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Coxeter–Dynkin diagram

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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. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras.[1]

Contents

DescriptionEdit

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 matrixEdit

Every Coxeter diagram has a corresponding Schläfli matrix (so named after Ludwig Schläfli), 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-negative, 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, after F. Lannér who enumerated the compact hyperbolic groups in 1950,[3] and Koszul (or quasi-Lannér) for the paracompact groups.

Rank 2 Coxeter groupsEdit

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. 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.

Geometric visualizationsEdit

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 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 3-space 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 groupsEdit

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  
               
1 A1=[ ]
 
 
2 A2=[3]
   
B2=[4]
   
D2=A1A1
 
  G2=[6]
   
H2=[5]
   
I2[p]
   
3 A3=[32]
     
B3=[3,4]
     
D3=A3
   
E3=A2A1
     
F3=B3
     
H3 
     
4 A4=[33]
       
B4=[32,4]
       
D4=[31,1,1]
     
E4=A4
     
F4
       
H4 
       
5 A5=[34]
         
B5=[33,4]
         
D5=[32,1,1]
       
E5=D5
       
   
6 A6=[35]
           
B6=[34,4]
           
D6=[33,1,1]
         
E6=[32,2,1]
         
7 A7=[36]
             
B7=[35,4]
             
D7=[34,1,1]
           
E7=[33,2,1]
           
8 A8=[37]
               
B8=[36,4]
               
D8=[35,1,1]
             
E8=[34,2,1]
             
9 A9=[38]
                 
B9=[37,4]
                 
D9=[36,1,1]
               
 
10+ .. .. .. ..

Application with uniform polytopesEdit

 
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.
 
Two orthogonal mirrors can be used to generate a square,    , seen here with a red generator point and 3 virtual copies across the mirrors. The generator has to be off both mirrors in this orthogonal case to generate an interior. The ring markup presumes active rings have generators equal distance from all mirrors, while a rectangle can also represent a nonuniform solution.

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. An unconnected diagram (subgroups separated by order-2 branches, or orthogonal mirrors) requires at least one active node in each subgraph.

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 k-ring generator points are on (k-1)-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 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.
 
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 slash-hole 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 tilingsEdit

For example, the B3 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.

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:

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.

In the hyperbolic plane [7,3],       family produces a parallel set of uniform tilings, 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.

Affine Coxeter groupsEdit

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.  : diagrams of this type are cycles. (Also Pn)
  2.   is associated with the hypercube regular tessellation {4, 3, ...., 4} family. (Also Rn)
  3.   related to C by one removed mirror. (Also Sn)
  4.   related to C by two removed mirrors. (Also Qn)
  5.  ,  ,  . (Also T7, T8, T9)
  6.   forms the {3,4,3,3} regular tessellation. (Also U5)
  7.   forms 30-60-90 triangle fundamental domains. (Also V3)
  8.   is two parallel mirrors. ( =   =  ) (Also W2)

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 3-space.

Affine Coxeter graphs up to (2 to 10 nodes)
Rank   (P2+)   (S4+)   (R2+)   (Q5+)   (Tn+1) /   (U5) /   (V3)
2  =[∞]
   
   =[∞]
   
   
3  =[3[3]]
*    
 =[4,4]
*      
 =[6,3]
*      
4  =[3[4]]
*      
 =[4,31,1]
*      
 =[4,3,4]
*        
 =[31,1,3−1,31,1]
      =  
5  =[3[5]]
*      
 =[4,3,31,1]
*        
 =[4,32,4]
*          
 =[31,1,1,1]
*      
 =[3,4,3,3]
*          
6  =[3[6]]
*        
 =[4,32,31,1]
*          
 =[4,33,4]
*            
 =[31,1,3,31,1]
*        
 
7  =[3[7]]
*        
 =[4,33,31,1]
           
 =[4,34,4]
             
 =[31,1,32,31,1]
         
 =[32,2,2]
         
8  =[3[8]]
*          
 =[4,34,31,1]
*              
 =[4,35,4]
               
 =[31,1,33,31,1]
*            
 =[33,3,1]
*          
9  =[3[9]]
*          
 =[4,35,31,1]
               
 =[4,36,4]
                 
 =[31,1,34,31,1]
             
 =[35,2,1]
*                
10  =[3[10]]
*            
 =[4,36,31,1]
                 
 =[4,37,4]
                   
 =[31,1,35,31,1]
               
11 ... ... ... ...

Hyperbolic Coxeter groupsEdit

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 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 H2Edit

Poincaré disk model of fundamental domain triangles
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)]
 
[(∞,∞,∞)]

Two-dimensional 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]

Compact hyperbolic Coxeter groups
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
  • [p,∞]      
  • [∞,∞]      
  • [(p,q,∞)]         
  • [(p,∞,∞)]         
  • [(∞,∞,∞)]         

Arithmetic triangle groupEdit

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)

       ,        ,        ,        ,        
(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 trianglesEdit

Fundamental domains of quadrilateral groups
 
        or    
[∞,3,∞]
[iπ/λ1,3,iπ/λ2]
(*3222)
 
       or    
[((3,∞,3)),∞]
[((3,iπ/λ1,3)),iπ/λ2]
(*3322)
 
      or    
[(3,∞)[2]]
[(3,iπ/λ1,3,iπ/λ2)]
(*3232)