞)] CDel 3.pngCDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel infin.pngCDel 3.png
  • [(p,∞,∞)] CDel 3.pngCDel node.pngCDel p.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel 3.png
  • [(∞,∞,∞)] CDel 3.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel 3.png


Arithmetic triangle group[edit]

A finite subset of hyperbolic triangle groups are arithmetic groups. By computer search the complete list was determined by Kisao Takeuchi in his 1977 paper Arithmetic triangle groups.[6] There are 85 total, 76 compact and 9 paracompact.

Right triangles (p q 2) General triangles (p q r)
Compact groups: (76)
CDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel 9.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel 10.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel 11.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel 12.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel 14.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel 16.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel 18.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel 2x.pngCDel 4.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel 3x.pngCDel 0x.pngCDel node.png
CDel node.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png, CDel node.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png, CDel node.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png, CDel node.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png, CDel node.pngCDel 4.pngCDel node.pngCDel 10.pngCDel node.png, CDel node.pngCDel 4.pngCDel node.pngCDel 12.pngCDel node.png, CDel node.pngCDel 4.pngCDel node.pngCDel 18.pngCDel node.png
CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png, CDel node.pngCDel 5.pngCDel node.pngCDel 6.pngCDel node.png, CDel node.pngCDel 5.pngCDel node.pngCDel 8.pngCDel node.png, CDel node.pngCDel 5.pngCDel node.pngCDel 10.pngCDel node.png, CDel node.pngCDel 5.pngCDel node.pngCDel 20.pngCDel node.png, CDel node.pngCDel 5.pngCDel node.pngCDel 3x.pngCDel 0x.pngCDel node.png
CDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png, CDel node.pngCDel 6.pngCDel node.pngCDel 8.pngCDel node.png, CDel node.pngCDel 6.pngCDel node.pngCDel 12.pngCDel node.png, CDel node.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node.png, CDel node.pngCDel 7.pngCDel node.pngCDel 14.pngCDel node.png
CDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png, CDel node.pngCDel 8.pngCDel node.pngCDel 16.pngCDel node.png, CDel node.pngCDel 9.pngCDel node.pngCDel 18.pngCDel node.png, CDel node.pngCDel 10.pngCDel node.pngCDel 10.pngCDel node.png, CDel node.pngCDel 12.pngCDel node.pngCDel 12.pngCDel node.png, CDel node.pngCDel 12.pngCDel node.pngCDel 2x.pngCDel 4.pngCDel node.png, CDel node.pngCDel 15.pngCDel node.pngCDel 3x.pngCDel 0x.pngCDel node.png, CDel node.pngCDel 18.pngCDel node.pngCDel 18.pngCDel node.png

Paracompact right triangles: (4)

CDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png, CDel node.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node.png, CDel node.pngCDel 6.pngCDel node.pngCDel infin.pngCDel node.png, CDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png
General triangles: (39)
CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 9.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 12.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 15.png
CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 6.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 12.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 18.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.pngCDel 8.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.pngCDel 2x.pngCDel 4.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 10.pngCDel node.pngCDel 3x.pngCDel 0x.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 12.pngCDel node.pngCDel 12.png
CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png, CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 5.png, CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 6.png, CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 9.png, CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.png, CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.png, CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.pngCDel 8.png, CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 16.pngCDel node.pngCDel 16.png
CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.png, CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.pngCDel 10.png, CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.pngCDel 15.png, CDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 10.pngCDel node.pngCDel 10.png
CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.png, CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 12.pngCDel node.pngCDel 12.png, CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2x.pngCDel 4.pngCDel node.pngCDel 2x.pngCDel 4.png, CDel 3.pngCDel node.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node.pngCDel 7.png, CDel 3.pngCDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.pngCDel 8.png, CDel 3.pngCDel node.pngCDel 9.pngCDel node.pngCDel 9.pngCDel node.pngCDel 9.png, CDel 3.pngCDel node.pngCDel 9.pngCDel node.pngCDel 18.pngCDel node.pngCDel 18.png, CDel 3.pngCDel node.pngCDel 12.pngCDel node.pngCDel 12.pngCDel node.pngCDel 12.png, CDel 3.pngCDel node.pngCDel 15.pngCDel node.pngCDel 15.pngCDel node.pngCDel 15.png

Paracompact general triangles: (5)

CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.png, CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.png, CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel infin.png, CDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.pngCDel infin.png, CDel 3.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.png
(2 3 7), (2 3 8), (2 3 9), (2 3 10), (2 3 11), (2 3 12), (2 3 14), (2 3 16), (2 3 18), (2 3 24), (2 3 30)
(2 4 5), (2 4 6), (2 4 7), (2 4 8), (2 4 10), (2 4 12), (2 4 18),
(2 5 5), (2 5 6), (2 5 8), (2 5 10), (2 5 20), (2 5 30)
(2 6 6), (2 6 8), (2 6 12)
(2 7 7), (2 7 14), (2 8 8), (2 8 16), (2 9 18)
(2 10 10) (2 12 12) (2 12 24), (2 15 30), (2 18 18)
(2 3 ∞) (2,4 ∞) (2,6 ∞) (2 ∞ ∞)
(3 3 4), (3 3 5), (3 3 6), (3 3 7), (3 3 8), (3 3 9), (3 3 12), (3 3 15)
(3 4 4), (3 4 6), (3 4 12), (3 5 5), (3 6 6), (3 6 18), (3 8 8), (3 8 24), (3 10 30), (3 12 12)
(4 4 4), (4 4 5), (4 4 6), (4 4 9), (4 5 5), (4 6 6), (4 8 8), (4 16 16)
(5 5 5), (5 5 10), (5 5 15), (5 10 10)
(6 6 6), (6 12 12), (6 24 24)
(7 7 7) (8 8 8) (9 9 9) (9 18 18) (12 12 12) (15 15 15)
(3,3 ∞) (3 ∞ ∞)
(4,4 ∞) (6 6 ∞) (∞ ∞ ∞)

Hyperbolic Coxeter polygons above triangles[edit]

Fundamental domains of quadrilateral groups
Hyperbolic domains 3222.png
CDel node.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png or CDel branch.pngCDel 2a2b-cross.pngCDel nodes.png
[∞,3,∞]
[iπ/λ1,3,iπ/λ2]
(*3222)
Hyperbolic domains 2233.png
CDel labelinfin.pngCDel branch.pngCDel split2.pngCDel node.pngCDel infin.pngCDel node.png or CDel branch.pngCDel 3a2b-cross.pngCDel nodes.png
[((3,∞,3)),∞]
[((3,iπ/λ1,3)),iπ/λ2]
(*3322)
H2chess 246a.png
CDel labelinfin.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel labelinfin.png or CDel branch.pngCDel 2a2b-cross.pngCDel branch.png
[(3,∞)[2]]
[(3,iπ/λ1,3,iπ/λ2)]
(*3232)
H2chess 248a.png
CDel labelinfin.pngCDel branch.pngCDel 4a4b.pngCDel branch.pngCDel labelinfin.png or CDel label4.pngCDel branch.pngCDel 2a2b-cross.pngCDel branch.pngCDel label4.png
[(4,∞)[2]]
[(4,iπ/λ1,4,iπ/λ2)]
(*4242)
H2chess 246b.png
CDel branch.pngCDel 3a3b-cross.pngCDel branch.png


(*3333)
Domains with ideal vertices
Hyperbolic domains i222.png
CDel labelinfin.pngCDel branch.pngCDel 2a2b-cross.pngCDel nodes.png
[iπ/λ1,∞,iπ/λ2]
(*∞222)
Hyperbolic domains ii22.png
CDel labelinfin.pngCDel branch.pngCDel ia2b-cross.pngCDel nodes.png

(*∞∞22)
H2chess 24ia.png
CDel labelinfin.pngCDel branch.pngCDel 2a2b-cross.pngCDel branch.pngCDel labelinfin.png
[(iπ/λ1,∞,iπ/λ2,∞)]
(*2∞2∞)
H2chess 24ib.png
CDel labelinfin.pngCDel branch.pngCDel iaib-cross.pngCDel branch.pngCDel labelinfin.png

(*∞∞∞∞)
H2chess 248b.png
CDel label4.pngCDel branch.pngCDel 4a4b-cross.pngCDel branch.pngCDel label4.png

(*4444)

Other H2 hyperbolic kaleidoscopes can be constructed from higher order polygons. Like triangle groups these kaleidoscopes can be identified by a cyclic sequence of mirror intersection orders around the fundamental domain, as (a b c d ...), or equivalently in orbifold notation as *abcd.... Coxeter–Dynkin diagrams for these polygonal kaleidoscopes can be seen as a degenerate (n-1)-simplex fundamental domains, with a cyclic of branches order a,b,c... and the remaining n*(n-3)/2 branches are labeled as infinite (∞) representing the non-intersecting mirrors. The only nonhyperbolic example is Euclidean symmetry four mirrors in a square or rectangle as CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png, [∞,2,∞] (orbifold *2222). Another branch representation for non-intersecting mirrors by Vinberg gives infinite branches as dotted or dashed lines, so this diagram can be shown as CDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png, with the four order-2 branches suppressed around the perimeter.

For example a quadrilateral domain (a b c d) will have two infinite order branches connecting ultraparallel mirrors. The smallest hyperbolic example is CDel node.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png, [∞,3,∞] or [iπ/λ1,3,iπ/λ2] (orbifold *3222), where (λ12) are the distance between the ultraparallel mirrors. The alternate expression is CDel branch.pngCDel 2a2b-cross.pngCDel nodes.png, with three order-2 branches suppressed around the perimeter. Similarly (2 3 2 3) (orbifold *3232) can be represented as CDel branch.pngCDel 2a2b-cross.pngCDel branch.png and (3 3 3 3), (orbifold *3333) can be represented as a complete graph CDel branch.pngCDel 3a3b-cross.pngCDel branch.png.

The highest quadrilateral domain (∞ ∞ ∞ ∞) is an infinite square, represented by a complete tetrahedral graph with 4 perimeter branches as ideal vertices and two diagonal branches as infinity (shown as dotted lines) for ultraparallel mirrors: CDel labelinfin.pngCDel branch.pngCDel iaib-cross.pngCDel branch.pngCDel labelinfin.png.

Compact (Lannér simplex groups)[edit]

Compact hyperbolic groups are called Lannér groups after Folke Lannér who first studied them in 1950.[7] They only exist as rank 4 and 5 graphs. Coxeter studied the linear hyperbolic coxeter groups in his 1954 paper Regular Honeycombs in hyperbolic space,[8] which included two rational solutions in hyperbolic 4-space: [5/2,5,3,3] = CDel node.pngCDel 5.pngCDel rat.pngCDel 2x.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png and [5,5/2,5,3] = CDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel 2x.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png.

Ranks 4–5[edit]

The fundamental domain of either of the two bifurcating groups, [5,31,1] and [5,3,31,1], is double that of a corresponding linear group, [5,3,4] and [5,3,3,4] respectively. Letter names are given by Johnson as extended Witt symbols.[9]

Compact hyperbolic Coxeter groups
Dimension
Hd
Rank Total count Linear Bifurcating Cyclic
H3 4 9
3:

{\bar{BH}}_3 = [4,3,5]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{K}}_3 = [5,3,5]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{J}}_3 = [3,5,3]: CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png

{\bar{DH}}_3 = [5,31,1]: CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png

{\widehat{AB}}_3 = [(33,4)]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.png 
{\widehat{AH}}_3 = [(33,5)]: CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.png 
{\widehat{BB}}_3 = [(3,4)[2]]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png
{\widehat{BH}}_3 = [(3,4,3,5)]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png
{\widehat{HH}}_3 = [(3,5)[2]]: CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png

H4 5 5
3:

{\bar{H}}_4 = [33,5]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{BH}}_4 = [4,3,3,5]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{K}}_4 = [5,3,3,5]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png

{\bar{DH}}_4 = [5,3,31,1]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png

{\widehat{AF}}_4 = [(34,4)]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png

Paracompact (Koszul simplex groups)[edit]

An example order-3 apeirogonal tiling, {∞,3} with one green apeirogon and its circumscribed horocycle

Paracompact (also called noncompact) hyperbolic Coxeter groups contain affine subgroups and have asymptotic simplex fundamental domains. The highest paracompact hyperbolic Coxeter group is rank 10. These groups are named after French mathematician Jean-Louis Koszul.[10] They are also called quasi-Lannér groups extending the compact Lannér groups. The list was determined complete by computer search by M. Chein and published in 1969.[11]

By Vinberg, all but eight of these 72 compact and paracompact simplices are arithmetic. Two of the nonarithmetic groups are compact: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png and CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png. The other six nonarithmetic groups are all paracompact, with five 3-dimensional groups CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png, CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.png, CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png, CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label6.png, and CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label6.png, and one 5-dimensional group CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png.

Ideal simplices[edit]

Ideal fundamental domains of CDel labelinfin.pngCDel branch.pngCDel split2-ii.pngCDel node.png, [(∞,∞,∞)] seen in the Poincare disk model

There are 5 hyperbolic Coxeter groups expressing ideal simplices, graphs where removal of any one node results in an affine Coxeter group. Thus all vertices of this ideal simplex are at infinity.[12]

Rank Ideal group Affine subgroups
3 [(∞,∞,∞)] CDel labelinfin.pngCDel branch.pngCDel split2-ii.pngCDel node.png [∞] CDel node.pngCDel infin.pngCDel node.png
4 [4[4]] CDel label4.pngCDel branch.pngCdel 4-4.pngCDel branch.pngCDel label4.png [4,4] CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
4 [3[3,3]] CDel tet.png [3[3]] CDel node.pngCDel split1.pngCDel branch.png
4 [(3,6)[2]] CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label6.png [3,6] CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
6 [(3,3,4)[2]] CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel label4.png [4,3,3,4], [3,4,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Ranks 4–10[edit]

Infinite Euclidean cells like a hexagonal tiling, properly scaled converge to a single ideal point at infinity, like the hexagonal tiling honeycomb, {6,3,3}, as shown with this single cell in a Poincaré disk model projection.
Further information: Paracompact uniform honeycombs

There are a total of 58 paracompact hyperbolic Coxeter groups from rank 4 through 10. All 58 are grouped below in five categories. Letter symbols are given by Johnson as Extended Witt symbols, using PQRSTWUV from the affine Witt symbols, and adding LMNOXYZ. These hyperbolic groups are given an overline, or a hat, for cycloschemes. The bracket notation from Coxeter is a linearized representation of the Coxeter group.

Hyperbolic paracompact groups
Rank Total
count
Groups
4 23

{\widehat{BR}}_3 = [(3,3,4,4)]: CDel label4.pngCDel branch.pngCDel 4-3.pngCDel branch.pngCDel 2.png
{\widehat{CR}}_3 = [(3,43)]: CDel label4.pngCDel branch.pngCDel 4-3.pngCDel branch.pngCDel label4.png
{\widehat{RR}}_3 = [4[4]]: CDel label4.pngCDel branch.pngCdel 4-4.pngCDel branch.pngCDel label4.png
{\widehat{AV}}_3 = [(33,6)]: CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel 2.png
{\widehat{BV}}_3 = [(3,4,3,6)]: CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png
{\widehat{HV}}_3 = [(3,5,3,6)]: CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png
{\widehat{VV}}_3 = [(3,6)[2]]: CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label6.png

{\bar{P}}_3 = [3,3[3]]: CDel branch.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{BP}}_3 = [4,3[3]]: CDel branch.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png
{\bar{HP}}_3 = [5,3[3]]: CDel branch.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{VP}}_3 = [6,3[3]]: CDel branch.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node.png
{\bar{DV}}_3 = [6,31,1]: CDel nodes.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node.png
{\bar{O}}_3 = [3,41,1]: CDel nodes.pngCDel split2-44.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{M}}_3 = [41,1,1]: CDel nodes.pngCDel split2-44.pngCDel node.pngCDel 4.pngCDel node.png

{\bar{R}}_3 = [3,4,4]: CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{N}}_3 = [43]: CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
{\bar{V}}_3 = [3,3,6]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
{\bar{BV}}_3 = [4,3,6]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
{\bar{HV}}_3 = [5,3,6]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
{\bar{Y}}_3 = [3,6,3]: CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{Z}}_3 = [6,3,6]: CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png

{\bar{DP}}_3 = [3[]x[]]: CDel node.pngCDel split1.pngCDel branch.pngCDel split2.pngCDel node.png
{\bar{PP}}_3 = [3[3,3]]: CDel tet.png

5 9

{\bar{P}}_4 = [3,3[4]]: CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png

{\bar{BP}}_4 = [4,3[4]]: CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png
{\widehat{FR}}_4 = [(32,4,3,4)]: CDel branch.pngCdel 4-4.pngCDel nodes.pngCDel split2.pngCDel node.png
{\bar{DP}}_4 = [3[3]x[]]: CDel node.pngCDel split1.pngCDel branchbranch.pngCDel split2.pngCDel node.png

{\bar{N}}_4 = [4,3,((4,2,3))]: CDel nodes.pngCDel split2-43.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\bar{O}}_4 = [3,4,31,1]: CDel nodes.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{S}}_4 = [4,32,1]: CDel nodes.pngCDel split2-43.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

{\bar{R}}_4 = [(3,4)2]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png

{\bar{M}}_4 = [4,31,1,1]: CDel node.pngCDel branch3.pngCDel splitsplit2.pngCDel node.pngCDel 4.pngCDel node.png
6 12

{\bar{P}}_5 = [3,3[5]]: CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
{\widehat{AU}}_5 = [(35,4)]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png

{\widehat{AR}}_5 = [(3,3,4)[2]]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel label4.png

{\bar{S}}_5 = [4,3,32,1]: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
{\bar{O}}_5 = [3,4,31,1]: CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{N}}_5 = [3,(3,4)1,1]: CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 4a4b.pngCDel nodes.png

{\bar{U}}_5 = [33,4,3]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{X}}_5 = [3,3,4,3,3]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{R}}_5 = [3,4,3,3,4]: CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\bar{Q}}_5 = [32,1,1,1]: CDel node.pngCDel branch3.pngCDel splitsplit2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

{\bar{M}}_5 = [4,3,31,1,1]: CDel node.pngCDel branch3.pngCDel splitsplit2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\bar{L}}_5 = [31,1,1,1,1]: CDel node.pngCDel branch3.pngCDel splitsplit2.pngCDel node.pngCDel split1.pngCDel nodes.png

7 3

{\bar{P}}_6 = [3,3[6]]:
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png

{\bar{Q}}_6 = [31,1,3,32,1]:
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
{\bar{S}}_6 = [4,32,32,1]:
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
8 4 {\bar{P}}_7 = [3,3[7]]:
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{Q}}_7 = [31,1,32,32,1]:
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
{\bar{S}}_7 = [4,33,32,1]:
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
{\bar{T}}_7 = [33,2,2]:
CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
9 4 {\bar{P}}_8 = [3,3[8]]:
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{Q}}_8 = [31,1,33,32,1]:
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
{\bar{S}}_8 = [4,34,32,1]:
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
{\bar{T}}_8 = [34,3,1]:
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
10 3 {\bar{Q}}_9 = [31,1,34,32,1]:
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
{\bar{S}}_9 = [4,35,32,1]:
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
{\bar{T}}_9 = [36,2,1]:
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Subgroup relations of paracompact hyperbolic groups[edit]

These trees represents subgroup relations of paracompact hyperbolic groups. Subgroup indices on each connection are given in red.[13] Subgroups of index 2 represent a mirror removal, and fundamental domain doubling. Others can be inferred by commensurability (integer ratio of volumes) for the tetrahedral domains.

H3 Hyperbolic subgroup tree 36.png Hyperbolic subgroup tree 336-direct.png Hyperbolic subgroup tree 363.png Hyperbolic subgroup tree 344.png
H4 Hyperbolic subgroup tree 3434.png
H5 Hyperbolic subgroup tree 33343.png

Hypercompact Coxeter groups (Vinberg polytopes)[edit]

Just like the hyperbolic plane H2 has nontrianglar polygonal domains, higher-dimensional reflective hyperbolic domains also exists. These nonsimplex domains can be considered degenerate simplices with non-intersecting mirrors given infinite order, or in a Coxeter diagram, such branches are given dotted or dashed lines. These nonsimplex domains are called Vinberg polytopes, after Ernest Vinberg for his Vinberg's algorithm for finding nonsimplex fundamental domain of a hyperbolic reflection group. Geometrically these fundamental domains can be classified as quadrilateral pyramids, or prisms or other polytopes with all edges having dihedral angles as π/n for n=2,3,4...

In a simplex-based domain, there are n+1 mirrors for n-dimensional space. In non-simplex domains, there are more than n+1 mirrors. The list is finite, but not completely known. Instead partial lists have been enumerated as n+k mirrors for k as 2,3, and 4.

Hypercompact Coxeter groups in three dimensional space or higher differ in two dimensional groups in one essential respect. Two hyperbolic n-gons having the same angles in the same cyclic order may have different edge lengths and are not in general congruent. In contrast Vinberg polytopes in 3 dimensions or higher are completely determined by the dihedral angles. This fact is based on the Mostow rigidity theorem, that two isomorphic groups generated by reflections in Hn for n>=3, define congruent fundamental domains (Vinberg polytopes).

Vinberg polytopes with rank n+2 for n dimensional space[edit]

The complete list of compact hyperbolic Vinberg polytopes with rank n+2 mirrors for n-dimensions has been enumerated by F. Esselmann in 1996.[14] A partial list was published in 1974 by I. M. Kaplinskaya.[15]

The complete list of paracompact solutions was published by P. Tumarkin in 2003, with dimensions from 3 to 17.[16]

The smallest paracompact form in H3 can be represented by CDel node.pngCDel ultra.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, or [∞,3,3,∞] which can be constructed by a mirror removal of paracompact hyperbolic group [3,4,4] as [3,4,1+,4]. The doubled fundamental domain changes from a tetrahedron into a quadrilateral pyramid. Another pyramids include [4,4,1+,4] = [∞,4,4,∞], CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node.png = CDel node.pngCDel ultra.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png. Removing a mirror from some of the cyclic hyperbolic Coxeter graphs become bow-tie graphs: [(3,3,4,1+,4)] = [((3,∞,3)),((3,∞,3))] or CDel branchu.pngCDel split2.pngCDel node.pngCDel split1.pngCDel branchu.png, [(3,4,4,1+,4)] = [((4,∞,3)),((3,∞,4))] or CDel branchu.pngCDel split2-43.pngCDel node.pngCDel split1-43.pngCDel branchu.png, [(4,4,4,1+,4)] = [((4,∞,4)),((4,∞,4))] or CDel branchu.pngCDel split2-44.pngCDel node.pngCDel split1-44.pngCDel branchu.png.

Other valid paracompact graphs with quadrilateral pyramid fundamental domains include:

Dimension Rank Graphs
H3 5
CDel node.pngCDel ultra.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel node.pngCDel ultra.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png, CDel node.pngCDel ultra.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png, CDel node.pngCDel ultra.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel ultra.pngCDel node.png, CDel node.pngCDel ultra.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel ultra.pngCDel node.png
CDel branchu.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-43.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-43.pngCDel node.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-44.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-44.pngCDel node.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png
CDel branchu.pngCDel split2-53.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-54.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-55.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-63.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-64.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-65.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png, CDel branchu.pngCDel split2-66.pngCDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node.png
CDel branchu.pngCDel split2.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-43.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-53.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-44.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-43.pngCDel node.pngCDel split1-43.pngCDel branchu.png, CDel branchu.pngCDel split2-44.pngCDel node.pngCDel split1-43.pngCDel branchu.png, CDel branchu.pngCDel split2-44.pngCDel node.pngCDel split1-44.pngCDel branchu.png, CDel branchu.pngCDel split2-54.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-55.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-63.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-64.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-65.pngCDel node.pngCDel split1.pngCDel branchu.png, CDel branchu.pngCDel split2-66.pngCDel node.pngCDel split1.pngCDel branchu.png

Another subgroup [1+,41,1,1] = [∞,4,1+,4,∞] = [∞[6]]. CDel node.pngCDel 4.pngCDel node h0.pngCDel split1-44.pngCDel nodes.png = CDel node.pngCDel ultra.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png = CDel node.pngCDel split1-uu.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.pngCDel split2-uu.pngCDel node.png. [17]

Vinberg polytopes with rank n+3 for n dimensional space[edit]

There are a finite number of degenerate fundamental simplices exist up to 8-dimensions. The complete list of Compact Vinberg polytopes with rank n+3 mirrors for n-dimensions has been enumerated by P. Tumarkin in 2004. These groups are labeled by dotted/broken lines for ultraparallel branches.

For 4 to 8 dimensions, rank 7 to 11 Coxeter groups are counted as 44, 16, 3, 1, and 1 respectively.[18] The highest was discovered by Bugganeko in 1984 in dimension 8, rank 11:[19]

Dimensions Rank Cases Graphs
H4 7 44
H5 8 16
H6 9 3 CDel node.pngCDel 5.pngCDel node.pngCDel split1-43.pngCDel nodes.pngCDel ua3b.pngCDel nodes u0.pngCDel ua3b.pngCDel nodes.pngCDel split2-43.pngCDel node.pngCDel 5.pngCDel node.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3aub.pngCDel branch.pngCDel 3a.pngCDel 10a.pngCDel nodea.png CDel nodea.pngCDel 5a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3aub.pngCDel nodes.pngCDel splitcross.pngCDel branch.pngCDel label5.png
H7 10 1 CDel node.pngCDel split1-53.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel ua3b.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2-53.pngCDel node.png
H8 11 1 CDel nodea.pngCDel 5a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3aub.pngCDel nodes 0u.pngCDel 3aub.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 5a.pngCDel nodea.png

Vinberg polytopes with rank n+4 for n dimensional space[edit]

There are a finite number of degenerate fundamental simplices exist up to 8-dimensions. Compact Vinberg polytopes with rank n+4 mirrors for n-dimensions has been explored by A. Felikson and P. Tumarkin in 2005.[20]

Lorentzian groups[edit]

Regular honeycombs with Lorentzian groups
H3 337 UHS plane at infinity.png
{3,3,7} in hyperbolic 3-space, rendered the intersection of the honeycomb with the plane-at-infinity, in the Poincare half-space model.
Heptagonal tiling honeycomb.png
{7,3,3} viewed outside of Poincare ball model.
This shows rank 5 Lorentzian groups arranged as subgroups from [6,3,3,3], and [6,3,6,3]. The highly symmetric group CDel pent.png, [3[3,3,3]] is an index 120 subgroup of [6,3,3,3].

Lorentzian groups for simplex domains can be defined as graphs beyond the hyperbolic forms. These can be considered to be related to a Lorentzian geometry, named after Hendrik Lorentz in the field of special and general relativity space-time, containing one (or more) time-like dimensional components whose self dot products are negative.[9]

A 1982 paper by George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, enumerates the finite list of Lorentzian of rank 5 to 11. He calls them level 2, meaning removal any permutation of 2 nodes leaves a finite or Euclidean graph. His enumeration is complete, but didn't list graphs that are a subgroup of another. All higher-order branch Coxeter groups of rank-4 are Lorentzian, ending in the limit as a complete graph 3-simplex Coxeter-Dynkin diagram with 6 infinite order branches, which can be expressed as [∞[3,3]]. Rank 5-11 have a finite number of groups 186, 66, 36, 13, 10, 8, and 4 Lorentzian groups respectively.[21] A 2013 paper by H. Chen and J.-P. Labbé, Lorentzian Coxeter groups and Boyd--Maxwell ball packings, recomputed and published the compete list.[22]

For the highest ranks 8-11, the complete lists are:

Lorentzian Coxeter groups
Rank Total
count
Groups
4 [3,3,7] ... [∞,∞,∞]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png ... CDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png

[4,3[3]] ... [∞,∞[3]]: CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.png ... CDel node.pngCDel infin.pngCDel node.pngCDel split1-ii.pngCDel branch.pngCDel labelinfin.png
[5,41,1] ... [∞1,1,1]: CDel node.pngCDel 5.pngCDel node.pngCDel split1-44.pngCDel nodes.png ... CDel node.pngCDel infin.pngCDel node.pngCDel split1-ii.pngCDel nodes.png
... [(5,4,3,3)] ... [∞[4]]: ... CDel label5.pngCDel branch.pngCDel 4a3b.pngCDel branch.png ... CDel labelinfin.pngCDel branch.pngCDel iaib.pngCDel branch.pngCDel labelinfin.png
... [4[]×[]] ... [∞[]×[]]: ... CDel node.pngCDel split1-ii-i.pngCDel branch.pngCDel split2-ii.pngCDel node.png
... [4[3,3]] ... [∞[3,3]]

5 186 ...[3[3,3,3]]:CDel pent.png ...
6 66
7 36 [31,1,1,1,1,1]: CDel node.pngCDel branch3.pngCDel splitsplit2.pngCDel node.pngCDel splitsplit1.pngCDel branch3.pngCDel node.png ...
8 13

[3,3,3[6]]:CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
[3,3[6],3]:CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
[3,3[2+4],3]:CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel 3a.pngCDel nodea.png
[3,3[1+5],3]:CDel nodes.pngCDel 3ab.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
[3[ ]e×[3]]:CDel node.pngCDel splitsplit1.pngCDel nodeabc.pngCDel 3abc.pngCDel nodeabc.pngCDel splitsplit2.pngCDel node.png

[4,3,3,33,1]:CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
[31,1,3,33,1]:CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
[3,(3,3,4)1,1]:CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png
[32,1,3,32,1]:CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

[4,3,3,32,2]:CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
[31,1,3,32,2]:CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png

9 10

[3,3[3+4],3]:CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
[3,3[9]]:CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
[3,3[2+5],3]:CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split5b.pngCDel nodes.png

[32,1,32,32,1]:CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png [33,1,33,4]:CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png

[33,1,3,3,31,1]:CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png

[33,3,2]:CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

[32,2,4]:CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[32,2,33,4]:CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[32,2,3,3,31,1]:CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png

10 8 [3,3[8],3]:CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png

[3,3[3+5],3]:CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel 3a.pngCDel nodea.png
[3,3[9]]:CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png

[32,1,33,32,1]:CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png [35,3,1]:CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

[33,1,34,4]:CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
[33,1,33,31,1]:CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png

[34,4,1]:CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
11 4 [32,1,34,32,1]:CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png [32,1,36,4]:CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png

[32,1,35,31,1]:CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png

[37,2,1]:CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

Very-extended Coxeter Diagrams[edit]

One usage includes a very-extended definition from the direct Dynkin diagram usage which considers affine groups as extended, hyperbolic groups over-extended, and a third node as over-extended simple groups. These extensions are usually marked by an exponent of 1,2, or 3 + symbols for the number of extended nodes. This extending series can be extended backwards, by sequentially removing the nodes from the same position in the graph, although the process stops after removing branching node. The E8 extended family is the most commonly shown example extending backwards from E3 and forwards to E11.

The extending process can define a limited series of Coxeter graphs that progress from finite to affine to hyperbolic to Lorentzian. The determinant of the Cartan matrices determine where the series changes from finite (positive) to affine (zero) to hyperbolic (negative), and ending as a Lorentzian group, containing at least one hyperbolic subgroup.[23] The noncrystalographic Hn groups forms an extended series where H4 is extended as a compact hyperbolic and over-extended into a lorentzian group.

The determinant of the Schläfli matrix by rank are:[24]

  • det(A1n=[2n-1]) = 2n (Finite for all n)
  • det(An=[3n-1]) = n+1 (Finite for all n)
  • det(BCn=[4,3n-2]) = 2 (Finite for all n)
  • det(Dn=[3n-3,1,1]) = 4 (Finite for all n)

Determinants of the Schläfli matrix in exceptional series are:

  • det(En=[3n-3,2,1]) = 9-n (Finite for E3(=A2A1), E4(=A4), E5(=D5), E6, E7 and E8, affine at E9 ({\tilde{E}}_{8}), hyperbolic at E10)
  • det([3n-4,3,1]) = 2(8-n) (Finite for n=4 to 7, affine ({\tilde{E}}_{7}), and hyperbolic at n=8.)
  • det([3n-4,2,2]) = 3(7-n) (Finite for n=4 to 6, affine ({\tilde{E}}_{6}), and hyperbolic at n=7.)
  • det(Fn=[3,4,3n-3]) = 5-n (Finite for F3(=B3) to F4, affine at F5 ({\tilde{F}}_{4}), hyperbolic at F6)
  • det(Gn=[6,3n-2]) = 3-n (Finite for G2, affine at G3 ({\tilde{G}}_{2}), hyperbolic at G4)
Smaller extended series
Finite A_2 C_2 G_2 A_3 B_3 C_3 H_4
Rank n [3[3],3n-3] [4,4,3n-3] Gn=[6,3n-2] [3[4],3n-4] [4,31,n-3] [4,3,4,3n-4] Hn=[5,3n-2]
2 [3]
A2
CDel branch.png
[4]
C2
CDel node.pngCDel 4.pngCDel node.png
[6]
G2
CDel node.pngCDel 6.pngCDel node.png
[2]
A12
CDel nodes.png
[4]
C2
CDel node.pngCDel 4.pngCDel node.png
[5]
H2
CDel node.pngCDel 5.pngCDel node.png
3 [3[3]]
A2+={\tilde{A}}_{2}
CDel branch.pngCDel split2.pngCDel node c1.png
[4,4]
C2+={\tilde{C}}_{2}
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node c1.png
[6,3]
G2+={\tilde{G}}_{2}
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node c1.png
[3,3]=A3
CDel node.pngCDel split1.pngCDel nodes.png
[4,3]
B3
CDel nodes.pngCDel split2-43.pngCDel node.png
[4,3]
C3
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
[5,3]
H3
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
4 [3[3],3]
A2++={\bar{P}}_3
CDel branch.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c2.png
[4,4,3]
C2++={\bar{R}}_3
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.png
[6,3,3]
G2++={\bar{V}}_3
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c2.png
[3[4]]
A3+={\tilde{A}}_3
CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node c1.png
[4,31,1]
B3+={\tilde{B}}_{3}
CDel nodes.pngCDel split2-43.pngCDel node.pngCDel 3.pngCDel node c1.png
[4,3,4]
C3+={\tilde{C}}_{3}
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node c1.png
[5,3,3]
H4
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5 [3[3],3,3]
A2+++
CDel branch.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
[4,4,3,3]
C2+++
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
[6,3,3,3]
G2+++
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
[3[4],3]
A3++={\bar{P}}_4
CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c2.png
[4,32,1]
B3++={\bar{S}}_4
CDel nodes.pngCDel split2-43.pngCDel node.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c2.png
[4,3,4,3]
C3++={\bar{R}}_4
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.png
[5,33]
H5={\bar{H}}_4
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6 [3[4],3,3]
A3+++
CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
[4,33,1]
B3+++
CDel nodes.pngCDel split2-43.pngCDel node.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
[4,3,4,3,3]
C3+++
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
[5,34]
H6
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Det(Mn) 3(3-n) 2(3-n) 3-n 4(4-n) 2(4-n)
Middle extended series
Finite A_4 B_4 C_4 D_4 F_4 A_5 B_5 D_5
Rank n [3[5],3n-5] [4,3,3n-4,1] [4,3,3,4,3n-5] [3n-4,1,1,1] [3,4,3n-3] [3[6],3n-6] [4,3,3,3n-5,1] [31,1,3,3n-5,1]
3 [4,3-1,1]
B2A1
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 2.pngCDel nodeb.png
[4,3]
BC3
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
[3-1,1,1,1]
A13
CDel nodeabc.png
[3,4]
BC3
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[4,3,3]
C3
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
4 [33]
A4
CDel branch.pngCDel 3ab.pngCDel nodes.png
[4,3,3]
B4
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel branch.png
[4,3,3]
C4
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[30,1,1,1]
D4
CDel node.pngCDel branch3.pngCDel splitsplit2.pngCDel node.png
[3,4,3]
F4
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
[4,3,3,3-1,1]
B3A1
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodeb.png
[31,1,3,3-1,1]
A3A1
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 2.pngCDel nodeb.png
5 [3[5]]
A4+={\tilde{A}}_{4}
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node c1.png
[4,3,31,1]
B4+={\tilde{B}}_{4}
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.png
[4,3,3,4]
C4+={\tilde{C}}_{4}
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[31,1,1,1]
D4+={\tilde{D}}_{4}
CDel node.pngCDel branch3.pngCDel splitsplit2.pngCDel node.pngCDel 3.pngCDel node c1.png
[3,4,3,3]
F4+={\tilde{F}}_{4}
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node c1.png
[34]
A5
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
[4,3,3,3,3]
B5
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.png
[31,1,3,3]
D5
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel branch.png
6 [3[5],3]
A4++={\bar{P}}_5
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c2.png
[4,3,32,1]
B4++={\bar{S}}_5
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.png
[4,3,3,4,3]
C4++={\bar{R}}_5
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.png
[32,1,1,1]
D4++={\bar{Q}}_5
CDel node.pngCDel branch3.pngCDel splitsplit2.pngCDel node.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c2.png
[3,4,33]
F4++={\bar{U}}_5
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c2.png
[3[6]]
A5+={\tilde{A}}_5
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node c1.png
[4,3,3,31,1]
B5+={\tilde{B}}_{5}
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.png
[31,1,3,31,1]
D5+={\tilde{D}}_5
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.png
7 [3[5],3,3]
A4+++
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
[4,3,33,1]
B4+++
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.pngCDel 3a.pngCDel nodea c3.png
[4,3,3,4,3,3]
C4+++
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
[33,1,1,1]
D4+++
CDel node.pngCDel branch3.pngCDel splitsplit2.pngCDel node.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
[3,4,34]
F4+++
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
[3[6],3]
A5++={\bar{P}}_6
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c2.png
[4,3,3,32,1]
B5++={\bar{S}}_6
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.png
[31,1,3,32,1]
D5++={\bar{Q}}_6
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.png
8 [3[6],3,3]
A5+++
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
[4,3,3,33,1]
B5+++
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.pngCDel 3a.pngCDel nodea c3.png
[31,1,3,33,1]
D5+++
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.pngCDel 3a.pngCDel nodea c3.png
Det(Mn) 5(5-n) 2(5-n) 4(5-n) 5-n 6(6-n) 4(6-n)
Some higher extended series
Finite A_6 B_6 D_6 E_6 A_7 B_7 D_7 E_7 E_8
Rank n [3[7],3n-7] [4,33,3n-6,1] [31,1,3,3,3n-6,1] [3n-5,2,2] [3[8],3n-8] [4,34,3n-7,1] [31,1,3,3,3,3n-7,1] [3n-5,3,1] En=[3n-4,2,1]
3 [3-1,2,1]
E3=A2A1
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodeb.png
4 [3-1,2,2]
A22
CDel nodes.pngCDel 3ab.pngCDel nodes.png
[3-1,3,1]
A3A1
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodeb.png
[30,2,1]
E4=A4
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.png
5 [4,3,3,3,3-1,1]
B4A1
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodeb.png
[31,1,3,3,3-1,1]
D4A1
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodeb.png
[30,2,2]
A5
CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
[30,3,1]
A5
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.png
[31,2,1]
E5=D5
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png
6 [35]
A6
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
[4,34]
B6
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.png
[31,1,3,3,3]
D6
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.png
[31,2,2]
E6
CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
[4,3,3,3,3,3-1,1]
B5A1
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodeb.png
[31,1,3,3,3,3-1,1]
D5A1
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodeb.png
[31,3,1]
D6
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png
[32,2,1]
E6 *
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
7 [3[7]]
A6+={\tilde{A}}_6
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node c1.png
[4,33,31,1]
B6+={\tilde{B}}_{6}
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.png
[31,1,3,3,31,1]
D6+={\tilde{D}}_6
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.png
[32,2,2]
E6+={\tilde{E}}_6
CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node c1.png
[36]
A7
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
[4,35]
B7
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.png
[31,1,3,3,3,30,1]
D7
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.png
[32,3,1]
E7 *
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
[33,2,1]
E7 *
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
8 [3[7],3]
A6++={\bar{P}}_7
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c2.png
[4,33,32,1]
B6++={\bar{S}}_7
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.png
[31,1,3,3,32,1]
D6++={\bar{Q}}_7
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.png
[33,2,2]
E6++={\bar{T}}_7
CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c2.png
[3[8]]
A7+={\tilde{A}}_7 *
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node c1.png
[4,34,31,1]
B7+={\tilde{B}}_{7} *
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.png
[31,1,3,3,3,31,1]
D7+={\tilde{D}}_7 *
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.png
[33,3,1]
E7+={\tilde{E}}_{7} *
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea c1.png
[34,2,1]
E8 *
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
9 [3[7],3,3]
A6+++
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
[4,33,33,1]
B6+++
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.pngCDel 3a.pngCDel nodea c3.png
[31,1,3,3,33,1]
D6+++
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.pngCDel 3a.pngCDel nodea c3.png
[34,2,2]
E6+++
CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
[3[8],3]
A7++={\bar{P}}_8 *
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c2.png
[4,34,32,1]
B7++={\bar{S}}_8 *
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.png
[31,1,3,3,3,32,1]
D7++={\bar{Q}}_8 *
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.png
[34,3,1]
E7++={\bar{T}}_8 *
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.png
[35,2,1]
E9=E8+={\tilde{E}}_{8} *
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea c1.png
10 [3[8],3,3]
A7+++ *
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
[4,34,33,1]
B7+++ *
CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.pngCDel 3a.pngCDel nodea c3.png
[31,1,3,3,3,33,1]
D7+++ *
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.pngCDel 3a.pngCDel nodea c3.png
[35,3,1]
E7+++ *
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.pngCDel 3a.pngCDel nodea c3.png
[36,2,1]
E10=E8++={\bar{T}}_9 *
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.png
11 [37,2,1]
E11=E8+++ *
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea c1.pngCDel 3a.pngCDel nodea c2.pngCDel 3a.pngCDel nodea c3.png
Det(Mn) 7(7-n) 2(7-n) 4(7-n) 3(7-n) 8(8-n) 2(8-n) 4(8-n) 2(8-n) 9-n

Geometric folding[edit]

Finite and affine foldings[25]
φA : AΓ --> AΓ' for finite types
Γ Γ' Folding description Coxeter–Dynkin diagrams
I2(h) Γ(h) Dihedral folding Geometric folding Coxeter graphs.png
BCn A2n (I,sn)
Dn+1, A2n-1 (A3,+/-ε)
F4 E6 (A3,±ε)
H4 E8 (A4,±ε)
H3 D6
H2 A4
G2 A5 (A5,±ε)
D4 (D4,±ε)
φ: AΓ+ --> AΓ'+ for affine types
{\tilde{A}}_{n-1} {\tilde{A}}_{kn-1} Locally trivial Geometric folding Coxeter graphs affine.png
{\tilde{B}}_{n} {\tilde{D}}_{2n+1} (I,sn)
{\tilde{D}}_{n+1}, {\tilde{D}}_{2n} (A3,±ε)
{\tilde{C}}_{n} {\tilde{B}}_{n+1}, {\tilde{C}}_{2n} (A3,±ε)
{\tilde{C}}_{2n+1} (I,sn)
{\tilde{C}}_{n} {\tilde{A}}_{2n+1} (I,sn) & (I,s0)
{\tilde{A}}_{2n} (A3,ε) & (I,s0)
{\tilde{A}}_{2n-1} (A3,ε) & (A3,ε')
{\tilde{C}}_{n} {\tilde{D}}_{n+2} (A3,-ε) & (A3,-ε')
{\tilde{C}}_{2} {\tilde{D}}_{5} (I,s1)
{\tilde{F}}_{4} {\tilde{E}}_{6}, {\tilde{E}}_{7} (A3,±ε)
{\tilde{G}}_{2} {\tilde{D}}_{6}, {\tilde{E}}_{7} (A5,±ε)
{\tilde{B}}_{3}, {\tilde{F}}_{4} (B3,±ε)
{\tilde{D}}_{4}, {\tilde{E}}_{6} (D4,±ε)

A (simply-laced) Coxeter–Dynkin diagram (finite, affine, or hyperbolic) that has a symmetry (satisfying one condition, below) can be quotiented by the symmetry, yielding a new, generally multiply laced diagram, with the process called "folding".[26][27]

For example, in D4 folding to G2, the edge in G2 points from the class of the 3 outer nodes (valence 1), to the class of the central node (valence 3).

Geometrically this corresponds to orthogonal projections of uniform polytopes and tessellations. Notably, any finite simply-laced Coxeter–Dynkin diagram can be folded to I2(h), where h is the Coxeter number, which corresponds geometrically to a projection to the Coxeter plane.

Geometric folding Coxeter graphs hyperbolic.png
A few hyperbolic foldings

See also[edit]

References[edit]

  1. ^ Hall, Brian C. (2003), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, ISBN 0-387-40122-9 
  2. ^ Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, Sec 7.7. page 133, Schläfli's Criterion
  3. ^ Lannér F., On complexes with transitive groups of automorphisms, Medd. Lunds Univ. Mat. Sem. [Comm. Sem. Math. Univ. Lund], 11 (1950), 1–71
  4. ^ Allcock, Daniel (11 July 2006). "Infinitely many hyperbolic Coxeter groups through dimension 19". Geometry & Topology 10: 737–758. doi:10.2140/gt.2006.10.737. 
  5. ^ The Geometry and Topology of Coxeter Groups, Michael W. Davis, 2008 p. 105 Table 6.2. Hyperbolic diagrams
  6. ^ "TAKEUCHI : Arithmetic triangle groups". Projecteuclid.org. Retrieved 2013-07-05. 
  7. ^ Folke Lannér, On complexes with transitive groups of automorphisms, Comm. Sém., Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 11 (1950) [1]
  8. ^ Regular Honeycombs in hyperbolic space, Coxeter, 1954
  9. ^ a b Norman Johnson, Geometries and Transformations, Chapter 13: Hyperbolic Coxeter groups, 13.6 Lorentzian lattices
  10. ^ J. L. Koszul, Lectures on hyperbolic Coxeter groups, University of Notre Dame (1967)
  11. ^ M. Chein, Recherche des graphes des matrices de Coxeter hyperboliques d’ordre ≤10, Rev. Française Informat. Recherche Opérationnelle 3 (1969), no. Ser. R-3, 3–16 (French). [2]
  12. ^ Subalgebras of hyperbolic Kay-Moody algebras, Figure 5.1, p.13
  13. ^ N.W. Johnson, R. Kellerhals, J.G. Ratcliffe,S.T. Tschantz, Commensurability classes of hyperbolic Coxeter groups H3: p130, H4: p137, H5: p 138. [3]
  14. ^ F. Esselmann, The classification of compact hyperbolic Coxeter d-polytopes with d+2 facets. Comment. Math. Helvetici 71 (1996), 229–242. [4]
  15. ^ I. M. Kaplinskaya, Discrete groups generated by reflections in the faces of simplicial prisms in Lobachevskian spaces. Math. Notes,15 (1974), 88–91. [5]
  16. ^ P. Tumarkin, Hyperbolic Coxeter n-polytopes with n+2 facets (2003)
  17. ^ Norman W. Johnson and Asia Ivic Weiss, Quadratic Integers and Coxeter Groups, Canad. J. Math. Vol. 51 (6), 1999 pp. 1307–1336 [6]
  18. ^ P. Tumarkin, Compact hyperbolic Coxeter (2004)
  19. ^ V. O. Bugaenko, Groups of automorphisms of unimodular hyperbolic quadratic forms over the ring Zh√5+12 i. Moscow Univ. Math. Bull. 39 (1984), 6-14.
  20. ^ Anna Felikson, Pavel Tumarkin, On compact hyperbolic Coxeter d-polytopes with d+4 facets, 2005 [7]
  21. ^ George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) [8]
  22. ^ Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, http://arxiv.org/abs/1310.8608
  23. ^ Kac-Moody Algebras in M-theory
  24. ^ Cartan–Gram determinants for the simple Lie groups, Wu, Alfred C. T, The American Institute of Physics, Nov 1982
  25. ^ John Crisp, 'Injective maps between Artin groups', in Down under group theory, Proceedings of the Special Year on Geometric Group Theory, (Australian National University, Canberra, Australia, 1996), Postscript, pp 13-14, and googlebook, Geometric group theory down under, p 131
  26. ^ Zuber, Jean-Bernard. "Generalized Dynkin diagrams and root systems and their folding". pp. 28–30. CiteSeerX: 10.1.1.54.3122. 
  27. ^ Dechant, Pierre-Philippe; Boehm, Celine; Twarock, Reidun (October 25, 2011). "Affine extensions of non-crystallographic Coxeter groups induced by projection". arXiv:1110.5228. 

Further reading[edit]

  • James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29 (1990)
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [9], Googlebooks [10]
    • (Paper 17) Coxeter, The Evolution of Coxeter-Dynkin diagrams, [Nieuw Archief voor Wiskunde 9 (1991) 233-248]
  • Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
  • Coxeter, Regular Polytopes (1963), Macmillian Company
  • H.S.M. Coxeter and W. O. J. Moser. Generators and Relations for Discrete Groups 4th ed, Springer-Verlag. New York. 1980
  • Norman Johnson, Geometries and Transformations, Chapters 11,12,13, preprint 2011
  • N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups 1999, Volume 4, Issue 4, pp 329–353 [11] [12]
  • Norman W. Johnson and Asia Ivic Weiss Quadratic Integers and Coxeter Groups PDF Canad. J. Math. Vol. 51 (6), 1999 pp. 1307–1336

External links[edit]