Branching of the WH(4 ) and Their Dual Polytopes

under the Coxeter Groups AW 4 )( and WH(3 )Represented by

Mehmet Koca 1, Nazife Ozdes Koca 2, and Mudhahir Al-Ajmi 3

1, 2, 3 Department of Physics, College of Science, Sultan Qaboos University P. O. Box 36, Al-Khoud 123, Muscat, Sultanate of Oman

E-mail: [email protected], 2 [email protected], [email protected]

Abstract. 4-dimensional H 4 polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl WH(4 )where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary WH(4 )orbit into three dimensions is made preserving the icosahedral WH(3 )and the tetrahedral subgroup (AW 3 ) , the latter follows a branching under the Coxeter groupW (A4 ) . The dual polytopes of the semi-regular and quasi-regular H 4 polytopes have been constructed.

1 1. Introduction

It seems that there exists an experimental evidence for the realization of the Coxeter- WE()8 . Radu Coldea from the University of Oxford and his colleagues, from Oxford, Bristol University, the ISIS Rutherford Laboratory and the Helmholtz Zentrum Berlin [1] performed a neutron scattering experiment on CoNb26 O (cobalt niobate) which describes the one dimensional quantum Ising chain. They have determined masses of the five emerging particles, first two are obeying the relation mm21 . Their results could be attributed to the Zamolodchikov model [2] which describes the one-dimensional Ising model at critical temperature perturbed by an external magnetic field leading to eight spinless bosons with the mass relations 74 m, mm 2 cos , mm 2 cos , mm  2 cos 131425230 30 30

mmmm2163 , = , mm 74 =  , mm 85=  (1)

1+ 5 where  = is the . Zamolodchikov model can be described by an affine 2

Toda Field Theory of E 8 [3]. The masses can be related to the radii of the Gosset's of

E 8 [4]. In the derivation of the mass relations in (1) the maximal subgroup WH(4 )[5] of the

Coxeter-Weyl group WE(8 ) plays a crucial role. This is evident from the relation (1) where four of the masses are the  times the other four masses. This motivates us to study the structure of the group WH(4 ) in terms of quaternions and its 4D polytopes.

The rank-4 WH(4 ) is unique in the sense that it has no correspondence in higher dimensions and describes the quasi crystallography in 4-dimensions. It is the extension of the icosahedral group WH()35A C2 Ih to 4-dimensions. The group WH(4 ) has four 4-dimensional irreducible representations. One of its 4-dimensional irreducible representations can be described by the transformations on the quaternions by performing a left-right multiplication of the elements I which constitute one of the finite of quaternions. The Coxeter group WH(4 ) has five maximal subgroups [6] described by the groups

WD()4 Aut( A442312222 ) W ( A ): C , W ( H A ), Aut ( H H ), Aut ( A A ), ( ): S3. (2) C 2 We use the notation (:) for the semi direct product of two groups. Importance of these groups will be discussed in the next section.

It is perhaps interesting to note that the Coxeter-Weyl group WE (8 ) seems to show itself as an affine Toda field theory in the zero temperature (the coldest regime) and perhaps as a Lie group in the form of EE8 8 describing the heterotic superstring theory at very hot regime

(Planck scale). The Coxeter-Weyl group WE ()8 includes the crystallographic (tetrahedral and

2 octahedral symmetries in 3D and 4D) as well as the quasi crystallographic symmetries ( in 3D and its generalization to 4D). The icosahedral group of rank-3 describes fully the structures of the fullerenes such as the C 60 molecule which is represented by a truncated , the icosahedral quasicrystals and the viral structures displaying the icosahedral symmetry. A short historical account of the development of the theory of polytopes could be in . The Platonic solids, , , , icosahedron and have been discovered by the people lived in Scotland nearly 1000 years earlier than the ancient Greeks. Their models curved on the stones are now kept in the Ashmolean Museum at Oxford [7]. They were used in old Greek as models to describe fundamental matter associating tetrahedron with fire, cube with earth, air with octahedron, and water with icosahedron and dodecahedron with the universe or ether. Archimedes discovered the semi-regular convex solids now known as Archimedean solids. Kepler completed the classifications of the regular polyhedra in 1620 by inventing prisms and anti-prisms as well as four regular non-convex polyhedra. In 1865 the Belgian mathematician Catalan constructed the dual solids of the Archimedean solids, now known as Catalan solids [8]. Extensions of the platonic solids to 4D dimensions have been made in 1855 by L. Schlaffli [9] and their generalizations to higher dimensions in 1900 by T. Gosset [10]. Further important contributions are made by W. A. Wythoff [11] among many others and, in particular, by the contemporary mathematicians H.S.M. Coxeter [12] and J.H. Conway [13]. In what follows we study the branching of the regular, semi regular and quasi regular 4D polytopes described as the orbits of the Coxeter group WH(4 ). Constructions of the defining representation of the Coxeter group WH(4 ) and its maximal subgroups in terms of quaternions provide an appropriate technique to pursue. For this reason we introduce the finite subgroups of quaternions in Section 2 and construct the group elements of the groups

WH(44 ), WA( ) and WH(3 ) in terms of quaternions. The Section 3 deals with the decomposition of the WH(4 ) polytopes under the subgroup WH (3 ). A similar analysis is carried in Section 4 for the decomposition under the subgroup WA ()4 . Noting that the tetrahedral group WA()3 is a subgroup of the group WA()4 we give a few examples of the projections of the WH()4 polytopes into 3D preserving the . Dual polytopes of the 4D semi have not been studied in mathematical literature. The section 5 deals with the construction of the dual polytopes of the regular and semi regular polytopes of the Coxeter group WH ()4 . Some concluding remarks are made in Section 6.

2. Finite subgroups of quaternions

This section deals with the quaternions and its relevance to the orthogonal transformations in 4- dimensions and gives the list of finite subgroups of quaternions.

2.1. Quaternions and O (4) Transformations

3 qq qei, ( 1,2,3) Let 0 ii  , be a real unit with its conjugate defined by qqqeand the norm qq qq 1. The imaginary units satisfy 0 ii

eei j ij  ijk e k , (,ijk, 1,2,3). (3) Let p, q be unit quaternions and r represents an arbitrary quaternion. Then the transformations [14] rprqpq :[ , ]; rprqpq :[ , ] (4) define the O(4) which preserves the norm rr rr . We shall use the abstract  notations[,]p qpq and [,] respectively for the transformations in (4). The first term in (3) represents a proper rotation and the second includes also the , generally called rotary reflection. In particular, the group element rprppp:[ ,  ] (5) represents the reflection with respect to the orthogonal to the unit quaternion p.

The orthogonal transformations in 3D can be simply written as rprppp:[,  ]where r is an imaginary quaternion.

2.2. Finite Subgroups of Quaternions

The finite subgroups of quaternions are well known and its classification can be found in the references [15]. They are given as follows.

(a) of order n with n an odd number.  (b) Cyclic group of order 2n is generated by  peexp( )and of order 4n can 1 n  be generated by the generators peexp( ),e . 12n (c) The can be represented by the set of 24 unit quaternions: 1 T  {,1 e ,  e ,  e , (1 e  e  e )}. (6) 1 2 3 2 1 2 3 The set T represents also the vertices of the 24-cell. (d) The consists of 48 unit quaternions. Let the set

Teeeeeeee {(1),(11 ),(1),( 1 1 ),(1),( 11e )} (7) 2212323131 2 2 222

represents the vertices of the 24-cell rotated with respect to the set T [16]. . The union of the set  TTO  represents the binary octahedral group of order 48.

4 (e) The binary icosahedral group I  bc, of order 120 can be generated by two unit 1 1 quaternions, say, be( e) 12(1) and ce()12e (1)belonging to the 2 12  2 12  same conjugacy classes. We display its elements in Table 1 as the sets of conjugacy classes which represent a number of icosahedra, dodecahedra and one icosidodecahedron.

Table 1. Conjugacy classes of the binary icosahedral group I represented as the orbits of

the group WA()3 .

Conjugacy classes denoted by the number of elements and order of elements 1 1 2 -1 10 111 12 : (ee ), (  ee ), (  ee )  22213 21 32 5 111 12 : (ee  ), (   ee  ), (   ee  )  22213 21 32 10 111 12 : ( ee ), (  ee ), (  ee )  22212 23 31 5 111 12 : ( ee  ), (  ee  ), (  ee  )  22212 23 31 6 1111 20:(1eee ),(1  e  e ),(1  e  e ),(1  e  e )  2222123 1 2 2 3 3 1 3 1111 20:(1eeeeeeee   ),(1   ),(1   ),(1  e )  2222123 1 2 2 3 3 1 4 111 30 :eee , , , (  e  ee  ), (   e  ee  ), (   e  ee  ) 123222 1 23 2 31 3 12

15 1 5 In Table 1 the golden ratio   and   satisfy the relations 1,   1, 2 2 2 1, 2  1 . Note that the 30 quaternions in the last line of Table 1 consist of imaginary quaternions and constitute the vertices of an icosidodecahedron.

The set of quaternions representing the binary icosahedral group I can also be written as the union of two sets,   STI }{ (8) where the set S is represented by the quaternions

5 1 1 1 S { ( τ e σe ), ( τ e σe ), ( τ e  σe ), 2 1 3 2 2 1 2 3 2 1 1 1 ( σ e τe ), ( σ e τe ), ( σ e  τe ), 2 1 2 2 2 3 2 3 1 1 1 1 τe1( σe ), τe1( σe ), τe1(  σe ), 2 1 2 2 2 3 2 3 1 1 1 1 σe( τe e ), σe( τe e ), σe( τe  e )}. 2 1 2 3 2 2 3 1 2 3 1 2 (9)

It is interesting to note that the set of quaternions I represents the vertices of 600-cell [17] and the set S represents the vertices of the snub 24-cell [18].

The Coxeter diagram H 4 is illustrated in Figure1 where the simple roots are given by the quaternions 22  2,eeeee   (   ),  2,   (  e  e ) (10) 11212332422 23

5

Figure 1. The Coxeter diagram of H 4 . .

Let i (i  1,2,3,4) be the basis vectors of the dual space defined by (,ij ) ij.The

Cartan matrix with the matrix elements ()C ij ( i , j )and its inverse with the matrix 1 elements ()(,C ij  i j ) are given by the respective matrices

200 432   210 36 4 2 CC, (14 )  . (11) HH440121 242(2)2   2  00 12 2 2 2

Note that the relations  (CC ) and (1 ) lead to the quaternionic iHijjiHij44j representations of the dual vectors 1 (42 ee  ),  2(3  e ), 112 32 3 (12) 1 ((  2) ee  ),  2. 322 34

6 Following equation (5) we define the generators of the group WH()4 rrrr 1234 ,,,  1 by r [,  ]* which leads to the quaternionic representation of the Coxeter group ii2 i  W()[,][,] H4  pq pq of order 14,400 where p,qI . We can use a symbolic notation  W()[,][,] H4  II II for the designation of the group. Now the subgroup WH(3 ) can be  represented by the set W(H3 ) [ pp , ] [ pp , ] with pp ,  I. It is obvious that the group can  be represented by the set W()[,][,]H3 II II  A5 C2 where A5 is the group of even permutations of five letters with an order of 60 and the C 2 stands for the group generated by inversion. An extended H 4 diagram allows us to represent an extended root represented by the quaternion0  2 . When the root4 is deleted the remaining diagram represents the group structure WH()31 A rrr12,,3r 0where the generator r0 can be represented by reflection r0 [1, 1] with respect to the hyperplane orthogonal to the unit quaternion 1. Then the representation of the maximal subgroup of order 240 is given by

 WHWA()31 ()[,][,]ACC II II 522 . (13)

WH()4 Since the index  60 then the maximal subgroup WH()3 WA (1)can be WH()31 WA () embedded in the Coxeter group WH(4 ) in 60 different ways which are all conjugate to each other. The group in (13) leaves the unit quaternion 1invariant. The other conjugate groups can be represented by the set

q  { WH(31 ) WA ( )} [ I , qIq ] [ I , qIq ] (14) which leaves q invariant where qIis an arbitrary element of the binary icosahedral group I.

The subgroup WA()4 can be generated by the reflections on the simple roots of the Weyl- Coxeter diagram shown in Fig. 2.

r1 r2 r3 r4

Figure 2. The Coxeter diagram of A4 . One possible choice of the simple roots can be taken from the set I as follows 22 1212,  (1 eee 2 331 ),   2 e , 41  ( e  e2  e3 ). 22 (15)

7 The Coxeter-Weyl group WA()4  rrrr 1234 ,,, then takes the form [6]  W(){[,][,]} A45 pcpc pcpc S AC5:2 where pI with p  p(  ) is an element of ~ the binary icosahedral group I obtained from I by interchanging  and and 1 23 )(  Teec  is a fixed quaternion. The symmetry 2

 :,1423can be represented by the quaternions 11 [ab , ] with a ( e e e) and b ( e e e) leading to the , 2212 3 1 23 namely, Aut() A W (): A C {[, I c I c][, I c I c]}(:* A C ): C . (16) 442 522

The group Aut() A4 is of the order 240 and can be embedded in the Coxeter group WH()4 in 60 different ways one for each qIpairs. Before we conclude this section we also mention that another 4-dimensional irreducible representation of the group WH(4 )can be described as   WH()[4  I,][,]III . Then the above analysis can be carried out with the interchange of two irreducible representations of the binary icosahedral group I  I . The other maximal subgroups of the Coxeter group WH(4 ) are also of special interest. The groups

WD()4 Aut( H22 H ) and ( ):S3are the symmetry groups of the Grand and the C 2 snub 24-cell respectively. The group Aut() A22 A[ W ()AWA2 ()]:2C 4 has some applications in particle physics where the group WA()2WA (2)is the skeleton of the chiral

Lie groupSU (3)LRSU (3) .

WH() WH() 3. The 4 polytopes and branching under the group 3

An arbitrary ‘highest’ weight vector  describing an irreducible representation of a rank-4 Lie group is given in the Dynkin basis [19] as   )(       aaaaaaaa 443322114321 where

i , (  ,,,i )4321 are the vectors of the dual basis satisfying the relation (,ij ) ij . For the

‘highest’ weight vector the Dynkin indices are non-negative integers ai  0 . Although the

Coxeter group WH(4 ) is not associated with any Lie group one can still use the same technique to determine the orbits of a given vector. Let then  aaaa )( a     aa   a  be a vector in the 4-dimensions with a  0any 4321 11 43322 4 i non-negative real number, not necessarily restricted to integers only.

We define an orbit of the group WH(4 ) by the group actionOWH()  (4 ).The size of the orbit OWH() (4 )  is 14,400 with ai  0 ,i  4,3,2,1 which is equal to the order of the group WH(). One can count the number of cells of the polytope OW() (H)by using 4 4

8 its rank-3 subgroups WH()3 , WA(352 ), D C and D 3 C2.The last two groups are the WH() symmetries of the pentagonal and triangular prisms respectively. In principle the 4 polytopes can be projected in 3-dimensions by preserving any one of these rank-3 subgroups.

We will give examples only for the case of the groups WH()3 and WA().3 A right-coset decomposition of the group W(H4) under its subgroup W(H3) can be given by 120 WH() WH(). gi (17) 43 i 1 120 i 0 Then the orbit OWH() ()4 decomposes as O()  WH( 3 )g. Here g  [1,1] is i1 the unit element. Let us define the vectors ()iigi , 1,2,...,120. It is clear from (14) that the set of coset representatives g i can be chosen from the set of group elements [,1].I

Indeed the set [,I 1]form an invariant subgroup of the group WH()4 . Thus the set of vectors ()ii g  are the set of 120 quaternionic vectors I  . Applying the group WH(3 )on these 120 vectors one generates 120 different orbits each consisting of 120 vectors in general.

The group WH()3 acts on an arbitrary vector

()bbb123 b 11 b 22b33where ( ij , ) ij with ij , 1,2,3 and the Cartan 1 matrix ()(,)C ij  ij and its inverse ()(,)C ij   ij of the group WH()3 are given by

20 32233 131 22 C   21, ()C  2 42 . (18) H3 H3 2  01 2 3222  

We have to express the WH(3 )orbits in the basis of the vectors i using the relations given in (19) 3  2  , 2,   . (19) 1122 422 433 4 Any vector ()ig i is now transformed to the form (,bb,bb, ) b  b b b. 1234 11 22 33 44 First three components of these do not in general satisfy the relation bbb0, 0, 0. However there is always a vector in the orbit WH()()i where the 123 3 ()i relations bb120, 0,b30 are satisfied. Each orbit WH()3  defines a set of discrete 2 points on a sphere S which is the intersection of the hyperplane orthogonal to the vectorb44 3 and the sphere S defined by the orbit WH()4  . Then the branching of an arbitrary orbit

WH()4  can be given as

9 1 (aaaa , , , ) ( aaa , , ){ (32 a 2 a ( 2) a 2 a )} 1234 1232 1 2 3 4 1 (,,){(aaaaaaaa     aaaa  )} 1234231 22 1 234 (20) (116 more vectors) From now on we simply use the vector notation  rather than W(G) for the designation of the orbit. The decomposition in (20) is obtained by a computer calculation. We will only list the decomposition of those polytopes where aa12 ,,aa34,take certain integer values. Before we proceed further we wish to classify the solids possessing the icosahedral symmetry WH()3 . The orbits obtained from the vectors (1,0,0), (0,1,0), (0,0,1) represent respectively dodecahedron, icosidodecahedron and icosahedron [20]. The orbits obtained from the vectors

(bb12 , ,0) and (0, bb23 , ) represent respectively a quasi regular with edges of lengths b1and b2 and a quasi regular truncated icosahedron with edge lengths bb23 and . If bb12 in the first vector and b2b3in the second vector then their orbits represent respectively the regular truncated dodecahedron and the regular truncated icosahedron. Finally the orbits of the vectors (,bb130,) and (,bbb 123 , )respectively represent a quasi regular small rhombicosidodecahedron with edge lengths b1and b3 and a quasi regular great rhombicosidodecahedron with three edge lengths bb12, and b 3.When the components of the vectors are equal we obtain the Archimedean solids including the small and great rhombicosidodecahedra.

Now we display the decompositions of some of the 4D solids under the WH(3 ) symmetry.

3.1. The 600-cell

The orbit WH(4 )(0,0,0,1)represents the 600-cell with 120 vertices. Each cell is a tetrahedron, 20 of which are meeting at one vertex. The orbit can be decomposed under the subgroup

WH (3 )as follows:   1 (0,0,0,1) (0,0,0)(  1) (0,0,1)( ) (0,0, )( ) (1,0,0)( ) (0,1,0)(0). (21) 222 From now on we assume that the orbit is represented by a vector where the components are taken to be positive numbers.

Actually the orbit WH(4 )(0,0,0,1)can be represented by the set of quaternions I when the  vector   2 is multiplied by , then, in turn, the orbits in (21) represent the conjugacy 4 2 classes of the group I given in Table 1. The orbits (0,0,0)( 1) represent two poles of the sphere S 3 , the other orbits respectively represent icosahedra, dodecahedra and an icosidodecahedron.

10

3.2. The 120-cell

It is the dual of the 600-cell and its cells are the dodecahedra, 4 of which are meeting at one point. It is represented by the orbit WH(4 )(1,0,0,0)which we have studied elsewhere in the context of quaternions [17] and [21]. It is decomposed as follows:  3  2 2 (1,0,0,0) (1,0,0)( ) ( ,0,0)( ) (2 ,0,0)( ) (0, ,0)( ) (0,1, )( ) 22 2 2(22)  1  (0, ,1)( ) ( ,0, )( ) (1,0,2 )(0). 22  All vectors can be converted to unit quaternions by multiplying each vector with then the 2 4 600 vertices are represented by the set of quaternions JpT i  pj where p 5 1. ij,0 3.3. The 720- cell

It is described by the orbit WH(4 )(0,1,0,0)which has 1,200 vertices. The polytope consists of two types of cells: tetrahedra (600) and icosidodecahedra (120). Its decomposition under the group WH(3 )is given by 323  (0,1,0,0) ( ,0,0)(  ) (0,1,0)(22 ) (0, ,0)( 1) (1,10)( ) (0, ,1)( ) 222  2 1   (0, , )( ) (1,0, 22 )(  ) (  ,0,1)( ) (  ,0,  2 )( ) (1,1,  )( ) 2222 (23)  (2 ,0,0)(0) (0,1,2 )(0).

 3.4. The 720 cell

It is described by the orbit WH(4 )(0,0,1,0)which has 720 vertices. It consists of 600 octahedra and 120 icosahedra. It is the unique 4D Archimedean polytope with WH(4 ) symmetry in the sense that its cells are made of Platonic solids. It decomposes under the group WH(3 ) as follows  2  (0,0,1,0) (0,0,1)( ) (0,0,2 )( ) (0,1,0)( ) (0,0,2 )(0) 22 2 1  (0,1, )( ) (1,0,1)( ) (1,0, )( ) (1,1,0)( ) (24) 2222  ( ,0,1)(0) (0, ,0)( 1). One can easily interpret the type of projected polyhedra by consulting the above discussions.

11 WH() WA() 4. The 4 polytopes and branching under the group 4

The right-coset decomposition of the group WH(4 )under the subgroup WA()4 can be given by 120 WH() WA() gi 4 4 (25) i1 where the right coset elements form the same subgroup gIi [,1]. The vectors to which ()ii WA()4 will act can be taken as gi , 1,2,...,120.

The group WA()4 acts on an arbitrary vector

(,bbbb1 2 , 3 , 4 ) b 11 b 22b33 b 44where ( ij , ) ij with ij , 1,2,3,4 and the Cartan 1 matrix ()(,)C ij  ij and its inverse ()C ij  (,) ij of the group WA()4 are given by

2100 4321  12 10 1 3642 C  , ()C 1  . (26) A4 0121 A4 5 2463  00 12 1234

The dual vectors  i can be written in terms of quaternions as follows 11  ( 5 ee ),  (2 e  2 e ), 1210322 10 3

1122 2 31 ( 5 ee 2 e3 ), 42  (2 ee  2)3c . 10 10 5 (27)

We have to express the WA()4 orbits in the basis of the vectors i . It is easy to obtain the relation iij D  jwhere the matrix D is given by  4221  2(31)0 32 D  . (28) (3 1)3 0 1   20 0 Any vector ()iig  is now in the form (,bbbb , , ) b  b b b . 1234 11 22 33 44 The components of these vectors do not in general satisfy the relation ()i bbbb12340, 0, 0, 0 . Nevertheless there is always a vector in the orbit WA()4  where the relations bb120, 0,bb340, 0 are satisfied. Then the branching of an arbitrary orbit WH() can be given as 4

12 43 (,,,){aaaa1 2 3 4 a 21 a 32 a 43 [(3  1) a 1  a 2  a 3  a 4]}{;}. 4  1 4 2 3 

 118 more vectors like this decomposition . (29) In the above decomposition all the vectors come in pairs as shown in (29). Now we display the decompositions of some of those 4D polytopes with WH(4 ) symmetry under the subgroup WA(). 4

4.1. The 600-cell (0,0,0,1) (0,0,1, ) ( ,1,0,0) ( ,0,0, ) (1,0, ,0) (0, ,0,1). (30) The first three are the polytopes with 20 vertices each and the last two polytopes have 30 vertices each. In particular the vertices of the polytope  (1,0,0,1)represent a scaled non-zero of the Coxeter –Weyl group WA()4 .

4.2. The 120-cell (1,0,0,0) (3 1,0,0,0) (0,0,0,3 1) (1,33 ,0,0) (0,0, ,1) ( ,0,0,2 22 ) (2 ,0,0, )  (0, 22 , ,0)  (1,  ,0,  3 ) (  3 ,0,  ,1) (  , 2 , ,0) (0,  , 2 , ) (  2 , ,1, ) (  ,1, , 2 ). (31)

4.3. The 720-cell (0,1,0,0) (0,3 1,0,0) (0,0,3 1,0) (44 ,0,0,1) (1,0,0,  ) (22 ,0,2 ,0) (0,2 ,0,22 ) 33222222 (,,1,0)(0,1,,)(1,2,,0)(0,,2,1)(,0,,2)(2,,0,)(,1,0,31)             22 22 3 3 2 2 (31,0,1,)(1,,,    )(,   ,,1)(,,1,)(,1,,)(,,,).         (32)

 4.4. The 720 cell (0, 01, 0) ( 33 , 0,1, 0) (0,1, 0, ) (0, 2 ,1, 0) (0,1, 2  , 0) ( ,22 , 0,1) (1, 0, , ) (2 , 0, , )  ( , ,0, 22 )  (  ,1,  ,0) (0,  ,1, 2 ) (2  ,1,0,  ) (  ,0,1,2  ) (1,  , ,1). (33) If we wish to decompose the WH(4 ) orbits under the subgroup WA()3 then we use the decomposition of the WA()orbits under the group WA()as shown below [22] 4 3 Oaaaa(1234 ,, , ) Oaaa ( 123 , , )(  ( a 1 2 a 2 3 a 3 4 a 4 ))  Oaa ( 12 , ,( a 3  a 4 ))(  ( a 1 2 a 2 3 a 3 a 4 ))

Oa (1234 ,( a  a ), a )(  ( a 1 2 a 2 2 a 34 a ))  O (( a 1234  a ), a, a )(  ( a 1 3 a 2 2 a 34 a )) Oa ( ,,)(43aa a a 2 a a ). 234 1 2 3 4 (34)

For example, the 600-cell with the 120 vertices decomposes under the group WA()3 as

13 (0,0,0,1) (1,0,0)(4 3) (0,0,1)( (4 3)) ( ,0,0)( 5  ) (0,0, )(5 ) (2 ,0,0)(2 )  (0,0,2 )( (2 )) (0, ,0)( (2 4)) (1,0,  )( (3 1)) ( ,0,1)(3 1)  (0, ,1)( 5) (1, ,0)( 5)  (,1,0)( ( 2))  (0,1,  )(  2)  ( ,0, )(0). (35) WH() 5. Dual polytopes of the uniform polytopes of the Coxeter-Weyl group 4

In the references [20] and [23] we have proved that all the Catalan solids can be derived from the Coxeter diagrams , , HBA 333 with a simple technique which we will also apply for the constructions of the dual polytopes of the 4D polytopes. To obtain the vertices of the dual polytope of a given polytope one determines the centers of the cells joined to the vertex(,aaaa1234 , , ). The relative magnitudes of these vectors are determined from the fact that the hyperplane formed by the vectors representing the centers of the cells are orthogonal to the vertex(,aaaa1234 , , ). Let us recall that the dual polytopes are cell transitive similar to the Catalan solids which are transitive. In what follows we study the duals of the

WH(4 )uniform polytopes in turn.

5.1. Dual polytope of the 600-cell OO(0001) ( )  4

The tetrahedral subgroup WA() generated by WA()  rrr ,,acting on the vector 3 3 234 (0,0,0,1)generates a tetrahedron whose can be represented by the vector as 1 the group WA()rrr ,, leaves the vector  invariant. The number of tetrahedra joining to 3 234 1 the vertex  is given by the formula 4 WH()number of vertices of the tedrahedron 120 120 4 4 20. (36)  WA ( ) number of vertices of the polytope 24 120 3 This formula has an interesting group theoretical interpretation. The orbit which includes the vector 1  (1,0,0,0) can be generated by the icosahedral group WH()312 rrr ,,3  and represents a dodecahedron. Note that the dihedral subgroup Dr32 ,r3leaves the vector1 20 i invariant. A left coset decomposition WH()3 gD3will generate the vertices of the i 1 i dodecahedron where gWHD ()33 . At each vertex of the 600-cell there exist 20 tetrahedra, ii1 each of which, is generated by a conjugate group gW( A3 )( g ) , i1,2,...,20. Similarly the left coset decomposition of the tetrahedral group can be written as 4 WA() fDi where the group Drr , leaves the vectors 3 3 323 i 1

1(1,0,0,0) and 4(0,0,0,1) invariant. It is those conjugate groups ii1 f WH(3 )( f ) (i1,2,3,4) generate four dodecahedra when applied on the vector 1

14 whose centers form four tetrahedra. The number of vertices of the polytope WH()41 is 600 which are equal to the number of left cossets (index) of the group WA()3 in the groupWH()4 .

The number of dodecahedral cells of the polytope WH()41 is 120 since it is equal to the number of conjugate groups of the group WH(3 )in the group WH().4 Therefore the 120-cell is the dual of the 600-cell and the vice versa.

5.2. Dual polytope of the 720-cell OO(0,1,0,0) ( )  2

This uniform polytope consists of the cells tetrahedra and the icosidodecahedra since the subgroup W (A3) = acting on the vector (0,1,0,0) generates a tetrahedron and the subgroup WH(32 )  WH ( 3 )(0,1,0,0) generates an icosidodecahedron with 30 vertices for the

WH()3 index of the groupCC22  rr 13,  leaving the vector 2 invariant is equal to  30 . CC22

Actually, the vector 2 is left invariant by the group of generators DC32 rrr13,,4 defining the symmetry of the of order 12. Therefore the number of the vertices of the WH() polytope is equal to the index 4 1200. Total number of cells are given by the sum of DC32 indices WH() WH() 44600 120. WA() WH( ) 33 (37)

The number of cells which share the vector 2 as a vertex is determined by the formula similar to the one in equation (36) which leads to 2 tetrahedra and 3 icosidodecahedra at the vertex2 . Then the centers of these polyhedra are given by the vectors up to a relative scale factor  :

The centers of the tetrahedra : 11, r 1

The centers of the icosidodecahedra : 4344, rr, r434r . (38)

These vertices define a hyperplane orthogonal to the vector2 . Therefore one can determine 2 the scale factor  from the equation ().  0leading to the value  . The set of 142 3 2 vertices of the dual polytope is the union of the orbits O(1,0,0,0)O (0,0,0,1).They define 3 3 two concentric S spheres with the fraction of radii RR411.061 where R1 and R 4represent the radii of the spheres determined by the set of vectors in the orbits respectively. The last three vectors in (38) define an equilateral whose vertices are permuted by 3 the generator rr34with ()rr34 1. The vector (111 r ) 1is orthogonal to the of the triangle. Therefore the solid is a dipyramid with triangular base with the edge

15 lengths 21.41 and 14  0.867 . Let us introduce new set of quaternionic unit

2 vectors p01102203, peppeppep,  , 30. Since the dipyramid lies in the hyperplane 2

2 orthogonal to the vector p0  the five vectors in (38), when expressed in terms of 2 quaternions in the basis p012, ppp, , 3, will have the same p0 components. If we disregard the

p0 components and an overall scale factor the vertices of the dipyramid are given by the set of vectors in 3-dimensional space as follows 11 1 1 (0,0,1), ( , , 1), ( ,  , 1), ( 1,22, 0), (1, ,0). (39) 22 3 3 Plots of this dipyramid are shown in Figure 3.

(a) (b) 2 Figure 3. (a) The dipyramid, a typical cell of the polytope OO(1,0,0,0) (0,0,0,1). 3 (b) Top view.

5.3. Dual polytope of the 720 cell O(0,0,1,0) O ( ) 3

This polytope has 720 vertices equal to the index of the group D  C25 of order 20 in the group W(H4). The cells of this polytope consist of two types of Platonic solids: icosahedral and octahedral cells.

The subgroup WH()312 rrr ,,3  acting on the vector 3 will generate the vertices of an icosahedron. Similarly the subgroup WA()3  rrr 234 ,,acting on the vector 3 describes an octahedron. Total number of cells is 720 with 120 icosahedral cells and 600 octahedral cells.

The number of cells sharing the vector 3 as a vertex consists of 2 icosahedra and 5 octahedra. The centers of the octahedral cells can be taken to be the 234 vectors11, rr21, (rr 121 ) , ( rr 12 ) 1, (rr 121 ) and the centers of the icosahedral cells are determined to be the vectors 4 and r44 up to a scale factor. Similar to the argument raised in Section 5.2 the scale factor of the last two vectors are determined from the relation

16 2 ().0 with   . These seven vectors describe a dipyramid with pentagonal 413 2  8 base with edge length 21.41 and the other edges of length   1.26 . The dual 41 5 polytope consist of 720 vertices represented by the union of two orbits 2 OO(1,0,0,0) (0,0,0,1). (40) 2  3 These two orbits define two concentric spheres S with the ratio of the radii RR411.023 . Expressing the vertices in the basis of unit quaternions

3 p01, peppeppep, 102203  , 30 and then deleting the common components of the 3

3 vector p0  , the vertices of the dipyramid are given by the set of vectors 3 555 (0,1, ), (0, 1, ), (  1,  , 22 ), ( ,1, ), ( 2 ,0,0), 222 (41) 55 (22 , 1, ), (  1,  , ). 22 Plots of these solid are shown in Figure 4.

(a) (b) 2 Figure 4. (a) The dipyramid, a typical cell of the polytopeOO(1,0,0,0) (0,0,0,1). 3 (b) Top view.

This polytope is the projection of the Voronoi cell of WE( 8 ) [24] with 19,440 vertices into 4D space with the residual symmetry WH()4 .

5.4. Dual polytope of the polytope OO(1, 0, 0,1) ( )  14

17 The generators 2 and rr 3 fixing the vector  14 form a of order 6 indicating that the polytope has 2400 vertices. There are four types of cells sharing the same vertex. The subgroup WH()312 rrr ,,3  acting on the vector   41 generates a dodecahedron as it leaves the vector4 invariant. The number of dodecahedral cells is then

120. The subgroup WA()3rrr 234 ,, generates a tetrahedron with a total of 600. The subgroup DC52rr 12,,r4generates a pentagonal prism. There are720 cells of this type.

Finally the subgroup DC32rrr 134,,  generates a cell with a total of 1200 triangular prisms. Therefore the total number of cells is 2640. The number of cells sharing the same vertex and the vectors representing their centers are determined using a formula similar to (36) as

Dodecahedral cell : 1 and its center represented by  4 Tetrahedral cell : 1 and its center represented by  1 Triangular prisms : 3 and their centers represented by  , (rr ) , ( rr )2  223223 2 (42) Pentagonal prisms : 3 and their centers represented by  , (rr ) , ( rr )2 323323 3

These vectors are defined up to some scale vectors. Now if we let ,(rr ) ,( rr )2 , and then the scale factors are determined from the 323323 314 , equations 44 4 ().0,().0 and ( ).0as , and   . 32 12 42  34 3 Above solutions are unique to determine the hyperplane orthogonal to the vector   . 14 The 2640 vertices of the dual polytope is defined as the union of the orbits

 OO (1,0,0,0) (0,1,0,0) O (0,0,1,0)  O (0,0,0,1). (43) These vertices lie on the concentric fourS 3 spheres with the ratio of the radii

RRRR2341: : : 2.45: 2.47 : 2.52 : 2.97. (44)

The dual polytope has 1200 cells each of which is a solid with 8 vertices. These 8 vertices can be expressed in the basis of unit quaternions defined by

14 p01102203, peppeppep, , 30. Excluding the common components in the 14 direction of p0 they can be determined in 3 dimensions as follows:

18 22 22 12 [  ,0,1],  [ 1,  , 2  ], rr 23 2 [  ,  2,  ], (rr 23 ) 2 [ 2  , 1,  ], 32 2 33 [ 1,1,   ], (rr2 ) 3   [  ,  (  2),0], rr 23 3   [ 2  ,  ,  ], 2 4  [,0,1].   (45)

The solid defined by these 8 vertices has a dihedral symmetry D 3 . Six faces consist of two types of kites, big and small. Three small kites meet at the top vertex and three big kites join to the bottom vertex. At the other six vertices either two small kites and one big or two big kites and one small kite get together. Plots of this solid are shown in Figure 5.

(a) (b) (c) Figure 5. (a) A typical cell of the dual polytope of the polytopeO (1,0,0,1) . (b) Big kite view. (c) Small kite view.

5.5. Dual polytope of the polytope OO(0,1,0,1) (   ) 24

This polytope has 3600 vertices and 1440 cells. The cells consist of 120 icosidodecahedra, 600 cuboctahedra and 720 pentagonal prisms.

The number of cells sharing the same vertex (0,1,0,1)  2 4 and the vectors representing their centers are determined as follows:

Icosidodecahedron : 1 and its center represented by  4 Cuboctahedra : 2 and their centers represented by , r 111 Pentagonal prisms : 2 and their centers represented by  , r  . 333 41378    The scale vectors can be determined as ,. 10 4 The vertices of the dual polytope can be represented by the union of the following orbits

 OO(1,0,0,0) (0,0,1,0)  O (0,0,0,1). (46) These vertices lie on the concentric threeS 3 spheres with the ratio of the radii

RRR314: : 1.6625:1.6631:1.7019. (47)

19 The 5 vertices can be expressed in the basis of unit quaternions defined

24 by p01, peppeppep10, 220 , 330. Excluding the common components in the 24 direction of p0 they can be determined as follows:

11 [ 5,  ,   ], rr 133  [ 5,   ,  ],   [  , 5,0], 34 [,  5,0],  [0,0,2]. (48) This cell is a dipyramid with a base of isosceles triangle possessing the symmetry

CC22 rr 13,. Plots of this solid are shown in Figure 6.

(a) (b) Figure 6. (a) A typical cell of the dual polytope of the polytopeO (0,1,0,1). (b) Top view.

5.6. Dual polytope of the polytope OO(0,0,1,1) (  ) 34

This polytope has 1440 vertices and 720 cells. The cells consist of 120 icosahedra and 600 truncated tetrahedra. The number of cells sharing the same vertex(0,0,1,1) 34 and the vectors representing their centers are determined as follows:

Icosahedra : 1 and its center represented by  4 Truncated tetrahedra : 5 and centers represented by  ,rr ,(),(),() rr234 rr rr . 1 12 1 12 1 12 1 12 1 21  9 The scale vector can be determined as   . 19 This is a pyramid with a pentagonal base and edge lengths 1.41 and 2.01. The vertices of the dual polytope can be represented as the union of the following orbits

OO(1,0,0,0)  (0,0,0,1). (49) These orbits define two concentric S 3 spheres with the ratio of the radii

RR14: 1.012. (50) The 6 vertices can be expressed in the basis of unit quaternions defined

34 by p01102203, peppeppep, , 30. Excluding the common components in the 34 direction of p0 they can be determined as follows:

20

22 3 2 11[ 5, 1, 5], rr211 [2 , , ],( rr2 )11  [ 3,1,1],( rr2 )1  [ ,2 ,  ], 422 (51) ()rr12 1 [,2,], 4 [0,,1].

It is a pyramid with a regular pentagonal base as shown in Figure 7.

(a) (b) Figure 7. (a) A typical cell of the dual polytope of the polytopeO (0,0,1,1). (b) Top view.

5.7. Dual polytope of the polytope OO(0,1,1,0) (23 )

This polytope has 3600 vertices and 720 cells. The cells consist of 120 truncated icosahedra and 600 truncated tetrahedra. The number of cells sharing the same vertex (0,1,1,0) 23 and the vectors representing their centers are determined as follows:

Truncated icosahedra : 2 and their centers represented by , r 444 Truncated tetrahedra : 2 and their centers represented by  ,.r  111 5 The scale factor can be determined as   .This is a solid with four vertices and with two 4  types of faces represented by two inequivalent isosceles possessing a symmetry

CC2214= rr , of order 4. The vertices of the dual polytope can be represented as the union of the following orbits OO(1,0,0,0)  (0,0,0,1). (52) These orbits define two concentric S 3 spheres with the ratio of the radii

RR14: 0.957. (53) The 4 vertices can be expressed in the basis of unit quaternions defined 23 by p01, peppeppep10, 220 , 330. Excluding the common components in the 23 direction of p0 they can be determined as follows:

 [ 22 ,  ,  1], rr  [  2,   2,   ],  [0,1, 3  ], [1,  (   2),  2  ]. 111444 (54)

21 A plot of this solid is shown in Figure 8.

Figure 8. A typical cell of the dual polytope of the polytopeO (0,1,1,0) .

5.8. Dual polytope of the polytope OO(1,0,1,0) (   ) 13

This polytope has 3600 vertices and 1920 cells. The cells consist of 120 small rhombicosidodecahedra, 600 octahedra and 1200 triangular prisms. The number of cells sharing the same vertex (1, 0,1, 0)  1 3 and the vectors representing their centers are determined as follows: Small rhombicosidodecahedra : 2 and centers represented by  , r  444 Octahedron : 1 and its center represented by  1 Triangular prisms : 2 and centers represented by ,.r 222 33 The scale factors can be determined as and .This is a dipyramid with an 42  43  isosceles triangular base left invariant under the symmetry CC2224 = rr , of order 4. The vertices of the dual polytope can be represented as the union of the following orbits

 OOO(1,0,0,0) (0,1,0,0) (0,0,0,1). (55) 3 These orbits define three concentric S spheres with the ratio of the radii RRR124: : 0.9991: 0.9495:1.0000. (56) The 5 vertices can be expressed in the basis of unit quaternions defined by

13 p01102203, peppeppep, , 30. Excluding the common component in the 13 direction of p0 they can be determined as follows: 2 23 12[ , ,  1],  [ ,  2,  ],r 224  [ 2 ,  ,1],   [1,1,  ], 2 (57) r44[, , 2].

Plots of this solid are shown in Figure 9.

22

(a) (b) Figure 9. (a) A typical cell of the dual polytope of the polytopeO (1,0,1,0) . (b) Another view.

5.9. Dual polytope of the polytope OO(1,1, 0, 0) (   ) 12

This polytope has 2400 vertices and 720 cells. The cells consist of 120 truncated dodecahedra and 600 tetrahedra. The number of cells sharing the same vertex (1,1, 0, 0) 12 and the vectors representing their centers are determined as follows: Truncated dodecahedra : 3 and their centers represented by  ,,()rr rr 2 4344 34 4 Tetrahedron : 1 and its center represented by  . 1   2 The scale factor can be determined as   . This cell is a pyramid with a triangular base 34  with two edge lengths 1.41 and 1.73. The cell is invariant under the dihedral symmetry

Drr334, . The vertices of the dual polytope can be represented as the union of the following orbits  OO(1,0,0,0) (0,0,0,1). (58) These orbits define two concentric S 3 spheres with the ratio of the radii

RR14: 0.935. (59) The 4 vertices can be expressed in the basis of unit quaternions defined by 12 p01102203, peppeppep, , 30. Excluding the common components in the 12 direction of p0 they can be determined as follows: [22 , 1,0], [1,0,3 1],rr [ 1,2 ,2 ],( rr )2 [2 ,3 , 2 ]. 143     4 43   4 4     (60) A plot of this solid is shown in Figure 10.

23

Figure 10. A typical cell of the dual polytope of the polytope O (1,1,0,0).

The next four polytopes have dual polytopes whose cells have only one reflection symmetry.

5.10. Dual polytope of the polytope OO(1,1,1,0) (  ) 123

This polytope has 7200 vertices and 1920 cells. The cells consist of 120 great rhombicosidodecahedra, 600 truncated tetrahedra and 1200 triangular prisms. The number of cells sharing the same vertex (1,1,1,0)  123 and the vectors representing their centers are determined as follows: Great rhombicosidodecahedra : 2 and their centers represented by  , r  444 : 1 and its center represented by  1 Triangular prism : 1 and its center represented by  . 2 55 The scale factors can be determined as and . This is a solid with 4 54 310 vertices possessing a reflection symmetry r4 . The faces of the solid consist of two unequal isosceles triangles and two equal scalene triangles. The vertices of the dual polytope can be represented as the union of the following orbits

 OO(1,0,0,0)  (0,1,0,0) (0,0,0,1). (61) These orbits define three concentric S 3 spheres with the ratio of the radii

RRR124: : 1.0032 : 0.9433:1.0000. (62) The 4 vertices can be expressed in the basis of unit quaternions defined by

123 p0110220, peppeppep, , 330. Excluding the common component in the 123 direction of p0 they can be determined as follows: [2,2 ,], [, 2,], [1,1,4 1],r [22 ,2 ,3]. 12444           (63) A plot of this solid is shown in Figure 11.

24

Figure 11. A typical cell of the dual polytope of the polytope O (1,1,1,0) .

5.11. Dual polytope of the polytope OO(1,1,0,1) (  ) 124

This polytope has 7200 vertices and 2640 cells. The cells consist of 120 truncated dodecahedra, 600 cuboctahedra, 720 decagonal prisms and 1200 triangular prisms. The number of cells sharing the same vertex (1,1,0,1)  124 and the vectors representing their centers are determined as follows: Truncated dodecahedron : 1 and its center represented by  4 Cuboctahedron : 1 and its center represented by  1 Decagonal prisms : 2 and their centers represented by  , r  333 Triangular prism : 1 and its center represented by  . 2 6  7 13 61 13 36 The scale factors can be determined as , and  . This is a 47929 solid with 5 vertices possessing a reflection symmetry r3. The vertices of the dual polytope can be represented as the union of the following orbits

 OO(1,0,0,0)  (0,1,0,0) (0,0,1,0) (0,0,0,1). (64) These orbits define four concentric S 3 spheres with the ratio of the radii

RRRR4123: : : 1.00 :1.33:1.88:1.90. (65) The 4 vertices can be expressed in the basis of unit quaternions defined by 124 p0110220, peppeppep, , 330. Excluding the common components 124

in the direction of p0 they can be determined as follows:

 2 124[2 3  , 2  ,  ],   [  ,  ,2],   [  ,0,   2], 2 (66)

333 [ 5,2 , ], r  [0,3  2,  1]. This is a non-regular pyramid based on a kite base with a r 3 symmetry. Plots of this solid are shown in Figure 12.

25

(a) (b) Figure 12. (a) A typical cell of the dual polytope of the polytope O (1,1,0,1) .(b) Bottom view.

5.12. Dual polytope of the polytope OO(1, 0,1,1) (  ) 134

This polytope has 7200 vertices and 2640 cells. The cells consist of 120 small rhombicosidodecahedra, 600 truncated tetrahedra, 720 pentagonal prisms and 1200 hexagonal prisms. The number of cells sharing the same vertex (1,0,1,1)  134 and the vectors representing their centers are determined as follows: Small rhombicosidodecahedra : 1 and its center represented by  4 Truncated tetrahedra : 1 and its center represented by  1 Pentagonal prism : 1 and its center represented by  3

Hexagonal prisms : 2 and the centers represented by 222,.r  33  36 6 27 33 51 The scale factors can be determined as , and  . This is a 31 19 71 pyramid with a kite base and a reflection symmetry r2 . The vertices of the dual polytope can be represented as the union of the following orbits

 OO(1,0,0,0) (0,1,0,0) (0,0,1,0) (0,0,0,1). (67) These orbits define four concentric S 3 spheres with the ratio of the radii

RRRR1234: : : 1.418:1.002 :1.000 :1.031. (68) The 5 vertices can be expressed in the basis of unit quaternions defined by 134 p0110220, peppeppep, , 330. Excluding the common component in the 134 direction of p0 they can be determined as follows: 22 134[5,1,5],  [,, ], [,,  ], (69) 222 [ 1,  5,3], r  [3, 5,1]. Plots of this solid are shown in Figure 13.

26

(a) (b) Figure 13. (a) A typical cell of the dual polytope of the polytope O (1,0,1,1) . (b) Bottom view.

5.13. Dual polytope of the polytope OO(0,1,1,1) (234 )

This polytope has 7200 vertices and 1440 cells. The cells consist of 120 truncated icosahedra, 600 truncated octahedra and 720 pentagonal prisms. The number of cells sharing the same vertex (0,1,1,1) 234 and the vectors representing their centers are determined as follows: Truncated icosahedron : 1 and its center represented by  4 Truncated octahedra : 2 and centers represented by  , r  111 Pentagonal prism : 1 and its center represented by  . 3 6  2 6(13  3) The scale factors can be determined as and . This solid consists of 5 121 two equal scalene triangles and two unequal isosceles triangles as faces possessing a reflection symmetry r2 . The vertices of the dual polytope can be represented as the union of the following orbits O (1,0,0,0)  (0,0,1,0)  (0,0,0,1). (70) These orbits define three concentric S 3 spheres with the ratio of the radii

RRR134: : 1.000 : 0.9906 :1.0233. (71) The 4 vertices can be expressed in the basis of unit quaternions defined by 234 p0110220, peppeppep, , 330. Excluding the common components in the 234 direction of p0 they can be determined as follows:

11[ (2  3),  ,  (  2)], r 13 [  3,  2,  (2  1)],  [1,  (  2),0],4  [0,1,3 ]. (72) A plot of this solid is shown in Figure 14.

27

Figure 14. A typical cell of the dual polytope of the polytope O (0,1,1,1) .

5.14. Dual polytope of the polytope OO(1,1,1,1) ( ) 1234

This polytope has 14,400 vertices equal to the order of the group WH(4 ) and 2640 cells. The cells consist of 120 great rhombicosidodecahedra, 600 truncated octahedra, 720 decagonal prisms and 1200 hexagonal prisms. The number of cells sharing the same vertex

(1,1,1,1) 1234 and the vectors representing their centers are determined as follows:

Great rhombicosidodecahedron : 1 and its center represented by  4 Truncated octahedron : 1 and its center represented by  1 Hexagonal prism : 1 and its center represented by  2 : 1 and its center represented by  . 3 31  20 17 30 17 89 The scale factors can be determined as , and   . This is 2 57 155 a solid with 4 vertices possessing no symmetry at all. All its faces are different scalene triangles. The vertices of the dual polytope can be represented as the union of the following orbits O (1,0,0,0)  (0,1,0,0) (0,0,1,0) (0,0,0,1). (73) These orbits define four concentric S 3 spheres with the ratio of the radii

RRRR1234: : : 0.9621: 0.9584 : 09632 :1.000. (74)

The 4 vertices can be expressed in the basis of unit quaternions defined by

1234 p0110220, peppeppep, , 330. Excluding the common component in 1234 the direction of p0 they can be determined as follows:  [ (5  2),1,  (3  1)],   [  1,3   1, 3(  1)], 12 (75) 34  [3  1, 2(2  1),  ],   [  ,  ,4  5  ]. A plot of this solid is shown in Figure 15.

28

Figure 15. A typical cell of the dual polytope of the polytope O (1,1,1,1) .

6. Conclusion

4D polytopes can be classified with respect to their symmetries represented by the Coxeter groups AW 4 )( , BW 4 )( , HW 4 )( and FW 4 )( ; the regular and semi regular polytopes are all well known in the mathematical literature. However their constructions in terms of quaternions are recently gaining some interest. Moreover their duals have not been constructed to date. In this paper we have first constructed the group WH(4 )in terms of quaternions and proved that its two 4-dimensional irreducible representations can be constructed with the use of two irreducible representations of the binary icosahedral group represented by the sets of ~ quaternions I and I I(   ) . We have constructed semi regular polytopes in terms of quaternions and projected them to 3D by decomposing the WH(4 )orbits under the icosahedral group WH(). Explicit decomposition of an arbitrary polytope of the group WH()under the 3 4 group AW 4 )( has been made. This decomposition allows the projection of the WH()4 polytopes into 3D by preserving the tetrahedral symmetry WA(3 ). The vertices of the dual polytopes of the semi regular WH()4 polytopes have been constructed and their cells have been determined.

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