COXETER GROUPS, QUATERNIONS, SYMMETRIES OF POLYHEDRA AND 4D POLYTOPES

August 17, 2012 11:7 8566: Mathematical Physics — Proceedings Trim size: 9in x 6in

v

ORGANIZING COMMITTEES

INTERNATIONAL ADVISORY COMMITTEE

Riazuddin – National Center for Physics, PakistanA. Qadir – National University of Science & Technology,

PakistanF. Ardalan – Sharif University of Technology & IPM, IranH. Arfaei – Sharif University of Technology & IPM, IranA. Chamssedine – CAMS & UAB, LebanonG. Jorjadze – Razmadze Mathematical Institute, GeorgiaG. Pogosyan – Yerevan State University & JINR, ArmeniaM. Koca – Sultan Qaboos University, OmanA. Mostafazadeh – Koc University, TurkeyI.H. Duru – Izmir Institute of Technology, TurkeyD.A. Demir – Izmir Institute of Technology, TurkeyO.T. Turgut – Bogazici University & FGI, TurkeyK. Ulker – Feza Gursey Institute, TurkeyN. Unal – Akdeniz University, TurkeyZ. Eker – Akdeniz University & TUG, TurkeyU. Camcı – Akdeniz University & TUG, Turkey


vii

LIST OF PARTICIPANTS

: Oral presentation

: Work included in the proceedings

Muhammad Afzal Quaid-i-Azam University, Islamabad,

PakistanBobomurat Ahmedov Institute of Nuclear Physics, Tashkent,

UzbekistanFarhad Ardalan Sharif University of Technology & IPM, Tehran,

IranEmrumiye Arlı Cukurova University, Adana, TurkeyOrhan Bayrak Akdeniz University, Antalya, TurkeyIsmail Boztosun Akdeniz University, Antalya, TurkeyCetin Camcı Canakkale Onsekiz Mart University, Canakkale,

Turkey

Ugur Camcı Akdeniz University & TUG, Antalya, Turkey

Diyadin Can Istanbul University, Istanbul, TurkeyAli Kazım Camlıbel Bogazici University, Istanbul, TurkeyTarık Celik Turkish Academy of Sciences, Ankara, TurkeyHakan Ciftci Gazi University, Ankara, TurkeyNaresh Dadhich Inter-University Center for Astronomy

and Astrophysics, Pune, IndiaDurmus Ali Demir Izmir Institute of Technology,

Izmir , TurkeyMustafa Dernek Akdeniz University, Antalya, TurkeyKoray Duztas Bogazici University, Istanbul, TurkeyMohammad Reza Sharif University of Technology,

Ejtehadi Tehran, IranZeki Eker Akdeniz University & TUG, Antalya, TurkeyGanim Gecim Akdeniz University, Antalya, Turkey


viii List of Participants

Cristiano Germani Ludwig-Maximilians-University,

Munchen, GermanyErtan Gudekli Istanbul University, Istanbul, TurkeySemra Gurtas Akdeniz University, Antalya, TurkeyRichard Lewis Hall Concordia University,

Montreal, CanadaMahmut Hortacsu Istanbul Technical University,

Istanbul, Turkey

Viqar Husain University of New Brunswick & AARMS,Canada

Ibrar Hussain National University of Science & Technology,Rawalpindi, Pakistan

George Jorjadze Razmadze Mathematical Institute, Tiblisi,Georgia

Mesut Karakoc Akdeniz University, Antalya, TurkeyVahid Karimipour Sharif University of Technology,

Tehran, Iran

Mehmet Koca Sultan Qaboos University, Oman

Burak Kurt Akdeniz University, Antalya, Turkey

Veli Kurt Akdeniz University, Antalya, TurkeyYusuf Kucukakca Akdeniz University, Antalya, Turkey

C. S. Lim Kobe University, Kobe, JapanBekir Can Lutfuoglu Akdeniz University, Antalya, TurkeyFazal Mahomed University of the Witwatersrand

& Centre for Differential Equations,Johannesburg, South Africa

Alexei Morozov Institute of Theoretical and ExperimentalPhysics, Moscow, Russia

Ali Mostafazadeh Koc University, Istanbul, TurkeyIsıl Basaran Oz Akdeniz University, Antalya, TurkeySacit Ozdemir Ankara University, Ankara, Turkey

Asghar Qadir National University of Science & Technology,Rawalpindi, Pakistan

Seifallah International Centre for Theoretical Physics,

Randjbar-Daemi Trieste, Italy


List of Participants ix

Muneer Ahmad National University of Science and Technology

Rashid & CAMP, Rawalpindi, Pakistan

Shahin Rouhani Sharif University of Technology, Tehran, IranMakoto Sakamoto Kobe University, Kobe, Japan

Ozgur Sevinc Istanbul University, Istanbul, Turkey

Ghulam Shabbir Ghulam Ishaq Khan Institute of EngineeringSciences and Technology, N.W.F.P, Pakistan

Muhammad Sharif Punjab University, Lahore, Pakistan

Azad Akhter Siddiqui National University of Science & Technology,

Rawalpindi, PakistanYusuf Sucu Akdeniz University, Antalya, TurkeyFerhat Taskın Erciyes University, Kayseri, TurkeyFrancesco Toppan Brazilian Center for Physics Research (CBPF),

Rio de Janeiro, Brazil

Gulcin Uluyazı Istanbul University, Istanbul, Turkey

Nuri Unal Akdeniz University, Antalya, TurkeyMaria A. H. Instituto de Ciencia de Materiales & CSIC,

Vozmediano Madrid, SpainFevziye Yasuk Erciyes University, Kayseri, TurkeyOzlem Yesiltas Gazi University, Ankara, TurkeyMustafa Kursat Yıldız Erciyes University, Kayseri, Turkey

Ihsan Yılmaz Canakkale Onsekiz Mart University, Canakkale,Turkey


xix

CONTENTS

Organizing Committees v

List of Participants vii

Preface by Asghar Qadir xi

Part A Formal Aspects 1

Geometric spectral inversion 3

Richard L. Hall

Heun functions and their uses in physics 23

M. Hortacsu

Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes 40

Mehmet Koca, N. Ozdes Koca

Hermite-Bernoulli 2D polynomials 61

Burak Kurt, Veli Kurt

Symmetry classification of coupled heat-diffusion systems vialow dimensional Lie algebras 71

F. M. Mahomed, M. Molati

Green’s functions and transition amplitudes for time-dependent linear harmonic oscillator with linear time-dependent terms added to the Hamiltonian 85

M. A. Rashid, M. U. Farooq

Time dependent harmonic oscillator and quasi-coherent states 94

Nuri Unal


xx Preface

Part B General Relativity and Cosmology 101

Integrability conditions for conformal Ricci collineation equations 103

M. Afzal, U. Camci, K. Saifullah

Noether symmetries of geodesic equations in spacetimes 116

Ibrar Hussain

Universal features of gravity and higher dimensions 122

Naresh Dadhich

Some interesting consequences of f(R) theory of gravity 136

Muhammad Sharif, Hafiza Rizwana Kausar

Part C Quantum Gravity 151

Stress-energy connection: Degravitating the vacuum energy 153

Durmus Ali Demir

The information loss paradox and the holographic principle 171

Asghar Qadir

Part D Quantum Field Theory 183

Generalized noncommutative gauge theory 185

F. Ardalan

CP violation and flavor physics in gauge-Higgs unification scenario 193

C. S. Lim

Hidden quantum-mechanical supersymmetry in extra dimensions 207

Makoto Sakamoto

On chiral and nonchiral 1D supermultiplets 227

Francesco Toppan


Preface xxi

Part E Applied Physics 239

Application of perturbation theory to elastic models of DNA 241

B. Eslami-Mossallam, M. R. Ejtehadi

Electromagnetic studies of ionospheric and magnetosphericperturbations associated with the earth, atmospheric and as-trophysical phenomena 254

S. R. Tojiev, V. S. Morozova, B. J. Ahmedov, H. E. Eshkuvatov


40

COXETER GROUPS, QUATERNIONS, SYMMETRIES OFPOLYHEDRA AND 4D POLYTOPES

MEHMET KOCA∗ and N. OZDES KOCA†

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

∗E-mail: [email protected]†E-mail: [email protected]

Emergence of the experimental evidence of E8 in the analysis of one di-mensional Ising-model invokes further studies of the Coxeter-Weyl groups gen-erated by reflections regarding their applications to polytopes. The Coxetergroup W (H4) which describes the Platonic Polytopes 600-cell and 120-cell in4D singles out in the mass relations of the bound states of the Ising modelfor it is a maximal subgroup of the Coxeter-Weyl group W (E8). There existsa one-to-one correspondence between the finite subgroups of quaternions andthe Coxeter-Weyl groups of rank 4 which facilitates the study of the rank-4Coxeter-Weyl groups. In this paper we study the systematic classifications ofthe 3D-polyhedra and 4D-polytopes through their symmetries described by therank-3 and rank-4 Coxeter-Weyl groups represented by finite groups of quater-nions. We also develop a technique on the constructions of the duals of thepolyhedra and the polytopes and give a number of examples. Applications ofthe rank-2 Coxeter groups have been briefly mentioned.

Keywords: Ising-model; Coxeter group; Polytopes; Quaternions.

1. Introduction

Zamolodchikov1 has proved that the one-dimensional Ising model atcritical temperature perturbed by an external magnetic field leads to eightspinless bosons with masses

m1, m3 = 2m1 cosπ

30, m4 = 2m2 cos

7π

30, m5 = 2m2 cos

4π

30,

m2 = τm1, m6 = τm3, m7 = τm4,m8 = τm5 (1)

where τ = (1 +√5)/2 is the golden ratio. It can be described by an affine

Toda Field Theory of E8 (Ref. 2). The masses can be related to the radii ofthe Gosset’s polytope of E8 (Ref. 3). In the derivation of the mass relationsin (1) the maximal subgroup W (H4) (Ref. 4) of the Coxeter-Weyl group


Mehmet Koca, N. Ozdes Koca 41

W (E8) plays a crucial role. This motivates us to study the structure of thegroup W (H4) in terms of quaternions and its 4D polytopes.

Radu Coldea of the University of Oxford and his colleagues, from Ox-ford, Bristol University, the ISIS Rutherford Laboratory and the HelmholtzZentrum Berlin5 performed a neutron scattering experiment on CoNb2O6

(cobalt niobate) which describes quantum Ising chain. They have deter-mined masses of the five emerging particles, first two are obeying the re-lation m2 = τm1. A modification of the Hamiltonian describing the Isingmodel has been suggested6 which confirms the mass relations in (1). Itseems that E8 displays itself as a Coxeter-Weyl group W (E8) at absolutezero temperature (the coldest regime) and perhaps as a Lie group in theform of E8 × E8 describing the heterotic superstring theory at very hotregime (Planck scale). The Coxeter-Weyl group W (E8) includes the crys-tallographic (tetrahedral and octahedral symmetries in 3D and 4D) as wellas the quasi crystallographic symmetries (icosahedral symmetry in 3D andits generalization to 4D). The icosahedral group of rank-3 describes fullythe structures of the fullerenes such as the C60 molecule which is repre-sented by a truncated icosahedrons, the icosahedral quasicrystals and theviral structures displaying the icosahedral symmetry.

The Platonic solids, tetrahedron, cube, octahedron, icosahedron and do-decahedron have been discovered by the people lived in Scotland nearly 1000years earlier than the ancient Greeks. Their models carved on the stonesare now kept in the Ashmolean Museum at Oxford7. They were used in oldGreek as models to describe fundamental matter associating tetrahedronwith fire, cube with earth, air with octahedron, and water with icosahe-dron. Archimedes discovered the semi-regular convex solids now known asArchimedean solids, and several centuries later they were rediscovered bythe renaissance mathematicians. The prisms and anti-prisms as well as fourregular non-convex polyhedra have been discovered in 1620 by Kepler com-pleting the classifications of the regular polyhedra. Nearly two centurieslater, in 1865, Catalan constructed the dual solids of the Archimedeansolids, now known as Catalan solids8 . Extensions of the platonic solidsto 4D dimensions have been made in 1855 by L. Schlaffli9 and their gener-alizations to higher dimensions in 1900 by T. Gosset10. Further importantcontributions are made by W. A. Wythoff11 among many others and inparticular by the contemporary mathematicians H.S.M. Coxeter12 and J.H.Conway13.

The Coxeter symmetries in 3D and 4D are richer than those in higherdimensions since the analogous structures to the Coxeter groups W (H3)


42 Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes

and W (H4) do not exist in higher dimensions. For example, the numberof Platonic solids is five in 3D and six in 4D contrary to the higher di-mensional cases where there exist only three platonic polytopes which arethe generalizations of tetrahedron, octahedron and cube. The Platonic andArchimedean solids14 as well as the Catalan solids15 can be described withthe rank-3 Coxeter groups W (A3),W (B3) and W (H3). The 4D polytopesare described by the rank-4 Coxeter groups W (A4),W (B4) and W (H4) andthe group W (F4).

In what follows we study the regular and semi regular 4D polytopesas the orbits of the rank-4 Coxeter groups expressed in terms of quater-nions. There is a one-to-one correspondence between the finite subgroupsof quaternions and the symmetries of the regular and semi regular 4D poly-topes. In Section 2 we introduce the finite subgroups of quaternions. Therank-4 Coxeter groups are constructed in terms of quaternions in Section3. The analysis of the 4D polytopes requires a study of the 3D polyhedrawhich will be introduced in Section 4. Examples of the 4D polytopes andtheir dual polytopes will be given in Section 5. Some concluding remarksare made in Section 7.

2. Finite subgroups of quaternions

This section deals with the introduction of quaternions and their rele-vance to the orthogonal transformations in four dimensions and gives thelist of finite subgroups of quaternions.

2.1. Quaternions and O(4) transformations

Let q = q0+qiei, (i = 1, 2, 3) be a real unit quaternion with its conjugatedefined by q = q0 − qiei and the norm qq = qq = 1. The imaginary unitssatisfy

eiej = −δij + εijkek, (i, j, k = 1, 2, 3). (2)

Let p, q be unit quaternions and r represents an arbitrary quaternion.Then the transformations16

r → prq : [p, q]; r → prq : [p, q]∗ (3)

define the orthogonal group O(4) which preserves the norm rr = rr. Thefirst term in (3) represents a proper rotation and the second includes alsothe reflection, generally called rotary reflection. In particular, the groupelement

r → prp : [p,−p]∗ (4)



represents the reflection with respect to the hyperplane orthogonal to thequaternion p. The orthogonal transformations in 3D can be simply writtenas r → ±prp : ±[p, p].

2.2. Finite subgroups of quaternions

The finite subgroups of quaternions are well known and their classifica-tion can be found in the reference 17. They are given as follows.

(a) Cyclic group of order n with n an odd number.(b) Cyclic group of order 2n is generated by �p = exp

�e1

πn

�� and

dicyclic group of order 4n is generated by the generators �p =exp

�e1

πn

�, e2�.

(c) The binary tetrahedral group can be represented by the set of 24unit quaternions:

T =

�±1,±e1,±e2,±e3,

1

2(±1,±e1,±e2,±e3)

�. (5)

(d) The binary octahedral group consists of 48 unit quaternions. Letthe set T � = (V1 ⊕ V2 ⊕ V3) represent the 24 unit quaternions,where

V1 =

�1√2(±1,±e1),

1√2(±e2,±e3)

�,

V2 =

�1√2(±1,±e2),

1√2(±e3,±e1)

�,

V3 =

�1√2(±1,±e3),

1√2(±e1,±e2)

�. (6)

Then the union of the set O� = T ⊕T � represents the binary octahedralgroup.

(e) The binary icosahedral group I = �b, c� of order 120 can be gen-erated by two unit quaternions b = 1

2 (τ + σe1 + e2) ∈ 12(1)+ andc = 1

2 (τ − σe1 + e2) ∈ 12(1)+. We display its elements in Table 1as the sets of conjugacy classes which represent a number of icosa-hedra, dodecahedra and one icosidodecahedron.

In Table 1 the golden ratio τ = 1+√5

2 and σ = 1−√5

2 satisfy the relations

τ + σ = 1, τ σ = −1, τ2 = τ + 1, σ2 = σ + 1.



Table 1. Conjugacy classes of the binary icosahedral group I represented by quaternions.

Order of the Conjugacy classes denoted by the number of elements and order ofelements the elements

1 12 −110 12+ : 1

2 (τ ± e1 ± σe3),12 (τ ± e2 ± σe1),

12 (τ ± e3 ± σe2)

5 12− : 12 (−τ ± e1 ± σe3),

12 (−τ ± e2 ± σe1),

12 (−τ ± e3 ± σe2)

10 12�+ : 12 (σ ± e1 ± τe2),

12 (σ ± e2 ± τe3),

12 (σ ± e3 ± τe1)

5 12�− : 12 (−σ ± e1 ± τe2),

12 (−σ ± e2 ± τe3),

12 (−σ ± e3 ± τe1)

6 20+ : 12 (1± e1 ± e2 ± e3),

12 (1± τe1 ± σe2),

12 (1± τe2 ± σe3),

12 (1± τe3 ± σe1)

3 20− : 12 (−1± e1 ± e2 ± e3),

12 (−1± τe1 ± σe2),12 (−1± τe2 ± σe3),

12 (−1± τe3 ± σe1)

4 30 : ±e1,±e2,±e3,12 (±σe1 ± τe2 ± e3),

12 (±σe2 ± τe3 ± e1),

12 (±σe3 ± τe1 ± e2)

3. Correspondence between the rank-4 Coxeter-Weylgroups and the finite subgroups of quaternions

Let I2(n) denotes the Coxeter diagram representing two vectors with theangle 2π

n between them. Then the Coxeter diagram I2(n)⊕ I2(n) of rank-4is represented by Figure 1.

Fig. 1. The Coxeter diagram I2(n)⊕ I2(n).

Let p = exp�e1

πn

�and q = exp

�e1

πn

�e2 be two orthogonal unit quater-

nions generating the dicyclic group of order 4n. Then the following set ofquaternions describes the root system of the diagram of Figure 1:

I2(n)⊕ I2(n) =�pk, qk

�; k = 1, 2, ..., 2n. (7)

If s, t ∈ (I2(n)⊕ I2(n)) are arbitrary elements of the root system thenthe group Aut (I2(n)⊕ I2(n)) = {[s, t]⊕ [s, t]∗} of order 4n × 4n is repre-sented by the elements of the dicyclic group18.

The F4 diagram shown in Figure 2 with its quaternionic roots19 leads toits automorphism group Aut(F4) = W (F4) : γ = {[O,O]⊕ [O,O]∗} whichis of the order 48 × 48 = 2304. Here γ is the Dynkin diagram symmetrygenerator of Figure 2.

The automorphism group of H4 with its quaternionic simple roots20 is



5

3. Correspondence between the rank-4 Coxeter-Weyl groups and the finite subgroups of quaternions Let )(2 nI denotes the Coxeter diagram representing two vectors with the

anglen!2 between them.

Then the Coxeter diagram 2 2( ) ( )I n I n! of rank-4 is represented by the Figure 1.

Figure 1. The Coxeter diagram )()( 22 nInI ! .

Let 1exp( )p en!

= and 1 2exp( )q e en!

= be two orthogonal unit quaternions generating

the dicyclic group of order 4n. Then the following set of quaternions describes the root system of the diagram of Figure1

2 2( ) ( ) { , }; 1, 2, ..., 2k kI n I n p q k n! = = . (7) If , ( ( ) ( ))2 2s t I n I n! " are arbitrary elements of the root system then the group

2 2( ( ) ( )) {[ , ] [ , ] }Aut I n I n s t s t !" = " of order 4n ! 4n is represented by the elements of the dicyclic group [18]. The 4F diagram shown in Figure 2 with its quaternionic roots [19] leads to its automorphism group 4 4( ) ( ) : {[ , ] [ , ] }Aut F W F O O O O! "= = # which is of the order 48! 48=2304. Here ! is the Dynkin diagram symmetry generator of Figure 2.

4

Figure 2. The Coxeter diagram of 4F .

The automorphism group of 4H with its quaternionic simple roots [20] is given as

4( ) {[ , ] [ , ] }W H I I I I != " with an order of 120! 120=14,400. The rank-4 Coxeter-Weyl groups 4( )W B of order 384 and

4( )W D of order 192 are the subgroups of the group 4( )W F . On the other hand the Coxeter-Weyl group 4 5( )W A S! of order 120 is a subgroup of the group 4( )W H [21].

4. Rank-3 Coxeter-Weyl groups and polyhedra

These groups are discussed extensively in the references [14-15]. Quaternionic representations of these groups can be classified as follows. The Coxeter-Dynkin diagram of the group 3( )W A is given as follows.

n!

n

Fig. 2. The Coxeter diagram of F4.

given as W (H4) = {[I, I]⊕ [I, I]∗} with an order of 120 × 120 = 14, 400.The rank-4 Coxeter-Weyl groups W (B4) of order 384 and W (D4) of order192 are the subgroups of the group W (F4) . On the other hand the Coxeter-Weyl group W (A4) ≈ S5 of order 120 is a subgroup of the group W (H4).21

4. Rank-3 Coxeter-Weyl groups and polyhedra

These groups are discussed extensively in the references 14 and 15.Quaternionic representations of these groups can be classified as follows.The Coxeter-Dynkin diagram of the group W (A3) is given in Figure 3.

Fig. 3. The Coxeter-Dynkin diagram of the group W (A3).

If the roots αi are chosen to be α1 = e1 + e2, α2 = e3 − e2, α3 =e2 − e1 then the group elements are given by the set W (A3) =�[T, T ]⊕ [T �, T �]∗

�≈ S4.

The Coxeter diagram of the group W (B3) is represented by the quater-nionic roots as in Figure 4.

Fig. 4. The Coxeter diagram of the group W (B3).

Note that all the roots in the Coxeter diagram in Figure 4 have thesame norm contrary to the Dynkin diagrams where some of the roots arerepresented as short roots. The group elements are represented by the set

W (B3) ≈ Aut(A3) =�[T, T ]⊕ [T �, T �]⊕ [T, T ]∗ ⊕ [T �, T �]∗

�≈ S4 × C2



which is isomorphic to the octahedral group Oh.The Coxeter diagram of the group W (H3) is represented by the unit

quaternions as shown in Figure 5.

Fig. 5. The Coxeter diagram of the group W (H3).

The group elements can be written in terms of the elements of thebinary icosahedral group as W (H3) =

�[I, I]⊕ [I, I]∗

�≈ A5 × C2 which is

isomorphic to the icosahedral group Ih. The symmetry of the prisms arerepresented by the group elements given by the set of quaternionsW (I2(n)⊕A1) ≈ Dn × C2.

Below we display some polyhedra as the orbits of these Coxeter groups.Let αi and ωi denote respectively the set of simple roots and the basisvectors of the dual space satisfying the relations

(αi,ωj) = δij , (αi,αj) = Cij , (ωi,ωj) = (C−1)ij . (8)

Note that the roots are normalized to√2. Define an arbitrary vector in the

dual space by Λ = a1ω1 + a2ω2 + a3ω3 ≡ (a1a2a3) and the Λ − orbit byO(Λ) = W (G)Λ.

The Platonic and Archimedean solids as well as Catalan solids can bedescribed as the orbits of the rank-3 Coxeter groups. Some examples aregiven below.

4.1. The tetrahedral group W (A3) ≈ S4

Let us consider the group W (A3) where the Cartan matrix C and itsinverse C−1 are given by

C =

2 −1 0−1 2 −10 −1 2

, C−1 =1

4

3 2 12 4 21 2 3

. (9)

The orbit O(100) = 12 (±e1 ± e2 ± e3) with even number of (−) sign and the

orbit O(001) = 12 (±e1 ± e2 ± e3) with odd number of (−) sign represent

two tetrahedra dual to each other.The union of two dual tetrahedra constitutes a cube. Similarly the orbits

O(110) and O(011) represent two truncated tetrahedra.



Fig. 6. Tetrahedron represented by the orbit O(100).

Fig. 7. The truncated tetrahedron represented by the orbit O(110).

4.2. The octahedral group W (B3) ≈ S4 × C2

The Cartan matrix and its inverse are given by the matrices

C =

2 −1 0−1 2 −

√2

0 −√2 2

, C−1 =

1 1 1√

2

1 2√2

1√2

√2 3

2

. (10)

The orbit O(100) = (±e1,±e2,±e3) represents an octahedron and the orbitO(001) = 1

2 (±e1 ± e2 ± e3) represents a cube which is the dual of octahe-dron.

Fig. 8. The octahedron (a) and its dual cube (b) as the respective orbits W (B3)(100)and W (B3)(001).



All non-chiral Archimedean solids with octahedral symmetry canbe constructed as the orbits of the octahedral group. The non-chiralArchimedean solids are shown in Figure 9.

Fig. 9. The Archimedean solids with octahedral symmetry (a) truncated octahedron asW (B3)(110), (b) truncated cube as W (B3)(011), (c) cuboctahedron as W (B3)(010),(d) small rhombicuboctahedron as W (B3)(101), (e) great rhombicuboctahedron asW (B3)(111).

4.3. The icosahedral symmetry W (H3) ≈ A5 × C2

The Cartan matrix and its inverse are given by the matrices

C =

2 −τ 0−τ 2 −10 −1 2

, C−1 =1

2

3τ2 2τ3 τ3

2τ3 4τ2 2τ2

τ3 2τ2 τ + 2

. (11)

The platonic as well as Archimedean solids possessing the icosahedral sym-metry can be constructed as the orbits of the Coxeter group W (H3). Forexample, two dual platonic solids, the icosahedron and dodecahedron, arerepresented by the orbits O(100) and O(001) respectively as shown in Fig-ure 10.

The non-chiral Archimedean solids with the icosahedral symmetryare represented by the orbits W (H3)(010), W (H3)(110), W (H3)(011),W (H3)(101) andW (H3)(111). They are respectively (a) icosidodecahedron,(b) truncated dodecahedron, (c) truncated icosahedron, (d) small rhombi-cosidodecahedron, and (e) great rhombicosidodecahedron as shown in theFigure 11.

There are two more Archimedean solids. They are the chiral solids: thesnub cube and the snub dodecahedron. They are generated by the propersubgroups of the octahedral group and the icosahedral group respectively.Construction of their vertices using the Coxeter diagrams of the groups



Fig. 10. The iocahedron (a) and its dual dodecahedron (b) as the respective orbitsW (H3)(001) and W (H3)(100).

Fig. 11. The non-chiral Archimedean solids with icosahedral symmetry.

W (B3) and W (H3) has been obtained in reference 22. The vertices of thesnub cube can be obtained as follows

(W (B3))/C2)ΛI =��

±xe1 ± e2 ± x−1e3�,�±xe2 ± e3 ± x−1e1

�,

�±xe3 ± e1 ± x−1e2

��(12)

(W (B3))/C2)ΛII =��

±e1 ± xe2 ± x−1e3�,�±e2 ± xe3 ± x−1e1

�,

�±e3 ± xe1 ± x−1e2

��

withW (B3)/C2 =�[T, T ]⊕ [T �, T �]

�and in terms of quaternionic units two

vectors are given by ΛI = (xe1 + e2 + x−1e3) and ΛII = (e1 + xe2 + x−1e3)where x3 − x2 − x− 1 = 0, x ≈ 1.8393. The left and the right oriented snubcubes are shown in Figure 12.

Snub dodecahedron can be obtained by the action of the proper sub-group of the icosahedral group to two vectors expressed in terms of quater-



Fig. 12. The snub cube (a) as the orbit (W (B3)/C2)ΛI and (b) (W (B3)/C2)ΛII .

nions as follows:

ΛI =1√2

�σ(x2 − 1)e1 − xe2 + (1− τx3)e3

�,

ΛII =1√2

�−σ(x2 − 1)e1 − xe2 + (1− τx3)e3

�,

x3 − x2 − x− τ = 0, x ≈ 1.94315. (13)

Now the orbits generated by these vectors (W (H3)/C2)ΛI and(W (H3)/C2)ΛII are represented by two snub dodecahedra as shown in Fig-ure 13.

Fig. 13. The snub dodecahedron (a) as the orbit of ΛI and its mirror image (b) as theorbit of ΛII.

4.4. Catalan solids

The Catalan solids are the duals of the Archimedean solids and areextensively studied in reference 15. Here we give a few examples which arequite useful in the crystallography or in the classification of viral structures.



They are the unions of the orbits of the relevant Coxeter groups. The dualof the cuboctahedron possessing the octahedral symmetry is the rhombicdodecahedron. Its 14 vertices are the union of the orbits W (B3)(100) ⊕1√2W (B3)(001) which is represented in Figure 14.

Fig. 14. The rhombic dodecahedron.

The dual of the icosidodecahedron is the rhombic triacontahedron. Its 32vertices is the union of the orbits W (H3)(100) ⊕ τW (H3)(001) which isdisplayed in Figure 15.

Fig. 15. The rhombic triacontahedron.

The duals of the snub cube and snub dodecahedron have been constructedin the reference 22 using the quaternionic representation of the chiral groupsand the vertices of the solids of concern. They are displayed in the Figure16.

5. Rank-4 Coxeter groups and 4D polytopes

As we have mentioned in the introduction there are five rank-4groups W (D4),W (B4),W (F4),W (A4), and W (H4) describing the regularand semiregular polytopes. We will discuss the group W (D4) with its rele-vance to the snub 24-cell later. Now let us start first with the group W (B4).



Fig. 16. Dual of the left oriented snub cube (a) and the dual of left oriented snubdodecahedron (b).

5.1. The Coxeter group W (B4) of order 384

This group is an extension of the octahedral group W (B3) to four di-mensions23 . As it is expected the generalizations of the octahedron andthe cube in terms of quaternions are straightforward. The hyper octa-hedron, also known as (16-cell), is given by the orbit W (B4)(1000) ={±1,±e1,±e2 ± e3}. It consists of 16 tetrahedra whose vertices are 16 dif-ferent combinations of the unit quaternions {1, e1, e2, e3} including the signchanges. Similarly the hyper cube, known as 8-cell, is represented by theorbit W (B4)(1000) =

12 (±1±e1±e2±e3). It consists of 8 cubic cells. Hyper-

octahedron and hyper cube are dual to each other as expected. The verticesof any quasi regular polytope can be obtained as the orbitW (B4)(a1a2a3a4)where ai ≥ 0, (i = 1, 2, 3, 4) , is a real number. For the values ai = 0 or 1many semi regular polytopes have been discussed in the reference 24.

5.2. The Coxeter group W (F4) of order 1152

It is a unique Coxeter group in the sense that it has correspondencesneither in three dimensions nor in any other higher dimensions. It has aself dual polytope with 24 vertices with 24 cells made of octahedra. It issimply called 24-cell which is represented in terms of quaternions either bythe orbit

W (F4)(1000) = T = {(±1,±e1,±e2,±e3),1

2(±1± e1 ± e2 ± e3)}

or by

W (F4)(0001) = T � = (V1 ⊕ V2 ⊕ V3).

The Dynkin diagram symmetry of F4 transforms these two sets of ver-tices to each other. Thus the polytope 24-cell is said to be self dual. The



quasi regular and semi regular polytopes can be constructed25 by the sametechnique explained as above for the group W (B4).

5.3. The Coxeter group W (H4) ⊂ W (E8)

The Coxeter group W (H4) is a maximal subgroup of the group W (E8).Its two dual platonic polytopes are called the 120-cell and 600-cell repre-sented by quaternions as follows:

120-cell: W (H4)(1000) =�4

i,j=0 ⊕piT �pj . It has 600 vertices. Here p ∈I, p5 = ±1.

Each cell is a dodecahedron. At each vertex there are 4 dodecahedra.The 600-cell is represented by the set of quaternions I which also repre-

sents the binary icosahedral group:600-cell: W (H4)(0001) = I =

�4i=0 ⊕piT .

Each cell is a tetrahedron. At each vertex there are 20 tetrahedra. Whenthe sphere S3 representing the vertices of the 600-cell is sliced by parallelhyperplanes one obtains 3D polyhedra represented by the conjugacy classesof the binary icosahedral group I in Table 1.

They consist of two points ±1 corresponding to the poles of the sphereS3, 4 icosahedra, 2 dodecahedra and one icosidodecahedron which can bereadily seen from the Table 1. The 120-cell and 600-cell are dual to eachother. In addition to these two platonic solids we will study two more poly-topes possessing the symmetry W (H4). Actually any polytope representedby the orbit W (H4)(a1a2a3a4) can be worked out and their branching un-der its maximal subgroups can be studied26. Here we will give the followingexamples.

720-cell = 120 + 600: This is the orbit represented by W (H4)(0100)consisting of 1200 vertices. Its cells are made of 120 icosidodecahedraand 600 tetrahedra. At any vertex there join 3 icosidodecahedra and2 tetrahedra whose centers are represented respectively by the vectorsω4, r3r4ω4, (r3r4)2ω4 up to a scale factor and by the vectors ω1, r1ω1.

The dual polytope of the 720-cell consists of 720 vertices which is theunion of the orbits W (H4)(0001) ⊕ 2

3τW (H4)(1000).26 Each cell of thedual polytope is a dipyramid with an equilateral triangular base. Thedual polytope is the projection of the Voronoi cell of E8 (see Ref. 27)with 19,440 vertices into 4D space. Another semi regular polytope of theCoxeter group W (H4) has 720 vertices and 720 cells which we denoteby 720� − cell = 120 + 600. The 720� − cell is represented by the orbitW (H4)(0010). It consists of the cells of 120 icosahedra and 600 octahedraand with this property it is the only Archimedean solid with the W (H4)



symmetry. At each vertex there are 2 icosahedra and 5 octahedra. Theircenters can be represented respectively by ω4, r4ω4 up to a scale factor2τ2+σ and ω1, r1r2ω1, (r1r2)2ω1, (r1r2)3ω1, (r1r2)4ω1. These 7 vertices form adipyramid with a pentagonal base. The 720 vertices of the dual polytopeconsist of the set 2τ

2+σW (H4)(0001)⊕W (H4)(1000).It is also interesting to note that the symmetries of the semi regular 4D

polytopes, the grand antiprism and the snub 24-cell, are represented by twomaximal subgroups of the Coxeter group W (H4). We display the maximalsubgroups of the Coxeter group W (H4) in Table 2.18,21

Table 2. Maximal subgroups of the Coxeter group W (H4).

Maximal subgroup Order Polytope

Aut(A2 ⊕A2) 144Aut(H2 ⊕H2) 400 Grand antiprism with 100 verticesW (H3 ⊕A1) 240

Aut(A4) 240(W (D4)/C2) : S3 576 Snub 24-cell with 96 vertices

One of these semi regular polytope is the Grand Antiprism (GA) whichwas first determined by a computer calculation by Conway and Guy in196528. This semi regular polytope has 100 vertices which can be obtainedas a subset of the set of quaternions I (600-cell) under the decompositionof its maximal subgroup Aut(H4 ⊕H2).18 Let us take the quaternions b =12 (τ + σe1 + e2) ∈ 12(1)+ and c = 1

2 (τ − σe1 + e2) ∈ 12(1)+. Then thevertices of the GA are given by the set of quaternions

GA = {bmcbn, bme3bn}, m, n = 0, 1, ..., 9 (14)

which consists of 100 elements of the set of quaternions I from which the setof 20 quaternionic roots of the Coxeter diagram H2⊕H �

2 = {bm, e3bm};m =0, 1, ..., 9 are removed. The GA has 100 vertices, 320 cells (20 pentagonalantiprisms and 300 tetrahedra), 500 edges and 720 faces (700 triangles and20 pentagons). The dual polytope of the GA consists of 320 vertices which



are given as the orbits of the group Aut(H2 ⊕H2) as follows:

200 vertices like J1 =4�

i.j=0

⊕biV1bj ;

100 vertices like J �3 =

4�

i.j=0

⊕bi1√2(±1± e3)b

j and

20 vertices like

�τ√2bm,

τ√2e3b

m

�; m = 0, 1, ..., 9. (15)

The typical cell of the dual polytope has 14 vertices with faces of 4 pen-tagons, 4 kites and 2 isosceles trapezoids as shown in Figure 17.

Fig. 17. A typical cell of the dual polytope of the grand antiprism.

Another interesting semi regular polytope is the snub 24-cell. The snub24-cell29 consists of 96 vertices S = I − T =

�4i=1 p

iT with the symmetrygroup (W (D4)/C2) : S3 = {[T, T ] ⊕ [T, T ]∗} of order 576. Its 144 cellsconsist of 24 icosahedra and 120 tetrahedra. The dual of the snub 24-cellhas the vertices T ⊕ T � ⊕ S� with S� =

�4i=1 p

ip†iT � where p† is obtainedfrom p by exchanging τ ↔ σ. One cell of the dual polytope is a solid with8 vertices as shown in Figure 18.

Fig. 18. A typical cell of the dual polytope of the snub 24-cell.



5.4. The Coxeter group W (A4) of order 120

The automorphism group Aut(A4) ≈ W (A4) : γ is an extension of theCoxeter group W (A4) by the Dynkin diagram symmetry γ. It is a maximalsubgroup of the group W (H4) and has its own Platonic and Archimedeanpolytopes. We discuss the following examples. The 5-cell is an extension oftetrahedron to 4D and can be represented either by the orbit W (A4)(1000)or W (A4)(0001). It has 5 vertices consisting of 5 tetrahedra, 4 of whichare meeting at one vertex. Similarly the orbit W (A4)(0100) represents therectified 5-cell consisting of 5 tetrahedral and 5 octahedral cells. The rec-tified cell has 10 vertices. At each vertex there are 2 tetrahedra and 3octahedra whose centers respectively consist of the vertices ω1, r1ω1 andω4, r3r4ω4, (r3r4)2ω4 up to a scale factor. These vertices form a dipyramidwhich is a typical cell of the dual polytope30,31 . The vertices of the dualpolytope consist of the union of the orbits W (A4)(1000)⊕ 3

2W (A4)O(0001).Another polytope with the symmetry group Aut(A4) is called runcinated5-cell consisting of 20 vertices and 30 cells composed of 10 tetrahedra and20 triangular prisms. It is represented by the orbit W (A4)(1001). 2 tetra-hedra and 6 triangular prisms join to each vertex. The dual polytope has30 vertices consisting of the union of the orbits

W (A4)(1000)⊕W (A4)(0100)⊕W (A4)(0010)⊕W (A4)(0001).

The dual polytope has 20 cells each is made of rhombohedron as shown inFigure 19. The rhombohedron has the symmetry of D3 × C2.

Fig. 19. The rhombohedron, one of the typical cell of the dual polytope of the polytopeW (A4)(1001).

6. Quasi regular polygonal tiling with the Coxetersymmetry Dn

Before we conclude we would also show how the plane can be tiled withthe isogonal and isotoxal hexagons with the dihedral symmetry D3. The



rank-2 Coxeter groups are the dihedral groups Dn which can be representedby the Coxeter diagram shown in Figure 20.

Fig. 20. The Coxeter diagram representing the dihedral symmetry.

The regular and quasi regular polygons possessing the dihedral sym-metry Dn has been worked out in the references 32 & 33. Here we giveisogonal and isotoxal hexagons having the symmetry D3 and the tiling ofthe plane with them. An isogonal hexagon has two types of lengths with120◦ interior angles as shown in Figure 21(a). Its dual isotoxal hexagon hasall edge lengths the same but two different interior angles alternating ateach vertex and complementing each other to 240◦. It is shown in Figure21(b).

Fig. 21. The isogonal hexagon (a) and its dual isotoxal hexagon (b) possessing thedihedral symmetry D3.

The tilings with isogonal hexagons are shown in Figure 22.Tiling of the plane with isotoxal hexagons and regular hexagons are shownin Figure 23.

7. Concluding remarks

We have presented a few examples of the 4D polytopes along withtheir duals. Their cells are made of Platonic and Archimedean polyhedra.A systematic study of the 4D polytopes and their dual polytopes will fol-low.The 4D polytopes and their duals with W (A4) symmetry has beenalready worked out in the reference 30 where we also projected the poly-topes into 3D with the symmetry W (A3). The analyses related with the



Fig. 22. Tiling of the plane with isogonal hexagons-regular hexagons (a-b), regularhexagons and triangles (c), tiling with triangles (d) and the honeycomb lattice (e).

Fig. 23. Tiling of the plane with isotoxal and regular hexagons.

W (H4),W (B4), and W (F4) symmetries are under preparation26,34,35. Thetilings of the plane with isogonal and isotoxal hexagons could be interestingfor the graphene physics if short and long bonds of carbon atoms are takeninto account.

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COXETER GROUPS, QUATERNIONS, SYMMETRIES OF POLYHEDRA AND 4D POLYTOPES

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