SQU Journal for Science, 2016, 21(2), 150-161 © 2016 Sultan Qaboos University 4D Pyritohedral Symmetry Nazife O. Koca*, Amal J.H. Al Qanobi and Mehmet Koca Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud 123, Muscat, Sultanate of Oman,* Email: [email protected]. ABSTRACT: We describe an extension of the pyritohedral symmetry in 3D to 4-dimensional Euclidean space and construct the group elements of the 4D pyritohedral group of order 576 in terms of quaternions. It turns out that it is a maximal subgroup of both the rank-4 Coxeter groups W (F4) and W (H4), implying that it is a group relevant to the crystallographic as well as quasicrystallographic structures in 4-dimensions. We derive the vertices of the 24 pseudoicosahedra, 24 tetrahedra and the 96 triangular pyramids forming the facets of the pseudo snub 24- cell. It turns out that the relevant lattice is the root lattice of W (D4). The vertices of the dual polytope of the pseudo snub 24-cell consists of the union of three sets: 24-cell, another 24-cell and a new pseudo snub 24-cell. We also derive a new representation for the symmetry group of the pseudo snub 24-cell and the corresponding vertices of the polytopes. Keywords: Pseudoicosahedron; Pyritohedron; Lattice; Coxeter groups and Quaternions. تماثل متعدد اﻷوجه ذو التركيب المشابه لمعدن البايريت في الفضاء اﻻقليدي ذو اﻷبعاد اﻷربعة نظيفة كوجا، أمل القنوبي و محمد كوجا مستخلص: وصفنا امتدادا لتماثل تركيب متعددات اﻷوجه من النوع المشابه لبلورة معدن البايرايت )pyritohedral( - و الذي يمكن صياغته في اﻻبعاد الثﻻثة - إلى الفضاء اﻻقليدي ذو اﻷبعاد اﻷربعة ثم بنينا عناصر المجموعة المتصلة ببلورة معدن البايرايت رباعية اﻻبعاد باستخدام العدد المركب التخيلي الرباعي )الكواتيرنيونات(، و اتضح أن هذه المجموعة من أكبر المجموعات الفرعية لكل من مجموعات الكوكستر (W (F4 و (W (H4 ذات الرتبة الرابعة مما يعني أنها مجموعة ذات صلة بالتركيبات البلورية فضﻻ عن اشباه البلورية في اﻷبعاد اﻷربعة، كما تم اشتقاق احداثيات رؤوس 24 ذوات العشرون وجها غير المنتظم و 24 رباعي اﻷسطح و 96 أهراما مثلثية و التي تمثل أسطح لمجسم رباعي اﻷبعاد يسمى pseudo snub 24-cell ، و اتضح أن الشبيكة الفراغية ذات الصلة هي جذر الشبيكة الفراغية لـ (W (D4 و أن رؤوس المزدوج لـ pseudo snub 24-cell هي اتحاد ثﻻث مجموعات: اربعة و عشرون خلية )24cell( و اربعة و عشرون خلية )cell-24( أخرى و pseudo snub 24-cell جديد، كذلك اشتققنا تمثيﻻ جديدا لمجموعة تماثل pseudo snub 24-cell و رؤوس اﻷجسام رباعية اﻷبعاد المقابلة. كلمات مفتاحية: شبيه ذي العشرين وجها المنتظم، متعدد اﻷوجه ذو التركيب المشابه لمعدن البايرايت، الشبيكة الفراغية، مجموعات كوكستر، الرباعيات. 1. Introduction attices in higher dimensions described by the affine Coxeter groups, when projected into lower dimensions, may L represent the quasicrystal structures [1-5]. It is known that the A4 lattice projects into the aperiodic lattice with 5- fold symmetry [1]. There is no doubt that the projections of the higher dimensional lattices may have some implications in physics. The exceptional Coxeter-Weyl group W(F4 ) describes the symmetry of the unique self-dual polytope, the 24-cell, which is the Voronoi cell (Wigner-Seitz cell) of the F4 lattice. The noncrystallographic Coxeter group W(H 4 ) is the symmetry of the famous 600-cell and its dual 120-cell [6-7]. WD() In this work we construct the 4D pyritohedral group from D diagram. In technical terms the group 4 :S of 4 C 3 2 order 576 [8] can be expressed in terms of quaternions and we will determine its orbits as the pseudo snub 24-cell and 150 4D PYRITOHEDRAL SYMMETRY its dual polytope, which are related to the lattice W(D4 ). All rank 4 Coxeter-Weyl groups can be represented, in compact forms, by quaternion pairs [9]. This paper is organized as follows. In Section 2 we introduce the 4D - pyritohedral symmetry derived from D4 WD() diagram. In Section 3 we construct the group 4 :S and apply it to a vector to generate the vertices of a C 3 2 polytope which we call “pseudo snub 24-cell”. We find the facets of the pseudo snub 24-cell which consist of the pseudoicosahedra, tetrahedra and triangular pyramids. The vertices of the dual polytope of the pseudo snub 24-cell are constructed. Finally, in Section 4 we present a brief discussion on the physical implications of our technique. 2. 4D crystals with the pyritohedral symmetry derived from D4 diagram In the paper [10] we discussed the pyritohedral group [T,T ][T,T ]* of order 24 which is derived from the Coxeter- Dynkin diagram D3 by the rotation generators and the Dynkin diagram symmetry. The straightforward generalization of this group to 4D is to start with the rotation generators of D4 and impose the Dynkin diagram symmetry. We will see that the generated group from the diagram is nothing other than the group [T,T][T,T]* of order 576. It represents the symmetry of the snub 24-cell [6], [8]. The snub 24-cell is a convex uniform polytope in four dimensions consisting of 120 regular tetrahedral and 24 icosahedral cells. It has 96 vertices at each of which five tetrahedra and three icosahedra meet. Snub 24-cell can be constructed from the 24-cell by dividing the edges in the golden ratio and truncating it in a certain way. This truncation transforms the 24 octahedral cells of the 24-cell to the 24 icosahedral cells of the snub 24-cell; the truncated vertices become 24 tetrahedral cells and the gaps in between are filled in by another 96 tetrahedra. W(D4 ) 3. Construction of the symmetry group of snub 24-cell : S3 C2 The Coxeter-Dynkin diagram is shown in Figure 1 with the quaternionic simple roots. α =e -e 3 2 3 α =1-e 1 1 α =e -e 2 1 2 α =e +e 4 2 3 Figure 1. The Coxeter-Dynkin diagram D with the simple roots. 4 The corresponding weights are determined as 11 1, 1e , (1 e e e ), (1 e e e ). (1) 1 2 1322 1234 123 Note that i T , i 1,2,3,4; 2 T; T , j=1,3,4. 22j The set T is given by the group elements 1 푇 = {±1, ±푒 , ±푒 , ±푒 , (±1 ± 푒 ± 푒 ± 푒 )}, (2) 1 2 3 2 1 2 3 and is called the binary tetrahedral group of order 24 . Another set of 24 quaternions is defined by 151 NAZIFE O. KOCA ET AL 1 1 1 (±1 ± 푒 ), (±1 ± 푒 ), (±1 ± 푒 ), √2 1 √2 2 √2 3 ′ 1 1 1 (3) 푇 = { (±푒 ± 푒 ), (±푒 ± 푒 ), (±푒 ± 푒 )}. √2 1 2 √2 2 3 √2 3 1 In terms of quaternionic simple roots, the group generators of W(D4 ) can be written as 11 r [ (1- e ), - (1- e )] , 122 1 1 11 r2[ ( e 1 e 2 ), - ( e 1 e 2 )] , 22 (4) 11 r[ ( e e ), - ( e e )] , 322 2 3 2 3 11 r[ ( e e ), - ( e e )] . 422 2 3 2 3 They generate the Coxeter-Weyl group W(D4 ) of order 192 [12]. The subsets of the quaternions T V0 V V and T V1 V2 V3 are defined as follows: 1 V {1, e , e , e } ,V (1 e e e ) , even number of (-) sign, 0 1 2 3 2 1 2 3 1 V (1 e e e ) , odd number of (-) sign, 2 1 2 3 1 1 V1 { (1 e1 ), (e2 e3 )} , 2 2 (5) 1 1 V2 { (1 e2 ), (e3 e1 )} , 2 2 1 1 V3 { (1 e3 ), (e1 e2 )} . 2 2 These subsets are useful to denote the Coxeter-Weyl group W(D4 ) in a compact form: * * * W(D4 ) {[V0 ,V0 ][V ,V ][V ,V ][V1 ,V1 ] [V2 ,V2 ] [V3 ,V3 ] } . (6) Note that the subset of the Coxeter-Weyl group WD()4 (7) {[,VVVVVV00 ][, ][, ]} C 2 represents the proper subgroup and can be directly generated by the rotation generators 푟2푟1, 푟2푟3, 푟2푟4. Let us impose the Dynkin diagram symmetry which is the permutation group 푆3 of the simple roots 훼1, 훼3 and 훼4 as shown in Figure 2. 152 4D PYRITOHEDRAL SYMMETRY α α 3 α α 1 α 1 α α α 4 4 Generator 1: [p, q] * Generator 2: [e3 ,e3 ] Figure 2 . The action of permutation group 푺ퟑ. * The permutation group of order 6 can be generated, for example, by two generators [,]pq and [,]ee33 where 1 1 p(1 e e e ) and q(1 e e e ) . They are the elements of [p , q ] [ V , V ] with [ p , q ]3 [1,1] and 2 1 2 3 2 1 2 3 W(D4 ) [,][,]e3 e 3 V 0 V 0 . The group is invariant under conjugation by the group 푆3. We first note that C2 the extension of the group of eq. (7) by the cyclic group of order 3 generated by the generator [푝, 푞] is a group of order 288 which can be denoted by WD() 4 :CTT [ , ] . (8) C 3 2 The extension of the group by full permutation group S 3 is given as the semi-direct product of two groups as: WD() * 4 :STTTT {[ , ] [ , ] } . (9) C 3 2 As we will see in the next section this is the symmetry group of the snub 24-cell as well as that of any pseudo snub 24- cell. 3.1 Construction of the vertices of the pseudo snub 24-cell The affine Coxeter group 푊푎(퐷4) =< 푟0, 푟1, 푟2, 푟3, 푟4 > can be generated by five generators by introducing 푟0 as shown in Figure 3.