Coxeter–Dynkin diagram
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 order3.
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. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras.^{[1]}
Description
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 forms studied by Ludwig Schläfli are the orthoschemes which have linear graphs that generate regular polytopes and regular honeycombs. Plagioschemes are simplices represented by branching graphs, and cycloschemes are simplices represented by cyclic graphs.
Schläfli matrix
Every Coxeter diagram has a corresponding Schläfli matrix with matrix elements a_{i,j} = a_{j,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 nonnegative, 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 nonequivalent 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, after F. Lannér who enumerated the compact hyperbolic groups in 1950,^{[3]} and Koszul (or quasiLannér) for the paracompact groups.
Rank 2 Coxeter groups
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 nodebranch graphic diagrams. Rational solutions [p/q], , also exist, with gcd(p,q)=1, which define overlapping fundamental domains. For example, 3/2, 4/3, 5/2, 5/3, 5/4. and 6/5.
Type  Finite  Affine  Hyperbolic  

Geometry  ...  
Coxeter  [ ] 
[2] 
[3] 
[4] 
[p] 
[∞] 
[∞] 
[iπ/λ] 
Order  2  4  6  8  2p  ∞  
Mirror lines are colored to correspond to Coxeter diagram nodes. Fundamental domains are alternately colored. 
Order p 
Group  Coxeter diagram  Schläfli matrix  

Determinant (4a_{21}*a_{12})  
Finite (Determinant>0)  
2  I_{2}(2) = A_{1}xA_{1}  [2]  4  
3  I_{2}(3) = A_{2}  [3]  3  
3/2  [3/2]  
4  I_{2}(4) = B_{2}  [4]  2  
4/3  [4/3]  
5  I_{2}(5) = H_{2}  [5]  ~1.38196601125  
5/4  [5/4]  
5/2  [5/2]  ~3.61803398875  
5/3  [5/3]  
6  I_{2}(6) = G_{2}  [6]  1  
6/5  [6/5]  
8  I_{2}(8)  [8]  ~0.58578643763  
10  I_{2}(10)  [10]  = ~0.38196601125  
12  I_{2}(12)  [12]  ~0.26794919243  
p  I_{2}(p)  [p]  
Affine (Determinant=0)  
∞  I_{2}(∞) = =  [∞]  0  
Hyperbolic (Determinant≤0)  
∞  [∞]  0  
∞  [iπ/λ] 
Geometric visualizations
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 90degree dihedral angles (order 2).
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 x is shown as a doubling of the , but can also be created as rectangular domains from doubling the triangles. The is a doubling of the triangle.  
Many Coxeter groups in the hyperbolic plane can be extended from the Euclidean cases as a series of hyperbolic solutions.  
Coxeter groups in 3space with diagrams. Mirrors (triangle faces) are labeled by opposite vertex 0..3. Branches are colored by their reflection order. fills 1/48 of the cube. fills 1/24 of the cube. fills 1/12 of the cube. 
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
 See also polytope families for a table of endnode 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 D_{n} groups is half or alternated version of the regular C_{n} groups.
 The bifurcated D_{n} and E_{n} groups are also labeled by a superscript form [3^{a,b,c}] where a,b,c are the numbers of segments in each of the three branches.
Rank  Simple Lie groups  Exceptional Lie groups  

1  A_{1}=[ ] 

2  A_{2}=[3] 
B_{2}=[4] 
D_{2}=A_{1}A_{1} 
G_{2}=[6] 
H_{2}=[5] 
I_{2}[p]  
3  A_{3}=[3^{2}] 
B_{3}=[3,4] 
D_{3}=A_{3} 
E_{3}=A_{2}A_{1} 
F_{3}=B_{3} 
H_{3} 

4  A_{4}=[3^{3}] 
B_{4}=[3^{2},4] 
D_{4}=[3^{1,1,1}] 
E_{4}=A_{4} 
F_{4} 
H_{4}  
5  A_{5}=[3^{4}] 
B_{5}=[3^{3},4] 
D_{5}=[3^{2,1,1}] 
E_{5}=D_{5} 

6  A_{6}=[3^{5}] 
B_{6}=[3^{4},4] 
D_{6}=[3^{3,1,1}] 
E_{6}=[3^{2,2,1}]  
7  A_{7}=[3^{6}] 
B_{7}=[3^{5},4] 
D_{7}=[3^{4,1,1}] 
E_{7}=[3^{3,2,1}]  
8  A_{8}=[3^{7}] 
B_{8}=[3^{6},4] 
D_{8}=[3^{5,1,1}] 
E_{8}=[3^{4,2,1}]  
9  A_{9}=[3^{8}] 
B_{9}=[3^{7},4] 
D_{9}=[3^{6,1,1}] 

10+  ..  ..  ..  .. 
Application with uniform polytopes
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 {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 kring generator points are on (k1)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. A truncated alternation is called a snub.
 A single node represents a single mirror. This is called group A_{1}. 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 ordern branch can create an ngon if the point is on one mirror, and a 2ngon if the point is off both mirrors. This forms the I_{1}(n) group.
 Two parallel mirrors can represent an infinite polygon I_{1}(∞) group, also called Ĩ_{1}.
 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.
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. 
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 slashhole for hole nodes of the snubs. For example, represents a rectangle (as two active orthogonal mirrors), and represents its dual polygon, the rhombus.
Example polyhedra and tilings
For example, the B_{3} Coxeter group has a diagram: . 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{3^{1,1}} 
t{3,4} t{3^{1,1}} 
{3,4} {3^{1,1}} 
rr{4,3} s_{2}{3,4} 
tr{4,3}  sr{4,3}  h{4,3} {3,3} 
h_{2}{4,3} t{3,3} 
s{3,4} s{3^{1,1}} 
= 
= 
= 
= or 
= or 
=  
Duals to uniform polyhedra  
V4^{3}  V3.8^{2}  V(3.4)^{2}  V4.6^{2}  V3^{4}  V3.4^{3}  V4.6.8  V3^{4}.4  V3^{3}  V3.6^{2}  V3^{5} 
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)  [6,2^{+}], (2*3)  
{6,2}  t{6,2}  r{6,2}  t{2,6}  {2,6}  rr{6,2}  tr{6,2}  sr{6,2}  s{2,6}  
Duals to uniforms  
V6^{2}  V12^{2}  V6^{2}  V4.4.6  V2^{6}  V4.4.6  V4.4.12  V3.3.3.6  V3.3.3.3 
In comparison, the [6,3], 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) 
[6,3^{+}] (3*3)  
{6,3}  t{6,3}  r{6,3}  t{3,6}  {3,6}  rr{6,3}  tr{6,3}  sr{6,3}  s{3,6}  
6^{3}  3.12^{2}  (3.6)^{2}  6.6.6  3^{6}  3.4.12.4  4.6.12  3.3.3.3.6  3.3.3.3.3.3  
Uniform duals  
V6^{3}  V3.12^{2}  V(3.6)^{2}  V6^{3}  V3^{6}  V3.4.12.4  V.4.6.12  V3^{4}.6  V3^{6} 
In the hyperbolic plane [7,3], 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)  
{7,3}  t{7,3}  r{7,3}  t{3,7}  {3,7}  rr{7,3}  tr{7,3}  sr{7,3}  
Uniform duals  
V7^{3}  V3.14.14  V3.7.3.7  V6.6.7  V3^{7}  V3.4.7.4  V4.6.14  V3.3.3.3.7 
Affine Coxeter groups
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)
 : diagrams of this type are cycles. (Also P_{n})
 is associated with the hypercube regular tessellation {4, 3, ...., 4} family. (Also R_{n})
 related to C by one removed mirror. (Also S_{n})
 related to C by two removed mirrors. (Also Q_{n})
 , , . (Also T_{7}, T_{8}, T_{9})
 forms the {3,4,3,3} regular tessellation. (Also U_{5})
 forms 306090 triangle fundamental domains. (Also V_{3})
 is two parallel mirrors. ( = = ) (Also W_{2})
Composite groups can also be defined as orthogonal projects. The most common use , like , represents square or rectangular checker board domains in the Euclidean plane. And represents triangular prism fundamental domains in Euclidean 3space.
Rank  (P_{2+})  (S_{4+})  (R_{2+})  (Q_{5+})  (T_{n+1}) / (U_{5}) / (V_{3}) 

2  =[∞] 
=[∞] 

3  =[3^{[3]}] * 
=[4,4] * 
=[6,3] *  
4  =[3^{[4]}] * 
=[4,3^{1,1}] * 
=[4,3,4] * 
=[3^{1,1},3^{−1},3^{1,1}] = 

5  =[3^{[5]}] * 
=[4,3,3^{1,1}] * 
=[4,3^{2},4] * 
=[3^{1,1,1,1}] * 
=[3,4,3,3] * 
6  =[3^{[6]}] * 
=[4,3^{2},3^{1,1}] * 
=[4,3^{3},4] * 
=[3^{1,1},3,3^{1,1}] * 

7  =[3^{[7]}] * 
=[4,3^{3},3^{1,1}] 
=[4,3^{4},4] 
=[3^{1,1},3^{2},3^{1,1}] 
=[3^{2,2,2}] 
8  =[3^{[8]}] * 
=[4,3^{4},3^{1,1}] * 
=[4,3^{5},4] 
=[3^{1,1},3^{3},3^{1,1}] * 
=[3^{3,3,1}] * 
9  =[3^{[9]}] * 
=[4,3^{5},3^{1,1}] 
=[4,3^{6},4] 
=[3^{1,1},3^{4},3^{1,1}] 
=[3^{5,2,1}] * 
10  =[3^{[10]}] * 
=[4,3^{6},3^{1,1}] 
=[4,3^{7},4] 
=[3^{1,1},3^{5},3^{1,1}]  
11  ...  ...  ...  ... 
Hyperbolic Coxeter groups
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 finitevolume Vinberg 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 H^{2}
Example right triangles [p,q]  

[3,7] 
[3,8] 
[3,9] 
[3,∞]  
[4,5] 
[4,6] 
[4,7] 
[4,8] 
[∞,4] 
[5,5] 
[5,6] 
[5,7] 
[6,6] 
[∞,∞] 
Example general triangles [(p,q,r)]  
[(3,3,4)] 
[(3,3,5)] 
[(3,3,6)] 
[(3,3,7)] 
[(3,3,∞)] 
[(3,4,4)] 
[(3,6,6)] 
[(3,∞,∞)] 
[(6,6,6)] 
[(∞,∞,∞)] 
Twodimensional hyperbolic triangle groups exist as rank 3 Coxeter diagrams, defined by triangle (p q r) for:
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]}
Linear  Cyclic  

∞ [p,q], : 2(p+q)<pq

∞ [(p,q,r)], : p+q+r>9

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



Arithmetic triangle group
The hyperbolic triangle groups that are also arithmetic groups form a finite subset. 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)
Paracompact right triangles: (4)

General triangles: (39)
Paracompact general triangles: (5)



Hyperbolic Coxeter polygons above triangles
or [∞,3,∞] [iπ/λ_{1},3,iπ/λ_{2}] (*3222) 
or [((3,∞,3)),∞] [((3,iπ/λ_{1},3)),iπ/λ_{2}] (*3322) 
or [(3,∞)^{[2]}] [(3,iπ/λ_{1},3,iπ/λ_{2})] (*3232) 
or [(4,∞)^{[2]}] [(4,iπ/λ_{1},4,iπ/λ_{2})] (*4242) 
(*3333) 
Domains with ideal vertices  

[iπ/λ_{1},∞,iπ/λ_{2}] (*∞222) 
(*∞∞22) 
[(iπ/λ_{1},∞,iπ/λ_{2},∞)] (*2∞2∞) 
(*∞∞∞∞) 
(*4444) 
Other H^{2} 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 (n1)simplex fundamental domains, with a cyclic of branches order a,b,c... and the remaining n*(n3)/2 branches are labeled as infinite (∞) representing the nonintersecting mirrors. The only nonhyperbolic example is Euclidean symmetry four mirrors in a square or rectangle as , [∞,2,∞] (orbifold *2222). Another branch representation for nonintersecting mirrors by Vinberg gives infinite branches as dotted or dashed lines, so this diagram can be shown as , with the four order2 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 , [∞,3,∞] or [iπ/λ_{1},3,iπ/λ_{2}] (orbifold *3222), where (λ_{1},λ_{2}) are the distance between the ultraparallel mirrors. The alternate expression is , with three order2 branches suppressed around the perimeter. Similarly (2 3 2 3) (orbifold *3232) can be represented as and (3 3 3 3), (orbifold *3333) can be represented as a complete graph .
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: .
Compact (Lannér simplex groups)
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 4space: [5/2,5,3,3] = and [5,5/2,5,3] = .
Ranks 4–5
The fundamental domain of either of the two bifurcating groups, [5,3^{1,1}] and [5,3,3^{1,1}], 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]}
Dimension H^{d} 
Rank  Total count  Linear  Bifurcating  Cyclic 

H^{3}  4  9  = [4,3,5]: 
= [5,3^{1,1}]: 
= [(3^{3},4)]: 
H^{4}  5  5  = [3^{3},5]: 
= [5,3,3^{1,1}]: 
= [(3^{4},4)]: 
Paracompact (Koszul simplex groups)
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 JeanLouis Koszul.^{[10]} They are also called quasiLannér 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: and . The other six nonarithmetic groups are all paracompact, with five 3dimensional groups , , , , and , and one 5dimensional group .
Ideal simplices
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  [(∞,∞,∞)]  [∞]  
4  [4^{[4]}]  [4,4]  
4  [3^{[3,3]}]  [3^{[3]}]  
4  [(3,6)^{[2]}]  [3,6]  
6  [(3,3,4)^{[2]}]  [4,3,3,4], [3,4,3,3]  , 
Ranks 4–10
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.
Rank  Total count  Groups  

4  23 
= [(3,3,4,4)]: 
= [3,3^{[3]}]: 
= [3,4,4]: 
= [3^{[]x[]}]: 
5  9 
= [3,3^{[4]}]: 
= [4,3,((4,2,3))]: 
= [(3,4)^{2}]: 
= [4,3^{1,1,1}]: 
6  12 
= [3,3^{[5]}]: 
= [4,3,3^{2,1}]: 
= [3^{3},4,3]: 
= [3^{2,1,1,1}]: = [4,3,3^{1,1,1}]: 
7  3 
= [3,3^{[6]}]: 
= [3^{1,1},3,3^{2,1}]: 
= [4,3^{2},3^{2,1}]: 

8  4  = [3,3^{[7]}]: 
= [3^{1,1},3^{2},3^{2,1}]: 
= [4,3^{3},3^{2,1}]: 
= [3^{3,2,2}]: 
9  4  = [3,3^{[8]}]: 
= [3^{1,1},3^{3},3^{2,1}]: 
= [4,3^{4},3^{2,1}]: 
= [3^{4,3,1}]: 
10  3  = [  = [3^{1,1},3^{4},3^{2,1}]: 
= [4,3^{5},3^{2,1}]: 
= [3^{6,2,1}]: 
Subgroup relations of paracompact hyperbolic groups
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.
H^{3}  

H^{4}  
H^{5} 
Hypercompact Coxeter groups (Vinberg polytopes)
Just like the hyperbolic plane H^{2} has nontriangular polygonal domains, higherdimensional reflective hyperbolic domains also exists. These nonsimplex domains can be considered degenerate simplices with nonintersecting 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 edges as the intersection of two mirrors having dihedral angles as π/n for n=2,3,4...
In a simplexbased domain, there are n+1 mirrors for ndimensional space. In nonsimplex 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 from two dimensional groups in one essential respect. Two hyperbolic ngons 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 H^{n} for n>=3, define congruent fundamental domains (Vinberg polytopes).
Vinberg polytopes with rank n+2 for n dimensional space
The complete list of compact hyperbolic Vinberg polytopes with rank n+2 mirrors for ndimensions 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 H^{3} can be represented by , 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,∞], = . Removing a mirror from some of the cyclic hyperbolic Coxeter graphs become bowtie graphs: [(3,3,4,1^{+},4)] = [((3,∞,3)),((3,∞,3))] or , [(3,4,4,1^{+},4)] = [((4,∞,3)),((3,∞,4))] or , [(4,4,4,1^{+},4)] = [((4,∞,4)),((4,∞,4))] or .
Other valid paracompact graphs with quadrilateral pyramid fundamental domains include:
Dimension  Rank  Graphs 

H^{3}  5 

Another subgroup [1^{+},4^{1,1,1}] = [∞,4,1^{+},4,∞] = [∞^{[6]}]. = = . ^{[17]}
Vinberg polytopes with rank n+3 for n dimensional space
There are a finite number of degenerate fundamental simplices exist up to 8dimensions. The complete list of Compact Vinberg polytopes with rank n+3 mirrors for ndimensions has been enumerated by P. Tumarkin in 2004. These groups are labeled by dotted/broken lines for ultraparallel branches. The complete list of nonCompact Vinberg polytopes with rank n+3 mirrors and with one nonsimple vertex for ndimensions has been enumerated by Mike Roberts.^{[18]}
For 4 to 8 dimensions, rank 7 to 11 Coxeter groups are counted as 44, 16, 3, 1, and 1 respectively.^{[19]} The highest was discovered by Bugaenko in 1984 in dimension 8, rank 11:^{[20]}
Dimensions  Rank  Cases  Graphs  

H^{4}  7  44  ...  
H^{5}  8  16  ..  
H^{6}  9  3  
H^{7}  10  1  
H^{8}  11  1 
Vinberg polytopes with rank n+4 for n dimensional space
There are a finite number of degenerate fundamental simplices exist up to 8dimensions. Compact Vinberg polytopes with rank n+4 mirrors for ndimensions has been explored by A. Felikson and P. Tumarkin in 2005.^{[21]}
Lorentzian groups
{3,3,7} in hyperbolic 3space, rendered the intersection of the honeycomb with the planeatinfinity, in the Poincare halfspace model. 
{7,3,3} viewed outside of Poincare ball model. 
Lorentzian groups for simplex domains can be defined as graphs beyond the paracompact hyperbolic forms. These are sometimes called superideal simplices and are also related to a Lorentzian geometry, named after Hendrik Lorentz in the field of special and general relativity spacetime, containing one (or more) timelike dimensional components whose self dot products are negative.^{[9]} Danny Calegari calls these convex cocompact Coxeter groups in ndimensional hyperbolic space.^{[22]}^{[23]}
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 higherorder branch Coxeter groups of rank4 are Lorentzian, ending in the limit as a complete graph 3simplex CoxeterDynkin diagram with 6 infinite order branches, which can be expressed as [∞^{[3,3]}]. Rank 511 have a finite number of groups 186, 66, 36, 13, 10, 8, and 4 Lorentzian groups respectively.^{[24]} A 2013 paper by H. Chen and J.P. Labbé, Lorentzian Coxeter groups and BoydMaxwell ball packings, recomputed and published the complete list.^{[25]}
For the highest ranks 811, the complete lists are:
Rank  Total count 
Groups  

4  ∞  [3,3,7] ... [∞,∞,∞]: ... [4,3^{[3]}] ... [∞,∞^{[3]}]: ...  
5  186  ...[3^{[3,3,3]}]: ...  
6  66  
7  36  [3^{1,1,1,1,1,1}]: ...  
8  13 
[3,3,3^{[6]}]: 
[4,3,3,3^{3,1}]: 
[4,3,3,3^{2,2}]:  
9  10 
[3,3^{[3+4]},3]: 
[3^{2,1},3^{2},3^{2,1}]:  [3^{3,1},3^{3},4]: [3^{3,1},3,3,3^{1,1}]: 
[3^{3,3,2}]: [3^{2,2,4}]:  
10  8  [3,3^{[8]},3]: [3,3^{[3+5]},3]: 
[3^{2,1},3^{3},3^{2,1}]:  [3^{5,3,1}]: [3^{3,1},3^{4},4]: 
[3^{4,4,1}]:  
11  4  [3^{2,1},3^{4},3^{2,1}]:  [3^{2,1},3^{6},4]: [3^{2,1},3^{5},3^{1,1}]: 
[3^{7,2,1}]: 
Veryextended Coxeter Diagrams
One usage includes a veryextended definition from the direct Dynkin diagram usage which considers affine groups as extended, hyperbolic groups overextended, and a third node as veryextended 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 E_{8} extended family is the most commonly shown example extending backwards from E_{3} and forwards to E_{11}.
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.^{[26]} The noncrystalographic H_{n} groups forms an extended series where H_{4} is extended as a compact hyperbolic and overextended into a lorentzian group.
The determinant of the Schläfli matrix by rank are:^{[27]}
 det(A_{1}^{n}=[2^{n1}]) = 2^{n} (Finite for all n)
 det(A_{n}=[3^{n1}]) = n+1 (Finite for all n)
 det(B_{n}=[4,3^{n2}]) = 2 (Finite for all n)
 det(D_{n}=[3^{n3,1,1}]) = 4 (Finite for all n)
Determinants of the Schläfli matrix in exceptional series are:
 det(E_{n}=[3^{n3,2,1}]) = 9n (Finite for E_{3}(=A_{2}A_{1}), E_{4}(=A_{4}), E_{5}(=D_{5}), E_{6}, E_{7} and E_{8}, affine at E_{9} (), hyperbolic at E_{10})
 det([3^{n4,3,1}]) = 2(8n) (Finite for n=4 to 7, affine (), and hyperbolic at n=8.)
 det([3^{n4,2,2}]) = 3(7n) (Finite for n=4 to 6, affine (), and hyperbolic at n=7.)
 det(F_{n}=[3,4,3^{n3}]) = 5n (Finite for F_{3}(=B_{3}) to F_{4}, affine at F_{5} (), hyperbolic at F_{6})
 det(G_{n}=[6,3^{n2}]) = 3n (Finite for G_{2}, affine at G_{3} (), hyperbolic at G_{4})
Geometric folding
φ_{A} : A_{Γ} > A_{Γ'} for finite types  

Γ  Γ'  Folding description  Coxeter–Dynkin diagrams 
I_{2}(h)  Γ(h)  Dihedral folding  
B_{n}  A_{2n}  (I,s_{n})  
D_{n+1}, A_{2n1}  (A_{3},+/ε)  
F_{4}  E_{6}  (A_{3},±ε)  
H_{4}  E_{8}  (A_{4},±ε)  
H_{3}  D_{6}  
H_{2}  A_{4}  
G_{2}  A_{5}  (A_{5},±ε)  
D_{4}  (D_{4},±ε)  
φ: A_{Γ}^{+} > A_{Γ'}^{+} for affine types  
Locally trivial  
(I,s_{n})  
,  (A_{3},±ε)  
,  (A_{3},±ε)  
(I,s_{n})  
(I,s_{n}) & (I,s_{0})  
(A_{3},ε) & (I,s_{0})  
(A_{3},ε) & (A_{3},ε')  
(A_{3},ε) & (A_{3},ε')  
(I,s_{1})  
,  (A_{3},±ε)  
,  (A_{5},±ε)  
,  (B_{3},±ε)  
,  (D_{4},±ε) 
A (simplylaced) 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".^{[29]}^{[30]}
For example, in D_{4} folding to G_{2}, the edge in G_{2} 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 simplylaced Coxeter–Dynkin diagram can be folded to I_{2}(h), where h is the Coxeter number, which corresponds geometrically to a projection to the Coxeter plane.
A few hyperbolic foldings 
Complex reflections
CoxeterDynkin diagram has been extended to Complex space, C^{n} where nodes are unitary reflections of period greater than 2. Nodes are labeled by an index, assumed to be 2 for ordinary real reflection if suppressed. Coxeter writes the complex group, p[q]r, as diagram . ^{[31]}
A 1dimensional regular complex polytope in is represented as , having p vertices. Its real representation is a regular polygon, {p}. Its symmetry is _{p}[] or , order p. A unitary operator generator for is seen as a rotation in by 2π/p radians counter clockwise, and a edge is created by sequential applications of a single unitary reflection. A unitary reflection generator for a 1polytope with p vertices is e^{2πi/p} = cos(2π/p) + i sin(2π/p). When p = 2, the generator is e^{πi} = –1, the same as a point reflection in the real plane.
In a higher polytope, _{p}{} or represents a pedge element, with a 2edge, {} or , representing an ordinary real edge between two vertices.
Complex 1polytopes represented in the Argand plane as regular polygons for p = 2, 3, 4, 5, and 6, with black vertices. The centroid of the p vertices is shown seen in red. The sides of the polygons represent one application of the symmetry generator, mapping each vertex to the next counterclockwise copy. These polygonal sides are not edge elements of the polytope, as a complex 1polytope can have no edges (it often is a complex edge) and only contains vertex elements. 
12 irreducible Shephard groups with their subgroup index relations.^{[32]} Subgroups index 2 relate by removing a real reflection: _{p}[2q]_{2} > _{p}[q]_{p}, index 2. _{p}[4]_{q} > _{p}[q]_{p}, index q. 
_{p}[4]_{2} subgroups: p=2,3,4... _{p}[4]_{2} > [p], index p _{p}[4]_{2} > _{p}[]×_{p}[], index 2 
Aa regular complex polygons in , has the form _{p}{q}_{r} or Coxeter diagram . The symmetry group of a regular complex polygon is not called a Coxeter group, but instead a Shephard group, a type of Complex reflection group. The order of _{p}[q]_{r} is .^{[33]}
The rank 2 Shephard groups are: _{2}[q]_{2}, _{p}[4]_{2}, _{3}[3]_{3}, _{3}[6]_{2}, _{3}[4]_{3}, _{4}[3]_{4}, _{3}[8]_{2}, _{4}[6]_{2}, _{4}[4]_{3}, _{3}[5]_{3}, _{5}[3]_{5}, _{3}[10]_{2}, _{5}[6]_{2}, and _{5}[4]_{3} or , , , , , , , , , , , , , of order 2q, 2p^{2}, 24, 48, 72, 96, 144, 192, 288, 360, 600, 1200, and 1800 respectively.
The symmetry group _{p1}[q]_{p2} is represented by 2 generators R_{1}, R_{2}, where: R_{1}^{p1} = R_{2}^{p2} = I. If q is even, (R_{2}R_{1})^{q/2} = (R_{1}R_{2})^{q/2}. If q is odd, (R_{2}R_{1})^{(q1)/2}R_{2} = (R_{1}R_{2})^{(q1)/2}R_{1}. When q is odd, p_{1}=p_{2}.
The group or [1 1 1]^{p} is defined by 3 period 2 unitary reflections {R_{1}, R_{2}, R_{3}}: R_{1}^{2} = R_{1}^{2} = R_{3}^{2} = (R_{1}R_{2})^{3} = (R_{2}R_{3})^{3} = (R_{3}R_{1})^{3} = (R_{1}R_{2}R_{3}R_{1})^{p} = 1. The period p can be seen as a double rotation in real .
A similar group or [1 1 1]^{(p)} is defined by 3 period 2 unitary reflections {R_{1}, R_{2}, R_{3}}: R_{1}^{2} = R_{1}^{2} = R_{3}^{2} = (R_{1}R_{2})^{3} = (R_{2}R_{3})^{3} = (R_{3}R_{1})^{3} = (R_{1}R_{2}R_{3}R_{2})^{p} = 1.
See also
 Coxeter group
 Schwarz triangle
 Goursat tetrahedron
 Dynkin diagram
 Uniform polytope
 Wythoff construction and Wythoff symbol
References
 ↑ Hall, Brian C. (2003), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, ISBN 0387401229
 ↑ Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0486614808, Sec 7.7. page 133, Schläfli's Criterion
 ↑ Lannér F., On complexes with transitive groups of automorphisms, Medd. Lunds Univ. Mat. Sem. [Comm. Sem. Math. Univ. Lund], 11 (1950), 1–71
 ↑ 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.
 ↑ The Geometry and Topology of Coxeter Groups, Michael W. Davis, 2008 p. 105 Table 6.2. Hyperbolic diagrams
 ↑ "TAKEUCHI : Arithmetic triangle groups". Projecteuclid.org. Retrieved 20130705.
 ↑ Folke Lannér, On complexes with transitive groups of automorphisms, Comm. Sém., Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 11 (1950)
 ↑ Regular Honeycombs in hyperbolic space, Coxeter, 1954
 1 2 Norman Johnson, Geometries and Transformations, Chapter 13: Hyperbolic Coxeter groups, 13.6 Lorentzian lattices
 ↑ J. L. Koszul, Lectures on hyperbolic Coxeter groups, University of Notre Dame (1967)
 ↑ M. Chein, Recherche des graphes des matrices de Coxeter hyperboliques d’ordre ≤10, Rev. Française Informat. Recherche Opérationnelle 3 (1969), no. Ser. R3, 3–16 (French).
 ↑