Report

Rank 3-4 Coxeter Groups, Quaternions and Quasicrystals Mehmet Koca Department of Physics College of Science Sultan Qaboos University Muscat-OMAN [email protected] Bangalore conference, 16-22 December, 2012 1 References Polyhedra obtained from Coxeter groups and quaternions. Koca M., Al-Ajmi M., Koc R. 11, November 2007, Journal of Mathematical Physics, Vol. 48. Catalan solids derived from 3D-root systems and quaternions. Koca M., Koca N.O, Koc R. 4, s.l. : Journal of Mathematical Physics, 2010, Vol. 51. Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions, Mehmet Koca, Nazife Ozdes Koca and Muna Al-Shueili, ", arXiv:1006.3149 [pdf], SQU Journal for Science, 16 (2011) 63-82, 2011. Bangalore conference, 16-22 December, 2012 2 Outline 1. Rank-3 Coxeter Groups with Quaternions and Polyhedra 1.1. Rank-3 Coxeter Groups with Quaternions 1.2. Quaternionic construction of vertices of Platonic and Archimedean polyhedra with tetrahedral, octahedral and icosahedral symmetries 1.3. Catalan solids as duals of the Archimedean solids 1.4. Novel construction of chiral polyhedra and their duals. Bangalore conference, 16-22 December, 2012 3 Outline 2. Rank-4 Coxeter Groups with Quaternions and 4D polytopes 2.1. Representations of the group elements of rank-4 Coxeter groups with quaternions 2.2. 4D polytopes with W ( A 4 ) symmetry 2.3. 4D polytopes with W ( B 4 ) symmetry 2.4. 4D polytopes with W ( F 4 ) symmetry 2.5. 4D polytopes with W ( H 4 ) symmetry 2.6. Maximal Subgroups of W ( H 4 ) and associated 4D polytopes, snub 24-cell and Grand antiprism Bangalore conference, 16-22 December, 2012 4 Outline 3. Quasicrystallography from higher dimensional lattices 3.1. Quasicrystals and aperiodic tiling of the plane 3.2. Maximal dihedral subgroups of the Coxeter groups 3.3. Projection of the lattices generated by the affine Coxeter groups onto the Coxeter plane 3.4. Affine A4 and decagonal quasicrystals 3.5. Affine D6 and Icosahedral quasicrystals 3.6. Conclusion Bangalore conference, 16-22 December, 2012 5 1.1. Rank-3 Coxeter Groups with Quaternions O(4) transformations with quaternions Let q q0 qiei , i 1,2,3 be unit quaternion qq qq 1 e1 i , e 2 j , e 3 k e i e j ij ijk e k , The transformations r p rq : [ p , q ] r prq : [ p , q ] ; with qq pp 1 define an orthogonal group O(4) leaving rr invariant. If q p then above transformations lead to O(3) transformations. Scalar product : ( p , q ) 1 ( pq q p) 2 Bangalore conference, 16-22 December, 2012 6 Reflections Quaternions can be used to represent reflections and rotations in Coxeter Groups. Let be an arbitrary quaternionic simple root. Then the reflection of an arbitrary vector with respect to the plane orthogonal to the simple root is given by 2 , The reflection can be represented as: r : 1 2 2 , 2 ʌ , 2 2 2 N o tatio n s :ro tary reflectio n [ p , q ] 2 -α α * p ro p er ro tatio n [ p , q ] w h ere p an d q are arb itrary u n it q u atern io n s Bangalore conference, 16-22 December, 2012 7 Finite Subgroups of Quaternions (a) Cyclic group of order 2n generated by p exp(e1 n ) , (b) Dicyclic group of order 4n generated by p exp(e1 ), e2 , n (c) Binary Tetrahedral Group: 1 T {1, e1, e2 , e3 , (1 e1 e2 e3 )} 2 Bangalore conference, 16-22 December, 2012 8 Finite Subgroups of Quaternions (d) Binary Octahedral Group Denote the set T (V1 V2 V3 ) with V1 { 12 (1 e1 ), 1 2 (e2 e3 )} , V2 { 12 (1 e2 ), 1 ( e e )} 3 1 , 2 V3 { 12 (1 e3 ), 12 (e1 e2 )} Then O T T represents the elements of the binary octahedral group (e) Binary icosahedral group I b, c 1 c ( e1 e2 ) , b 1 ( e1 e2 ) 2 2 1 5 2 , 1 5 2 Bangalore conference, 16-22 December, 2012 9 Coxeter Diagrams A3, B3 and H3 with quaternionic roots A3 : W ( A3 ) {[T , T ] [T , T ]}:Symmetry of Tetrahedron B 3: W (B 3 ) A ut (A 3 ) {[T ,T ] [T ,T ] [T ,T ] [T ,T ]}:Symmetry of Cube H3 : W ( H 3 ) {[ I , I ] [ I , I ]}: Symmetry of Icosahedron Bangalore conference, 16-22 December, 2012 10 Platonic solids (regular polyhedra) Platonic solids are the five convex regular polyhedra. They consist of regular polygons (triangle, square or pentagon) meeting in identical vertices. They have identical faces of regular polygons and the same number of faces meeting at each corner In geometry, polyhedra are formed in pairs called duals, where the vertices of one correspond to the faces of the other. The dual of each platonic solid is another platonic solid: Tetrahedron is self dual Cube and Octahedron form a dual pair Dodecahedron and Icosahedron form a dual pair Bangalore conference, 16-22 December, 2012 11 Archimedean Solids (semi-regular polyhedra) Two or more types of regular polygons meet in identical vertices. There are 13 Archimedean solids. 7 of the Archimedean solids can be obtained by truncation of the platonic solids. 4 of the Archimedean solids are obtained by expansion of platonic solids and previous Archimedean solids. The remaining 2 chiral solids are snub cube and snub dodecahedron Bangalore conference, 16-22 December, 2012 12 , Construction of polyhedra with Tetrahedral Symmetry , For any Coxeter diagram, the simple roots α i and their dual vectors ( i , j ) C ij , ( i , j ) ( C r1 [ 1 2 ( e1 e 2 ), 1 2 1 ) ij , ( i , j ) ij ; i, j 1,2,3. ( e1 e 2 )] r2 [ 1 2 ( e3 e 2 ), ω i satisfy th e scalar product i C ij j , i ( C 1 2 ( e3 e 2 )] Bangalore conference, 16-22 December, 2012 r3 [ 1 2 1 ) ij j . ( e 2 e1 ), 1 2 ( e 2 e1 )] 13 A method used to construct the polyhedra We construct polyhedra using a method based on applying the group elements of Coxeter-Weyl groups W(A3), W(B3) and W(H3) on a vector representing one vertex of the polyhedron in the dual space denoted as a1 1 a 2 2 a 3 3 ( a1 a 2 a 3 ) This vector is called “highest weight” . It can be expressed as a linear combination of imaginary quaternionic units. Certain choices of the parameters of the highest weight vector lead to the Platonic, Archimedean solids as well as the semi-regular polyhedra. The set of vertices obtained by the action of Coxeter-Weyl group elements on the highest weight defines a polyhedron and is called the “orbit”. Denote by W(G)(a1a2a3)=(a1a2a3)G , the orbit of W(G) Bangalore conference, 16-22 December, 2012 14 Construction of polyhedra with Tetrahedral Symmetry Bangalore conference, 16-22 December, 2012 15 Construction of polyhedra with Octahedral Symmetry T he C artan m atrix o f the C o xeter d iag ra m B 3 and its in verse m atrix are g iv e n b y 2 C 1 0 1 2 2 , 2 0 2 C 1 1 1 1 2 1 2 2. 3 2 1 2 2 T he g en erato rs, r1 [ 1 2 ( e1 e 2 ), 1 2 ( e1 e 2 )] , r2 [ 1 2 ( e 2 e3 ), 1 2 ( e 2 e 3 )] , r3 [ e 3 , e 3 ] ) g enerate the o ctahed ra l g ro u p w h ic h ca n be w r itten as W ( B 3 ) A ut ( A3 ) S 4 C 2 {[ p , p ] [ p , p ] [ t , t ] [ t , t ] } , p T , t T . Bangalore conference, 16-22 December, 2012 16 Construction of polyhedra with Octahedral Symmetry Bangalore conference, 16-22 December, 2012 17 Construction of polyhedra with Icosahedral symmetry 1 = 2 C 0 1 , −1 ∗ , 2 1 2 2 1 = 0 1 2 3 −1 1 + 2 + 3 , − 2 1 3 = 2 3 2 3 1 (1 2 2 3 4 2 2 2 ∗ 3 2 2 . +2 + 2 + 3 ) , 3 = 2 , −2 Bangalore conference, 16-22 December, 2012 ∗ . 18 Construction of polyhedra with Icosahedral Symmetry Bangalore conference, 16-22 December, 2012 19 Catalan Solids (Duals of Archimedean Solids) Face transitive (faces are transformed to each other by the Coxeter-Weyl group) . Faces are non regular polygons: scalene triangles, isosceles triangles, rhombuses, kites or irregular pentagons. Two Catalan solids are Chiral. Bangalore conference, 16-22 December, 2012 20 The method to generate the dual polyhedra It is based on examining the CoxeterDynkin diagram of the polyhedron which helps in determining the type of its faces and the center of a representative face which corresponds to the dual’s vertex. The center is a vector that is left invariant under the action of the dihedral subgroup that generated the face For example, the Coxeter-Dynkin diagram of (3 ) can generate polyhedra with faces listed in Table depending on the components (Dynkin indices) of the highest weight Λ and the dihedral subgroups of (3 ). Example: 1 is left unchanged by 2 , 3 because 2 1 = 1 3 1 = 1 . Bangalore conference, 16-22 December, 2012 21 The method to generate the dual polyhedra Bangalore conference, 16-22 December, 2012 22 Catalan Solid Possessing Tetrahedral Symmetry There is only one Catalan solid that possesses tetrahedral symmetry. It is the triakis tetrahedron (dual of truncated tetrahedron)which has 8 faces (4 triangles and 4 hexagons). The vertices of the truncated tetrahedron were obtained as the orbit (). A triangle of the truncated tetrahedron is generated as the orbit , (), so its representative center will be . On the other side a hexagon is generated by , ()and its center will be . The line joining the two centers should be orthogonal to Λ = (110), so one of the centers should be scaled by . 1 − 3 . Λ = 0 . + The triakis tetrahedron vertices are the union of the following two orbits 1 .( + ) We can determine = . = .(+) = + = . 3 = 5 1 1 1 + 2 + 3 , 1 − 2 − 3 , 2 2 1 1 −1 − 2 + 3 , −1 + 2 − 3 2 2 , 3 = 1 1 −1 + 2 + 3 , 1 − 2 + 3 , 2 2 . 1 1 + 2 − 3 , − 1 + 2 + 3 2 1 2 the two orbits comprising the triakis tetrahedron are the vertices of two mirror images tetrahedra. Bangalore conference, 16-22 December, 2012 23 Catalan Solid Possessing Octahedral Symmetry There are 5Catalan solids that possesses octahedral symmetry. Example:Rhombic dodecahedron (dual of cuboctahedron) The cuboctahedron was obtained as the orbit 3 (010). A triangle will be generated as 1 , 2 (01) with a center represented by 3 Asquare is obtained as 2 , 3 (10) with a center 1 . For the orthogonality condition, the scale factor multiplying 3 can be calculated to be = 1 . 2 2 1 = = . 3 . 2 2 2 2 The vertices of the dual are given as the union of the following two orbits 1 = ±1 , ±2 , ±3 , 3 = 1 2 ±1 ± 2 ± 3 . The first orbit contains vertices of an octahedron and the second orbit contains vertices of a cube. The two orbits lie on two concentric spheres. Since 4 faces of the cuboctahedron meet at one vertex, then the dual’s face will be of 4 vertices (that is a rhombus). Bangalore conference, 16-22 December, 2012 24 Chiral Archimedean and Catalan solids Bangalore conference, 16-22 December, 2012 25 Chirality Objects or molecules which cannot be superimposed with their mirror image are called chiral. Human hands are one of the example of chirality. Achiral (not chiral) objects are objects that are identical to their mirror image. In three dimensional Euclidean space the chirality is defined as follows: The object which can not be transformed to its mirror image by proper rotations and translations is called a chiral object. Chirality is a very interesting topic in i) molecular chemistry A number of molecules display one type of chirality; they are either left-oriented or right-oriented molecules. ii) In fundamental physics chirality plays very important role. For example: A massless Dirac particle has to be either in the left handed state or in the right handed state. The weak interactions which is described by the standard model of high energy physics is invariant under one type of chiral transformations Bangalore conference, 16-22 December, 2012 26 Snub Cube T he proper rotational subgroup is W ( B 3 ) C 2 S 4 (S ym m etric group of order 4!= 24) T hey can be generated by the generators a r1 r2 and b r2 r3 , a b 3 4 ( ab ) 1. 2 Let a1 1 a 2 2 a 3 3 ( a1a 2 a 3 ) be a vector in the d ual space. T he follow ing sets of vertices form an equ ilateral triangle and a sq uare respecti vely ( , r1 r2 , ( r1 r2 ) ), ( , r2 r3 , ( r2 r3 ) ), ( r2 r3 ) ) w ith the respective 2 2 3 r3 r2 r1 r3 square of edge len g t hs: 2( a1 + a1a 2 a 2 ) and 2( a 2 + 2 2 2 5 2 a 2 a 3 a 3 ). 2 W e have another vertex r1 r3 r3 r1 . T o have all edg e r1 r2 4 2 2 2 Factoring by a 2 a nd d efining x 2 x 3 x 2 a1 a2 2 r2 r1 and y 2 a3 one obtains y a2 (r3r2 ) 2 2 2 a 2 a 3 a 3 ) ( a1 a 3 ) 2 1 3 length equal the fol low ing equations m ust be satisfied ( a 1 + a 1a 2 a 2 ) ( a 2 + x 2 1 r2 r3 and the cubic equation 2 x 1 0 w hich has one real solu tio n x 1 .83 9 3. Bangalore conference, 16-22 December, 2012 27 Snub Cube The first vertex and its mirror image r can be derived from the vector a ( x + y ) and II r1 I a 2 ( x (1 - 1 ) + 2 y 3 ) II I I 2 1 2 1 I 3 can be written in terms of quaternionic units as a 2 ( x 1) I 2 a 2 ( x 1) 2 2 1 ( xe1 +e 2 x e3 ) II 2 deleting the overall scale factor the orbits can easily be determined as 1 1 ( e1 xe 2 x e 3 ) W ( B 3 ) / C 2 ( I ) and W ( B 3 ) / C 2 ( II ) 1 1 O ( I ) {( xe1 e 2 x e 3 ), ( xe 2 e 3 x e1 ), ( xe 3 e 1 x e 2 )} 1 1 1 O ( II ) {( e1 x e 2 x e3 ), ( e 2 x e 3 x e1 ), ( e 3 x e 1 x e 2 )} The snub cubes represented by these sets of vertices are shown Bangalore conference, 16-22 December, 2012 28 Dual solid of the snub cube We can determine the centers of the faces in figure below: The faces 1 and 3 are represented by the vectors and up to some scale factors. is invariant under the rotation represented by r1r2 . In other words the triangle 3 is rotated to itself by a rotation around the vector . 1 3 3 3 The vectors representing the centers of the faces 2, 4 and 5 can be determined by averaging the vertices representing these faces and they lie in the same orbit under the proper octahedral group. r3 r2 r1 r3 The vector representing the center of the face 2 is c 2 (2 x 1) e1 +e 2 x e3 2 r1 r2 The scale factors multiplying the vectors and can be determined as 2 plane containing these five points . 4 2 2 when 1 3 r2 r1 x (r3r2 ) 2 2 1 , 3 and c 2 x 5 I r2 r3 represents the normal of the Bangalore conference, 16-22 December, 2012 29 Dual solid of the snub cube Then 38 vertices of the dual solid of the snub cube, the pentagonal icositetrahedron, are given in three orbits x O ( 1 ) 2 O ( 3 ) 1 O ( c 2 ) x 2 { e1 , e 2 , e 3 } ( e1 e 2 e 3 ) 2 {[ (2 x 1) e1 e 2 x e 3 ],[ (2 x 1) e 2 e 3 x e1 ],[ (2 x 1) e 3 e1 x e 2 ]} 2 2 2 Bangalore conference, 16-22 December, 2012 2 30 Snub Dodecahedron The proper rotational subgroup W(H3)/C2 is the simple finite subgroup of order 60. They can be generated by the generators a r1 r2 and b r2 r3 , a 5 b 3 ( ab ) 2 1 Let ( a1 a 2 a 3 ) be a general vector in the dual basis. The following sets of vertices form a pentagon and an equilateral triangle r2 r1 ( r1 r2 ) r2 r3 3 ( , r1 r2 , ( r1 r2 ) , ( r1 r2 ) , ( r1 r2 ) ), 2 3 ( , r2 r3 , ( r2 r3 ) ), 4 2 with the respective square of edge lengths: 2( a1 a1 a 2 a 2 ) and 2( a 2 a 2 a 3 a 3 ) 2 2 2 2 2 1 3 4 ( r1 r2 ) 2 rr r3 r2 5 r1 r3 1 2 We have another vertex : r1 r3 r3 r1 Let all edge lengths be the same. The following equation is satisfied ( a1 2 a1 a 2 a 2 2 ) ( a 2 2 a 2 a 3 a 3 2 ) ( a1 2 a 3 2 ) The equation x x x 0 3 2 has the real solution Bangalore conference, 16-22 December, 2012 x 1 .9 4 3 1 5 31 Snub Dodecahedron The first orbit and its mirror image can be obtained from the vectors expressed in terms of quaternionic units as I a2 2 [ ( x 1) e1 x e 2 (1 x ) e3 ] 2 3 II a2 2 [ ( x 1) e1 x e 2 (1 x ) e3 ] 2 3 The snub dodecahedrons represented by the orbits of these vectors W ( H 3 ) / C 2 ( I ) and W ( H 3 ) / C 2 ( II ) are shown: Bangalore conference, 16-22 December, 2012 32 Dual solid of the snub dodecahedron The vertices of the dual solid of the snub dodecahedron represented byW ( H can be given as the union of three orbits of the group W(H3)/C2. The first orbit O ( ) consists of 20 vertices of a dodecahedron. The second orbit consists of 12 vertices of an icosahedron O ( 3 ) where 3 ) / C 2 ( I ) 1 r2 r1 3 x x 2 2 ( r1 r2 ) 2 x (2 ) x 1 2 3 The third orbit O ( c 2 ) involves the vertices including the centers of the faces 2, 4 and 5 where the vector c 2 is given by c2 1 3 x x 2 2 1 5 4 ( r1 r2 ) 2 r3 r2 r1 r2 r1 r3 3 {[(2 1) x x 1]e1 ( x 3 x 3) e 2 ( x ) e 3 } 2 2 (21 20) x (21 17 ) x 21 11 2 r2 r3 3 Bangalore conference, 16-22 December, 2012 2 3 3 33 Dual solid of the snub dodecahedron Applying the group A [ I , I ] on the vector c 2 one generates an orbit of size 60. The 92 vertices consisting of these three orbits constitute dual solid of the snub dodecahedron, pentagonal hexecontahedron. It is one of the face transitive Catalan solid which has 92 vertices, 180 edges and 60 faces. 5 Bangalore conference, 16-22 December, 2012 34 Summary In this work a systematic construction of all Platonic, Archimedean and Catalan solids and chiral polyhedra, the snub cube, snub dodecahedron and their duals have been presented. The Coxeter diagrams A3, B3 and H3 were used to represent the symmetries of the polyhedra. A number of programs were developed to generate Coxeter group elements in terms of quaternions, quaternionic vertices of polyhedra and to plot the polyhedra. Bangalore conference, 16-22 December, 2012 35