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
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 . CONTENTS * 1 Description * 2 Schläfli matrix * 2.1 Rank 2 Coxeter groups * 2.2 Geometric visualizations * 3 Finite Coxeter groups * 4 Application with uniform polytopes * 4.1 Example polyhedra and tilings * 5 Affine Coxeter groups * 6 Hyperbolic Coxeter groups * 6.1 Hyperbolic groups in H2 * 6.1.1 Arithmetic triangle group * 6.1.2 Hyperbolic Coxeter polygons above triangles * 6.2 Compact (Lannér simplex groups) * 6.2.1 Ranks 4–5 * 6.3 Paracompact (Koszul simplex groups) * 6.3.1 Ideal simplices * 6.3.2 Ranks 4–10 * 6.3.2.1 Subgroup relations of paracompact hyperbolic groups * 6.4 Hypercompact Coxeter groups (Vinberg polytopes) * 6.4.1 Vinberg polytopes with rank n+2 for n dimensional space * 6.4.2 Vinberg polytopes with rank n+3 for n dimensional space * 6.4.3 Vinberg polytopes with rank n+4 for n dimensional space * 7 Lorentzian groups * 7.1 Very-extended Coxeter Diagrams * 8 Geometric folding * 9 Complex reflections * 10 See also * 11 References * 12 Further reading * 13 External links DESCRIPTION Branches of a Coxeter–
Diagrams can be labeled by their graph structure. The first forms
studied by
SCHLäFLI MATRIX 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
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. 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
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, and Koszul (or quasi-Lannér) for the paracompact groups. RANK 2 COXETER GROUPS For rank 2, the type of a
TYPE FINITE AFFINE HYPERBOLIC GEOMETRY ... COXETER ORDER 2 4 6 8 2P ∞
Rank 2
{DISPLAYSTYLE LEFT {displaystyle left {displaystyle left {displaystyle left {displaystyle left {displaystyle left {displaystyle left {displaystyle left {displaystyle left {displaystyle left {displaystyle left {displaystyle left {displaystyle left {displaystyle left {displaystyle left {displaystyle left {displaystyle left {displaystyle left {displaystyle left The Coxeter–
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 I 1 {displaystyle {tilde {I}}_{1}} x I 1 {displaystyle {tilde {I}}_{1}} is shown as a doubling of the C 2 {displaystyle {tilde {C}}_{2}} , but can also be created as rectangular domains from doubling the G 2 {displaystyle {tilde {G}}_{2}} triangles. The A 2 {displaystyle {tilde {A}}_{2}} is a doubling of the G 2 {displaystyle {tilde {G}}_{2}} 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. C 3 {displaystyle {tilde {C}}_{3}} fills 1/48 of the cube. B 3 {displaystyle {tilde {B}}_{3}} fills 1/24 of the cube. A 3 {displaystyle {tilde {A}}_{3}} 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 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 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 + {DISPLAYSTYLE {A}_{1+}} B 2 + {DISPLAYSTYLE {B}_{2+}} D 2 + {DISPLAYSTYLE {D}_{2+}} E 3 8 {DISPLAYSTYLE {E}_{3-8}} F 3 4 {DISPLAYSTYLE {F}_{3-4}} G 2 {DISPLAYSTYLE {G}_{2}} H 2 4 {DISPLAYSTYLE {H}_{2-4}} I 2 ( P ) {DISPLAYSTYLE {I}_{2}(P)} 1 A1= 2 A2= B2= D2=A1A1 G2= H2= I2 3 A3= B3= D3=A3 E3=A2A1 F3=B3 H3 4 A4= B4= D4= E4=A4 F4 H4 5 A5= B5= D5= E5=D5 6 A6= B6= D6= E6= 7 A7= B7= D7= E7= 8 A8= B8= D8= E8= 9 A9= B9= D9= 10+ .. .. .. .. APPLICATION WITH UNIFORM POLYTOPES 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
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
* 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 TILINGS For example, the B3
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
UNIFORM OCTAHEDRAL POLYHEDRA SYMMETRY : , (*432) + (432) = (*332) (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} = = = = or = or = 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 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 ×[] or family: UNIFORM HEXAGONAL DIHEDRAL SPHERICAL POLYHEDRA SYMMETRY : , (*622) +, (622) , (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 V62 V122 V62 V4.4.6 V26 V4.4.6 V4.4.12 V3.3.3.6 V3.3.3.3 In comparison, the , 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 : , (*632) + (632) (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} 63 3.122 (3.6)2 6.6.6 36 3.4.12.4 4.6.12 3.3.3.3.6 3.3.3.3.3.3 UNIFORM DUALS V63 V3.122 V(3.6)2 V63 V36 V3.4.12.4 V.4.6.12 V34.6 V36 In the hyperbolic plane , 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: , (*732) +, (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 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 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. (
* A n 1 {displaystyle {tilde {A}}_{n-1}} : diagrams of this type are cycles. (Also Pn) * C n 1 {displaystyle {tilde {C}}_{n-1}} is associated with the hypercube regular tessellation {4, 3, ...., 4} family. (Also Rn) * B n 1 {displaystyle {tilde {B}}_{n-1}} related to C by one removed mirror. (Also Sn) * D n 1 {displaystyle {tilde {D}}_{n-1}} related to C by two removed mirrors. (Also Qn) * E 6 {displaystyle {tilde {E}}_{6}} , E 7 {displaystyle {tilde {E}}_{7}} , E 8 {displaystyle {tilde {E}}_{8}} . (Also T7, T8, T9) * F 4 {displaystyle {tilde {F}}_{4}} forms the {3,4,3,3} regular tessellation. (Also U5) * G 2 {displaystyle {tilde {G}}_{2}} forms 30-60-90 triangle fundamental domains. (Also V3) * I 1 {displaystyle {tilde {I}}_{1}} is two parallel mirrors. ( = A 1 {displaystyle {tilde {A}}_{1}} = C 1 {displaystyle {tilde {C}}_{1}} ) (Also W2) Composite groups can also be defined as orthogonal projects. The most common use A 1 {displaystyle {tilde {A}}_{1}} , like A 1 2 {displaystyle {tilde {A}}_{1}^{2}} , represents square or rectangular checker board domains in the Euclidean plane. And A 1 G 2 {displaystyle {tilde {A}}_{1}{tilde {G}}_{2}} represents triangular prism fundamental domains in Euclidean 3-space. Affine Coxeter graphs up to (2 to 10 nodes) RANK A 1 + {DISPLAYSTYLE {TILDE {A}}_{1+}} (P2+) B 3 + {DISPLAYSTYLE {TILDE {B}}_{3+}} (S4+) C 1 + {DISPLAYSTYLE {TILDE {C}}_{1+}} (R2+) D 4 + {DISPLAYSTYLE {TILDE {D}}_{4+}} (Q5+) E N {DISPLAYSTYLE {TILDE {E}}_{N}} (TN+1) / F 4 {DISPLAYSTYLE {TILDE {F}}_{4}} (U5) / G 2 {DISPLAYSTYLE {TILDE {G}}_{2}} (V3) 2 A 1 {displaystyle {tilde {A}}_{1}} = C 1 {displaystyle {tilde {C}}_{1}} = 3 A 2 {displaystyle {tilde {A}}_{2}} =] * C 2 {displaystyle {tilde {C}}_{2}} = * G 2 {displaystyle {tilde {G}}_{2}} = * 4 A 3 {displaystyle {tilde {A}}_{3}} =] * B 3 {displaystyle {tilde {B}}_{3}} = * C 3 {displaystyle {tilde {C}}_{3}} = * D 3 {displaystyle {tilde {D}}_{3}} = = A 3 {displaystyle {tilde {A}}_{3}} 5 A 4 {displaystyle {tilde {A}}_{4}} =] * B 4 {displaystyle {tilde {B}}_{4}} = * C 4 {displaystyle {tilde {C}}_{4}} = * D 4 {displaystyle {tilde {D}}_{4}} = * F 4 {displaystyle {tilde {F}}_{4}} = * 6 A 5 {displaystyle {tilde {A}}_{5}} =] * B 5 {displaystyle {tilde {B}}_{5}} = * C 5 {displaystyle {tilde {C}}_{5}} = * D 5 {displaystyle {tilde {D}}_{5}} = * 7 A 6 {displaystyle {tilde {A}}_{6}} =] * B 6 {displaystyle {tilde {B}}_{6}} = C 6 {displaystyle {tilde {C}}_{6}} = D 6 {displaystyle {tilde {D}}_{6}} = E 6 {displaystyle {tilde {E}}_{6}} = 8 A 7 {displaystyle {tilde {A}}_{7}} =] * B 7 {displaystyle {tilde {B}}_{7}} = * C 7 {displaystyle {tilde {C}}_{7}} = D 7 {displaystyle {tilde {D}}_{7}} = * E 7 {displaystyle {tilde {E}}_{7}} = * 9 A 8 {displaystyle {tilde {A}}_{8}} =] * B 8 {displaystyle {tilde {B}}_{8}} = C 8 {displaystyle {tilde {C}}_{8}} = D 8 {displaystyle {tilde {D}}_{8}} = E 8 {displaystyle {tilde {E}}_{8}} = * 10 A 9 {displaystyle {tilde {A}}_{9}} =] * B 9 {displaystyle {tilde {B}}_{9}} = C 9 {displaystyle {tilde {C}}_{9}} = D 9 {displaystyle {tilde {D}}_{9}} = 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 finite-volume Vinberg polytopes for dimension up to 19, 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 Further information:
EXAMPLE GENERAL TRIANGLES Two-dimensional hyperbolic triangle groups exist as rank 3 Coxeter diagrams, defined by triangle (p q r) for: 1 p + 1 q + 1 r 3 {displaystyle {overline {K}}_{3}} = : J 3 {displaystyle {overline {J}}_{3}} = : D H 3 {displaystyle {overline {DH}}_{3}} = : A B 3 {displaystyle {widehat {AB}}_{3}} = : A H 3 {displaystyle {widehat {AH}}_{3}} = : B B 3 {displaystyle {widehat {BB}}_{3}} = ]: B H 3 {displaystyle {widehat {BH}}_{3}} = : H H 3 {displaystyle {widehat {HH}}_{3}} = ]: H4 5 5 3: H 4 {displaystyle {overline {H}}_{4}} = : B H 4 {displaystyle {overline {BH}}_{4}} = : K 4 {displaystyle {overline {K}}_{4}} = : D H 4 {displaystyle {overline {DH}}_{4}} = : A F 4 {displaystyle {widehat {AF}}_{4}} = : PARACOMPACT (KOSZUL SIMPLEX GROUPS) 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
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 3-dimensional groups , , , , and , and one 5-dimensional group . Ideal Simplices Ideal fundamental domains of , seen in the
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. RANK IDEAL GROUP AFFINE SUBGROUPS 3 4 ] 4 ] ] 4 ] 6 ] , , Ranks 4–10 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
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 B R 3 {displaystyle {widehat {BR}}_{3}} = : C R 3 {displaystyle {widehat {CR}}_{3}} = : R R 3 {displaystyle {widehat {RR}}_{3}} = ]: A V 3 {displaystyle {widehat {AV}}_{3}} = : B V 3 {displaystyle {widehat {BV}}_{3}} = : H V 3 {displaystyle {widehat {HV}}_{3}} = : V V 3 {displaystyle {widehat {VV}}_{3}} = ]: P 3 {displaystyle {overline {P}}_{3}} = ]: B P 3 {displaystyle {overline {BP}}_{3}} = ]: H P 3 {displaystyle {overline {HP}}_{3}} = ]: V P 3 {displaystyle {overline {VP}}_{3}} = ]: D V 3 {displaystyle {overline {DV}}_{3}} = : O 3 {displaystyle {overline {O}}_{3}} = : M 3 {displaystyle {overline {M}}_{3}} = : R 3 {displaystyle {overline {R}}_{3}} = : N 3 {displaystyle {overline {N}}_{3}} = : V 3 {displaystyle {overline {V}}_{3}} = : B V 3 {displaystyle {overline {BV}}_{3}} = : H V 3 {displaystyle {overline {HV}}_{3}} = : Y 3 {displaystyle {overline {Y}}_{3}} = : Z 3 {displaystyle {overline {Z}}_{3}} = : D P 3 {displaystyle {overline {DP}}_{3}} = x[]]: P P 3 {displaystyle {overline {PP}}_{3}} = ]: 5 9 P 4 {displaystyle {overline {P}}_{4}} = ]: B P 4 {displaystyle {overline {BP}}_{4}} = ]: F R 4 {displaystyle {widehat {FR}}_{4}} = : D P 4 {displaystyle {overline {DP}}_{4}} = x[]]: N 4 {displaystyle {overline {N}}_{4}} = : O 4 {displaystyle {overline {O}}_{4}} = : S 4 {displaystyle {overline {S}}_{4}} = : R 4 {displaystyle {overline {R}}_{4}} = : M 4 {displaystyle {overline {M}}_{4}} = : 6 12 P 5 {displaystyle {overline {P}}_{5}} = ]: A U 5 {displaystyle {widehat {AU}}_{5}} = : A R 5 {displaystyle {widehat {AR}}_{5}} = ]: S 5 {displaystyle {overline {S}}_{5}} = : O 5 {displaystyle {overline {O}}_{5}} = : N 5 {displaystyle {overline {N}}_{5}} = : U 5 {displaystyle {overline {U}}_{5}} = : X 5 {displaystyle {overline {X}}_{5}} = : R 5 {displaystyle {overline {R}}_{5}} = : Q 5 {displaystyle {overline {Q}}_{5}} = : M 5 {displaystyle {overline {M}}_{5}} = : L 5 {displaystyle {overline {L}}_{5}} = : 7 3 P 6 {displaystyle {overline {P}}_{6}} = ]: Q 6 {displaystyle {overline {Q}}_{6}} = : S 6 {displaystyle {overline {S}}_{6}} = : 8 4 P 7 {displaystyle {overline {P}}_{7}} = ]: Q 7 {displaystyle {overline {Q}}_{7}} = : S 7 {displaystyle {overline {S}}_{7}} = : T 7 {displaystyle {overline {T}}_{7}} = : 9 4 P 8 {displaystyle {overline {P}}_{8}} = ]: Q 8 {displaystyle {overline {Q}}_{8}} = : S 8 {displaystyle {overline {S}}_{8}} = : T 8 {displaystyle {overline {T}}_{8}} = : 10 3 P 9 {displaystyle {overline {P}}_{9}} = ]: Q 9 {displaystyle {overline {Q}}_{9}} = : S 9 {displaystyle {overline {S}}_{9}} = : T 9 {displaystyle {overline {T}}_{9}} = : Subgroup Relations Of Paracompact Hyperbolic Groups These trees represents subgroup relations of paracompact hyperbolic groups. Subgroup indices on each connection are given in red. 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 H4 H5 HYPERCOMPACT COXETER GROUPS (VINBERG POLYTOPES) Just like the hyperbolic plane H2 has nontriangular 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 edges as the intersection of two mirrors 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 from 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 The complete list of compact hyperbolic Vinberg polytopes with rank n+2 mirrors for n-dimensions has been enumerated by F. Esselmann in 1996. A partial list was published in 1974 by I. M. Kaplinskaya. The complete list of paracompact solutions was published by P. Tumarkin in 2003, with dimensions from 3 to 17. The smallest paracompact form in H3 can be represented by , or which can be constructed by a mirror removal of paracompact hyperbolic group as . The doubled fundamental domain changes from a tetrahedron into a quadrilateral pyramid. Another pyramids include = , = . Removing a mirror from some of the cyclic hyperbolic Coxeter graphs become bow-tie graphs: = or , = or , = or . Other valid paracompact graphs with quadrilateral pyramid fundamental domains include: DIMENSION RANK GRAPHS H3 5 , , , , , , , , , , , , , , , , , , , , , , , , , , , Another subgroup = = ]. = = . Vinberg Polytopes With Rank N+3 For N Dimensional Space 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. The complete list of non-Compact Vinberg polytopes with rank n+3 mirrors and with one non-simple vertex for n-dimensions has been enumerated by Mike Roberts. For 4 to 8 dimensions, rank 7 to 11 Coxeter groups are counted as 44, 16, 3, 1, and 1 respectively. The highest was discovered by Bugaenko in 1984 in dimension 8, rank 11: DIMENSIONS RANK CASES GRAPHS H4 7 44 ... H5 8 16 .. H6 9 3 H7 10 1 H8 11 1 Vinberg Polytopes With Rank N+4 For N Dimensional Space 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. LORENTZIAN GROUPS Regular honeycombs with Lorentzian groups {3,3,7} in hyperbolic 3-space, rendered the intersection of the honeycomb with the plane-at-infinity, in the Poincare half-space model . {7,3,3} viewed outside of Poincare ball model. This shows rank 5 Lorentzian groups arranged as subgroups from , and . The highly symmetric group , ] is an index 120 subgroup of . Lorentzian groups for simplex domains can be defined as graphs beyond
the paracompact hyperbolic forms. These are sometimes called
super-ideal simplices and are also related to a Lorentzian geometry ,
named after
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-
For the highest ranks 8-11, the complete lists are: Lorentzian Coxeter groups RANK Total count GROUPS 4 ∞ ... : ... ] ... ]: ... ... : ... ... ... ]: ... ... ... ×[]] ... ×[]]: ... ... ] ... ] 5 186 ...]:... 6 66 7 36 : ... 8 13 ]: ,3]: ,3]: ,3]: e×]: : : : : : : 9 10 ,3]: ]: ,3]: : : : : : : : 10 8 ,3]: ,3]: ]: : : : : : 11 4 : : : : VERY-EXTENDED COXETER DIAGRAMS 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 VERY-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. 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: * det(A1n=) = 2n (Finite for all n) * det(An=) = n+1 (Finite for all n) * det(Bn=) = 2 (Finite for all n) * det(Dn=) = 4 (Finite for all n) Determinants of the Schläfli matrix in exceptional series are: * det(En =) = 9-n (Finite for E3(=A2A1), E4(=A4), E5(=D5), E6 , E7 and E8 , affine at E9 ( E 8 {displaystyle {tilde {E}}_{8}} ), hyperbolic at E10) * det() = 2(8-n) (Finite for n=4 to 7, affine ( E 7 {displaystyle {tilde {E}}_{7}} ), and hyperbolic at n=8.) * det() = 3(7-n) (Finite for n=4 to 6, affine ( E 6 {displaystyle {tilde {E}}_{6}} ), and hyperbolic at n=7.) * det(Fn=) = 5-n (Finite for F3(=B3) to F4 , affine at F5 ( F 4 {displaystyle {tilde {F}}_{4}} ), hyperbolic at F6) * det(Gn=) = 3-n (Finite for G2 , affine at G3 ( G 2 {displaystyle {tilde {G}}_{2}} ), hyperbolic at G4) Smaller extended series FINITE A 2 {DISPLAYSTYLE A_{2}} C 2 {DISPLAYSTYLE C_{2}} G 2 {DISPLAYSTYLE G_{2}} A 3 {DISPLAYSTYLE A_{3}} B 3 {DISPLAYSTYLE B_{3}} C 3 {DISPLAYSTYLE C_{3}} H 4 {DISPLAYSTYLE H_{4}} RANK N ,3N-3] GN= ,3N-4] HN= 2 A2 C2 G2 A12 C2 H2 3 ] A2+= A 2 {displaystyle {tilde {A}}_{2}} C2+= C 2 {displaystyle {tilde {C}}_{2}} G2+= G 2 {displaystyle {tilde {G}}_{2}} =A3 B3 C3 H3 4 ,3] A2++= P 3 {displaystyle {overline {P}}_{3}} C2++= R 3 {displaystyle {overline {R}}_{3}} G2++= V 3 {displaystyle {overline {V}}_{3}} ] A3+= A 3 {displaystyle {tilde {A}}_{3}} B3+= B 3 {displaystyle {tilde {B}}_{3}} C3+= C 3 {displaystyle {tilde {C}}_{3}} H4 5 ,3,3] A2+++ C2+++ G2+++ ,3] A3++= P 4 {displaystyle {overline {P}}_{4}} B3++= S 4 {displaystyle {overline {S}}_{4}} C3++= R 4 {displaystyle {overline {R}}_{4}} H5= H 4 {displaystyle {overline {H}}_{4}} 6 ,3,3] A3+++ B3+++ C3+++ H6 DET(MN) 3(3-n) 2(3-n) 3-n 4(4-n) 2(4-n) Middle extended series FINITE A 4 {DISPLAYSTYLE A_{4}} B 4 {DISPLAYSTYLE B_{4}} C 4 {DISPLAYSTYLE C_{4}} D 4 {DISPLAYSTYLE D_{4}} F 4 {DISPLAYSTYLE F_{4}} A 5 {DISPLAYSTYLE A_{5}} B 5 {DISPLAYSTYLE B_{5}} D 5 {DISPLAYSTYLE D_{5}} RANK N ,3N-5] ,3N-6] 3 B2A1 B3 A13 B3 C3 4 A4 B4 C4 D4 F4 B3A1 A3A1 5 ] A4+= A 4 {displaystyle {tilde {A}}_{4}} B4+= B 4 {displaystyle {tilde {B}}_{4}} C4+= C 4 {displaystyle {tilde {C}}_{4}} D4+= D 4 {displaystyle {tilde {D}}_{4}} F4+= F 4 {displaystyle {tilde {F}}_{4}} A5 B5 D5 6 ,3] A4++= P 5 {displaystyle {overline {P}}_{5}} B4++= S 5 {displaystyle {overline {S}}_{5}} C4++= R 5 {displaystyle {overline {R}}_{5}} D4++= Q 5 {displaystyle {overline {Q}}_{5}} F4++= U 5 {displaystyle {overline {U}}_{5}} ] A5+= A 5 {displaystyle {tilde {A}}_{5}} B5+= B 5 {displaystyle {tilde {B}}_{5}} D5+= D 5 {displaystyle {tilde {D}}_{5}} 7 ,3,3] A4+++ B4+++ C4+++ D4+++ F4+++ ,3] A5++= P 6 {displaystyle {overline {P}}_{6}} B5++= S 6 {displaystyle {overline {S}}_{6}} D5++= Q 6 {displaystyle {overline {Q}}_{6}} 8 ,3,3] A5+++ B5+++ D5+++ 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 {DISPLAYSTYLE A_{6}} B 6 {DISPLAYSTYLE B_{6}} D 6 {DISPLAYSTYLE D_{6}} E 6 {DISPLAYSTYLE E_{6}} A 7 {DISPLAYSTYLE A_{7}} B 7 {DISPLAYSTYLE B_{7}} D 7 {DISPLAYSTYLE D_{7}} E 7 {DISPLAYSTYLE E_{7}} E 8 {DISPLAYSTYLE E_{8}} RANK N ,3N-7] ,3N-8] EN= 3 E3=A2A1 4 A22 A3A1 E4=A4 5 B4A1 D4A1 A5 A5 E5=D5 6 A6 B6 D6 E6 B5A1 D5A1 D6 E6 * 7 ] A6+= A 6 {displaystyle {tilde {A}}_{6}} B6+= B 6 {displaystyle {tilde {B}}_{6}} D6+= D 6 {displaystyle {tilde {D}}_{6}} E6+= E 6 {displaystyle {tilde {E}}_{6}} A7 B7 D7 E7 * E7 * 8 ,3] A6++= P 7 {displaystyle {overline {P}}_{7}} B6++= S 7 {displaystyle {overline {S}}_{7}} D6++= Q 7 {displaystyle {overline {Q}}_{7}} E6++= T 7 {displaystyle {overline {T}}_{7}} ] A7+= A 7 {displaystyle {tilde {A}}_{7}} * B7+= B 7 {displaystyle {tilde {B}}_{7}} * D7+= D 7 {displaystyle {tilde {D}}_{7}} * E7+= E 7 {displaystyle {tilde {E}}_{7}} * E8 * 9 ,3,3] A6+++ B6+++ D6+++ E6+++ ,3] A7++= P 8 {displaystyle {overline {P}}_{8}} * B7++= S 8 {displaystyle {overline {S}}_{8}} * D7++= Q 8 {displaystyle {overline {Q}}_{8}} * E7++= T 8 {displaystyle {overline {T}}_{8}} * E9=E8+= E 8 {displaystyle {tilde {E}}_{8}} * 10 ,3,3] A7+++ * B7+++ * D7+++ * E7+++ * E10=E8++= T 9 {displaystyle {overline {T}}_{9}} * 11 E11=E8+++ * 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 Finite and affine foldings φA : AΓ --> AΓ\' FOR FINITE TYPES Γ Γ\' FOLDING DESCRIPTION COXETER–DYNKIN DIAGRAMS I2(h ) Γ(h) Dihedral folding Bn 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 A n 1 {displaystyle {tilde {A}}_{n-1}} A k n 1 {displaystyle {tilde {A}}_{kn-1}} Locally trivial B n {displaystyle {tilde {B}}_{n}} D 2 n + 1 {displaystyle {tilde {D}}_{2n+1}} (I,sn) D n + 1 {displaystyle {tilde {D}}_{n+1}} , D 2 n {displaystyle {tilde {D}}_{2n}} (A3,±ε) C n {displaystyle {tilde {C}}_{n}} B n + 1 {displaystyle {tilde {B}}_{n+1}} , C 2 n {displaystyle {tilde {C}}_{2n}} (A3,±ε) C 2 n + 1 {displaystyle {tilde {C}}_{2n+1}} (I,sn) C n {displaystyle {tilde {C}}_{n}} A 2 n + 1 {displaystyle {tilde {A}}_{2n+1}} (I,sn) "> A 2 n {displaystyle {tilde {A}}_{2n}} (A3,ε) "> A 2 n 1 {displaystyle {tilde {A}}_{2n-1}} (A3,ε) "> C n {displaystyle {tilde {C}}_{n}} D n + 2 {displaystyle {tilde {D}}_{n+2}} (A3,-ε) "> C 2 {displaystyle {tilde {C}}_{2}} D 5 {displaystyle {tilde {D}}_{5}} (I,s1) F 4 {displaystyle {tilde {F}}_{4}} E 6 {displaystyle {tilde {E}}_{6}} , E 7 {displaystyle {tilde {E}}_{7}} (A3,±ε) G 2 {displaystyle {tilde {G}}_{2}} D 6 {displaystyle {tilde {D}}_{6}} , E 7 {displaystyle {tilde {E}}_{7}} (A5,±ε) B 3 {displaystyle {tilde {B}}_{3}} , F 4 {displaystyle {tilde {F}}_{4}} (B3,±ε) D 4 {displaystyle {tilde {D}}_{4}} , E 6 {displaystyle {tilde {E}}_{6}} (D4,±ε) See also:
A (simply-laced) Coxeter–
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–
A few hyperbolic foldings COMPLEX REFLECTIONS Coxeter–
A 1-dimensional regular complex polytope in C 1 {displaystyle mathbb {C} ^{1}} 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 R 2 {displaystyle mathbb {R} ^{2}} 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 1-polytope with p vertices is e2πi/p = cos(2π/p) + i sin(2π/p). When p = 2, the generator is eπi = –1, the same as a point reflection in the real plane. In a higher polytope, p{} or represents a p-edge element, with a 2-edge, {} or , representing an ordinary real edge between two vertices. Regular complex 1-polytopes Complex 1-polytopes represented in the
12 irreducible Shephard groups with their subgroup index relations. Subgroups index 2 relate by removing a real reflection: p2 --> pp, index 2. pq --> pp, index q. p2 subgroups: p=2,3,4... p2 --> , index p p2 --> p[]×p[], index 2 Aa regular complex polygons in C 2 {displaystyle mathbb
{C} ^{2}} , has the form p{q}r or Coxeter diagram . The symmetry
group of a regular complex polygon is not called a
The rank 2 Shephard groups are: 22, p2, 33, 32, 33, 44, 32, 42, 43, 33, 55, 32, 52, and 53 or , , , , , , , , , , , , , of order 2q, 2p2, 24, 48, 72, 96, 144, 192, 288, 360, 600, 1200, and 1800 respectively. The symmetry group p1p2 is represented by 2 generators R1, R2, where: R1p1 = R2p2 = I. If q is even, (R2R1)q/2 = (R1R2)q/2. If q is odd, (R2R1)(q-1)/2R2 = (R1R2)(q-1)/2R1. When q is odd, p1=p2. The C 3 {displaystyle mathbb {C} ^{3}} group or p is defined by 3 period 2 unitary reflections {R1, R2, R3}: R12 = R12 = R32 = (R1R2)3 = (R2R3)3 = (R3R1)3 = (R1R2R3R1)p = 1. The period p can be seen as a double rotation in real R 4 {displaystyle mathbb {R} ^{4}} . A similar C 3 {displaystyle mathbb {C} ^{3}} group or (p) is defined by 3 period 2 unitary reflections {R1, R2, R3}: R12 = R12 = R32 = (R1R2)3 = (R2R3)3 = (R3R1)3 = (R1R2R3R2)p = 1. SEE ALSO *
*
*
REFERENCES * ^ Hall, Brian C. (2003), Lie Groups, Lie Algebras, and
Representations: An Elementary Introduction, Springer, ISBN
0-387-40122-9
* ^ Coxeter, Regular Polytopes , (3rd edition, 1973), Dover
edition, ISBN 0-486-61480-8 , Sec 7.7. page 133, Schläfli's Criterion
* ^ Lannér F., On complexes with transitive groups of
automorphisms, Medd. Lunds Univ. Mat. Sem. , 11 (1950), 1–71
* ^ Allcock, Daniel (11 July 2006). "Infinitely many hyperbolic
Coxeter groups through dimension 19".
FURTHER READING * 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 , Googlebooks * (Paper 17) Coxeter , The Evolution of Coxeter-Dynkin diagrams, * 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 * Regular Polytopes , Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter 5: The Kaleidoscope, and Section 11.3 Representation by graphs) * 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 hyperbol |