Grand Antiprism and Quaternions Mehmet Koca1, Mudhahir Al-Ajmi1, Nazife Ozdes Koca1 1Department of Physics, College of Science, Sultan Qaboos University, P. O. Box 36, Al-Khoud, 123 Muscat, Sultanate of Oman Email: [email protected], [email protected], [email protected] Abstract Vertices of the 4-dimensional semi-regular polytope, the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It follows from the icosian representation of the E8 root system which decomposes into two copies of the root system of H4. The symmetry of the grand antiprism is a maximal subgroup of the Coxeter group W (H4). It is the group Aut(H2 H′ ) which is constructed in terms of 20 quaternionic roots ⊕ 2 of the Coxeter diagram H2 H2′ . The root system of H4 represented by the binary icosahedral group I ⊕of order 120, constitutes the regular 4D polytope 600-cell. When its 20 quaternionic vertices corresponding to the roots of the diagram H2 H2′ are removed from the vertices of the 600-cell the remaining 100 quaternions⊕ constitute the vertices of the grand antiprism. We give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions. The dual polytope of the grand antiprism has been also constructed. 1 Introduction Six regular 4D polytopes and their semi-regular Archimedean polytopes are very interesting because their symmetries W (A4), W (B4), W (F4) and W (H4) can be represented with the use of finite subgroups of quaternions. There is a close correspondence between the finite subgroups of quaternions and the arXiv:0906.2117v2 [math-ph] 18 Aug 2009 symmetries of the above polytopes [1]. Some of them occur in high energy physics in model building either as a gauge symmetry like SU(5) [2] or as a little group SO(9) [3] of the M-theory. The Coxeter group W (H4) arises as the symmetry group of the polytope 600-cell, 3, 3, 5 [4], vertices of which can be represented by 120 quaternions of the binary{ icosahedral} group [5, 6]. The dual polytope 120-cell, 5, 3, 3 with 600 vertices, can be constructed from 600-cell in terms of quaternions.{ } Another observation is that they are all nested in the root system of E8 represented by icosians [7, 8]. 1 The grand antiprism was first constructed in 1965 by Conway and Guy [9] with a computer analysis. An antiprism is defined to be a polyhedron com- posed of two parallel copies of some particular n sided polygon,connected by an alternating band of triangles. In this paper− we study the symme- try group of the grand antiprism and construct its 100 vertices in terms of quaternions. We explicitly show how these vertices form 20 pentagonal an- tiprisms and 300 tetrahedra. We organize the paper as follows. In Section 2 we briefly discuss the correspondence between the root system of H4 and the quaternionic representation of the binary icosahedral group I and decompose it as 120=20+100 where first 20 quaternions represent the root system of the Coxeter diagram H2 H2′ and the remaining quaternions are the vertices of the grand antiprism⊕. The symmetry group of the grand antiprism turns out to be the group Aut(H2 H′ ) of order 400 which can be constructed ⊕ 2 directly from the quaternionic roots of the Coxeter diagram H2 H2′ . A simple technique has been employed in Section 3 for the construction⊕ of the cell structures of the grand antiprism. A projection of the grand antiprism into three dimensional space is discussed in Section 4. The dual polytope of the grand antiprism has been constructed in Section 5 where 320 vertices decompose as 320=20+200+100 under the group Aut(H2 H′ ) where 20 ⊕ 2 vertices are scaled copies of the root system of the Coxeter diagram H2 H′ ⊕ 2 lying on a sphere S3 with radius 1+√5 1.14 and 300 vertices lie on an 2√2 ≈ inner sphere S3 with a radius of unit length. Conclusion is given in Section 6 and an Appendix lists the decompositions of the vertices of the Archimedian W (H4) solids in terms of the orbits of Aut(H2 H′ ). ⊕ 2 2 Construction of the root system of H H′ 2 ⊕ 2 as the maximal subset of the root system of H4 and the vertices of the grand antiprism Let q = q0 + qiei, (i =1, 2, 3) be a real quaternion with its conjugate defined byq ¯ = q0 qiei where the quaternionic imaginary units satisfy the relations − eiej = δij + εijkek, (i, j, k =1, 2, 3). (1) − Here δijand εijk are the Kronecker and Levi-Civita symbols respectively and summation over the repeated indices is implicit. Quaternions generate the four dimensional Euclidean space where the quaternionic scalar product is defined by 2 1 1 (p, q)= (¯pq +¯qp)= (pq¯ + qp¯). (2) 2 2 The imaginary quaternionic units ei can be related to the Pauli matrices σi by ei = iσi and the unit quaternion is represented by 2 2 unit matrix. This correspondence− proves that the group of quaternions× is isomorphic to SU(2) which is a double cover of the proper rotation group SO(3). The root system of the Coxeter diagram H4 can be represented by the quaternionic elements of the binary icosahedral group I [10] which are given as the set of conjugacy classes in Table 1. The sets of conjugacy classes are indeed the orbits of the Coxeter group W (H3)= [p, p¯] [p, p¯]∗, p I (3) { ⊕ ∈ } where we use the notation [8] [a, b]: q aqb and [a, b]∗: q aqb.¯ (4) → → Table 1: Conjugacy classes of the binary icosahedral group I represented by quaternions for the cases of 1 being fixed. (Sets of conjugacy classes of N elements fixing the elements q are denoted as N(q) where additional subscripts are used to distinguish different sets of the same type.) Orders of elements The sets N(1) of the conjugacy classes denoted with N elements 1 1 2 1 − 1 1 10 12(1)+ : 2 (τ e1 σe3), 2 (τ e2 σe1), 1 ± ± ± ± 2 (τ e3 σe2) 1 ± ± 1 5 12(1) : 2 ( τ e1 σe3), 2 ( τ e2 σe1), − 1 − ± ± − ± ± 2 ( τ e3 σe2) 1 − ± ± 1 10 12(1)+′ : 2 (σ e1 τe2), 2 (σ e2 τe3), 1 ± ± ± ± 2 (σ e3 τe1) 1 ± ± 1 5 12(1)′ : 2 ( σ e1 τe2), 2 ( σ e2 τe3), − 1 − ± ± − ± ± 2 ( σ e3 τe1) 1 − ± ± 1 6 20(1)+ : 2 (1 e1 e2 e3), 2 (1 τe1 σe2), 1 ± ± ± 1 ± ± 2 (1 τe2 σe3), 2 (1 τe3 σe1) 1 ± ± ±1 ± 3 20(1) : 2 ( 1 e1 e2 e3), 2 ( 1 τe1 σe2), − 1 − ± ± ± 1 − ± ± 2 ( 1 τe2 σe3), 2 ( 1 τe3 σe1) − ± ± 1 − ± ± 4 30(1) : e1, e2, e3, 2 ( σe1 τe2 e3), ±1 ± ± ± 1 ± ± ( σe2 τe3 e1), ( σe3 τe1 e2) 2 ± ± ± 2 ± ± ± 3 1+√5 1 √5 2 In Table 1 τ = 2 , σ= −2 and they satisfy the relations τ = τ + 1, σ2 = σ +1, τ+σ=1, τσ= 1. The notation # (1) denote the set of ele- ments of the conjugacy classes which− also represent the orbits of the Coxeter group W (H3) fixing the element 1. 1 5 Let b = (τ + σe1 + e2) 12(1)+ and b = 1 represent the two simple 2 ∈ − roots of the Coxeter group H2 with unit norm. Then the quaternions e3b = 1 5 ( e1 + σe2 + τe3) and e3b = e3 represent the simple roots of the Coxeter 2 − − diagram H′ orthogonal to the roots of H2. The Coxeter diagram of H2 H′ 2 ⊕ 2 with its simple roots are depicted in Figure 1. The reflection generators at these simple roots are given by [11] [b, b]∗, [1, 1]∗, [e3b, e3b]∗, [e3, e3]∗. (5) − − − − These generators will lead to the following 20 quaternionic roots of the Cox- eter diagram H2 H′ ⊕ 2 m m H2 H′ = b , e3b , (m =0, 1, ..., 9). (6) ⊕ 2 { } This set of quaternions form a maximal subgroup, the dicyclic group of order 20, in the binary icosahedral group I [4]. The generators in (5) generate the group W (H2) W (H′ ), each of which, is a dihedral group of order 10. × 2 It is more appropriate to represent each dihedral group with one rotation generator and one reflection generator which can be written as W (H2)= [b, b], [1, 1]∗ ; W (H′ )= [¯b, b], [e3, e3]∗ . (7) { − } 2 { − } The direct product group is of order 100. One may simply note that the generator [e3, 1] or [1, e3] permutes the simple roots of H2 H′ . Actually ⊕ 2 these generators leave the Cartan matrix of the diagram H2 H′ invariant. ⊕ 2 Without loss of generality we take the generator [e3, 1] as the new generator which generates the cyclic group C4 of order 4. The group C4 is the normalizer of the group W (H2) W (H′ ) in the Coxeter group W (H4). An extension of × 2 the group W (H2) W (H′ ) by the group C4 is a semi-direct product of the × 2 two groups which is the group Aut(H2 H′ ) W (H2) W (H′ ) : C4 (8) ⊕ 2 ≈{ × 2 } ⊕ ′ ′ α1 = b α2 = −1 α1 = e3b α2 = −e3 Figure 1: Coxeter diagram of H2 H′ with quaternionic simple roots. ⊕ 2 4 of order 400 and is one of the five maximal subgroups of the Coxeter group W (H4) [11]. The group elements now can be written more concisely as Aut(H2 H′ )= [p, q] [p, q]∗; p, q (H2 H′ ) . (9) ⊕ 2 { ⊕ ∈ ⊕ 2 } It is represented by the pairs of quaternions taking values from the root sys- tem of the Coxeter diagram H2 H2′ .