A Highly Symmetric Four-Dimensional Quasicrystal * Veit Elser and N. J. A. Sloane AT&T Bell Laboratories Murray Hill, New Jersey

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1 A Highly Symmetric Four-Dimensional Quasicrystal * Veit Elser and N. J. A. Sloane AT&T Bell Laboratories Murray Hill, New Jersey 7974 Abstract A quasiperiodic pattern (or quasicrystal) is constructed in real four-dimensional Euclidean space, having the noncrystallographic reflection group [3,3,5] of order 44 as its point group. It is obtained as a projection of the eight-dimensional lattice E, and has as a cross-section a three-dimensional quasicrystal with icosahedral symmetry. Physics Abstracts Classification Numbers: 65E, 22, 24. * This paper appeared in J. Phys. A: Math. Gen., Vol. 2 (97), pp

2 . Introduction We use a version of de Bruijn s projection method ( ) ( 5 ) (as developed especially in its group-theoretic sense by Kramer and Neri ( 2 ) ) to construct a quasiperiodic pattern (or quasicrystal, for short) in four-dimensional Euclidean space, having the reflection group G = [ 3, 3, 5 ] as its point group. This group is a noncrystallographic group of order 44, and is the symmetry group of the regular four-dimensional polytope known as the 6-cell, usually denoted by the Schlȧ. fli symbol {3, 3, 5} ( 6 ),( 7 ). G is the largest finite real four-dimensional group. ( ) The new quasicrystal is obtained as a projection of the -dimensional lattice E ; it has as a cross-section a three-dimensional quasicrystal with icosahedral symmetry. 2. The lattice E and the group G = [ 3, 3, 5 ] Let V be -dimensional (-D) Euclidean space with an orthonormal basis e,..., e. The lattice E consists of all points P = Σ i = v i e i where Σ i = v i is even and either all v i Z or all v i Z + 2. ( 6 ), ( 9 ), ( ) (Z denotes the integers.) The point group G of this lattice is the Weyl group W(E ), of order = G has a subgroup isomorphic to G ; to make this subgroup visible we give a second definition of E using quaternions. The unit icosians consist of the 2 quaternions obtained from (±,,, ), (±, ±, ±, ± ), (, ±, ± σ, ± τ) 2 2 by allowing any choice of signs and any even permutation of the coordinates, where σ = 2 ( 5 ), τ = 2 ( + 5 ), and (a, b, c, d) is an abbreviation for the

3 - 2 - quaternion a + bi + c j + dk. The set ( of icosians consists of all finite sums of unit ( ),( ),( 2 ) icosians. A quaternion q = a + bi + c j + dk has quaternionic conjugate q _ = a bi c j dk. The quaternionic norm of an icosian q = (a, b, c, d) is Q(q) = q q _ = a 2 + b 2 + c 2 + d 2, which is a real number of the form x + y 5 where x,y are rational; the Euclidean norm of q is N(q) = x + y, which can be shown to be a nonnegative rational number. The unit icosians are precisely the icosians of quaternionic norm. The icosians equipped with the quaternionic norm lie in a real 4-D space, in which the unit icosians form the 2 vertices of the regular polytope {3, 3, 5} ( 6 ),( ). But the icosians equipped with the Euclidean norm lie in a real -D space, and form a lattice isomorphic to the E lattice. ( ),( 2 ) We shall use the particular isomorphism between ( and E defined by the mappings shown in Table I. There are 24 icosians of Euclidean norm, consisting of the unit icosians and σ times the unit icosians, and these correspond to the 24 minimal vectors of the E lattice. This correspondence can be deduced from Table I, and is written out in full in Table. of Ref. and Table I of Ref. 2. We define the group G = [ 3, 3, 5 ] to consist of all transformations q r q s and q r q _ s, q (, of the icosians, where r and s are unit icosians. ( ) These transformations preserve Q(q) and N(q). There are 2 choices for each of r and s, but r, s and r,s produce the same transformations, so G has order 44. G is generated by the particular ( ),( 2 ) transformations

4 - 3 - L i : q iq, L ω : q ωq, B: q q _, R i : q qi, R ω : q qω, where ω = 2 (, τ, σ, ) ( (and ω 3 = ). The identification of ( with E shows that G acts on V as a subgroup of G. Each element A G is represented by an matrix (a i j ), where A(e i ) = Σ j = a i j e j ; the a i j may be obtained from Table I. Consider L i for example. From Table I we find that e = 4 ( 2 σ, σ, σ, σ), i e = 4 (σ, 2 σ, σ, σ) = 2 (e 2 + e 4 + e 6 e ), which gives the first row of the matrix L i = 2. There are two 4-D subspaces X and X of V that are invariant under the action of G. The space X is spanned by the vectors f, f 2, f 3, f 4, and X by f 5, f 6, f 7, f, where f j = Σ i = φ i j e i and Φ = (φ i j ) is the orthogonal matrix

5 - 4 - Φ = c(i + σ H) c(i σ H) c (I + τ H), c (I τ H) where I = I 4 = diag{,,, }, c = = , H = _ σ 2, and tilde is the algebraic conjugation operation that changes the sign of 5 (i.e. interchanges σ and τ). Thus c = ( τ) 2 = To see that X, X are invariant under G we represent points of V in the f i basis. If P = Σ i = v i e i = Σ i = v i f i V, where v = (v,..., v ), v = (v,..., v ), then v = v Φ. This changes A = (a i j ) G to A = Φ tr A Φ. The generators for G in this basis are L i = diag{ T, T, T, T }, R i = diag{ T, T, T, T }, T =, L ω = diag{ U, Ũ }, R ω = diag{ U 2, Ũ 2 }, where U = 2 τ σ τ σ σ τ σ, U τ 2 = _ 2 τ σ τ σ σ τ σ, τ and B = diag{ +,,,, +,,, }. In this basis the matrices have the form g g, where the g are a 4-D representation of G ; this shows that X and X are invariant subspaces.

6 - 5 - The projection map π X from V to X sends P = Σ i = v i e i V to Q = Σ i = w i e i X, where w = v Π, Π = Φ I Φ tr = σ I H. 5 H σ I In the f i basis π X simply sends P = Σ i = v i f i to Q = 4 Σ i = v i f i X. The projection map π X from V to X is described by the matrix Π, and sends P = Σ i = v i f i to Q = Σ i = 5 v i f i X. We note that E has only the origin in common with either of the spaces X or X. 3. The Voronoi cell of E and its projection onto X The Voronoi cell Ω of E is defined by Ω = { Q V: Q Q P for all P E }. The definition of the new quasicrystal involves the 4-D polytope Ψ = π X (Ω) obtained by projecting Ω onto the subspace X. The Voronoi cell Ω is a convex -D polytope, described in Refs.,, 3. It has 944 vertices, shown in the e i basis in Table II. (All permutations of the coordinates are permitted. The third column indicates when all sign combinations are permitted, or if there must be an even or odd number of minus signs.) The polytope Ψ is the convex hull of the projection of these vertices onto X. As discussed in 2, to project onto X we postmultiply these vectors by Φ and take the last four coordinates.

7 - 6 - From this it can be shown (we omit the details) that Ψ has 72 vertices. There are 2 vertices with coordinates τ 2 c (±2,,, ), τ 2 c (±, ±, ±, ± ), τ 2 c (, ±, ± τ, ± σ) (with all choices of signs and any even permutation of the coordinates), which are obtained from the projections of the first three rows of Table II. These are the 2 vertices of a copy of the polytope {3, 3, 5}, having edge-length τ c = , circumradius R = τ 2 c = , inradius R 3 = 2 3/2 τ 4 c =. 9..., and in which the distance from the center to the midpoint of a 2-D face is R 2 = 3 /2 τ 3 c = (cf. Ref. 6, pp. 57, 293). The other 6 vertices arise from the projections of the last four rows of Table II, and have coordinates τ 2 c (± τ 2, ± τ 2, ±, ), 3 τ 2 c (± 5, ± τ, ± τ, ), 3 τ 2 c (± τ 2, ± τ, ± τ, ± τ ), 3 τ 2 c (± 5, ±, ±, ± ), 3 τ 2 c (±2, ±2,, ), τ 2 c (±2, ±, ± τ, ± τ ), 3 3 τ 2 c (± τ, ± τ, ± τ, ± τ 2 ) 3 (again with all choices of signs and any even permutation of the coordinates). These are the 6 vertices of a copy of the reciprocal polytope {5, 3, 3} (the 2-cell), having edge-length 2c /3 = , circumradius R = 2 3/2 3 τ 2 c =. 9..., and in which the distance from the center to the midpoint of an edge is R = 3 /2 τ 3 c = (cf. Ref. 6, pp. 57, 293).

8 - 7 - Thus Ψ is the convex hull of reciprocal (and concentric) polytopes {3, 3, 5} and {5, 3, 3}, arranged so that the midpoints of the edges of the {5, 3, 3} pass through the centers of the triangular 2-D faces of the {3, 3, 5}. Ψ is a 4-D analog of the triacontahedron encountered in the investigation of 3-D quasicrystals, the convex hull of concentric polyhedra {3, 5} (an icosahedron) and {5, 3} (a dodecahedron) arranged so that the midpoints of their edges coincide. 4, 5 4. The four-dimensional quasicrystal # The 4-D quasicrystal # is obtained by projecting the lattice E onto the subspace X, subject to the requirement that the projection onto X lies in the polytope Ψ: # = { π X (P) : P E, π X (P) Ψ }. Properties of the quasicrystal. (i) # is invariant under a point group (fixing the origin) isomorphic to G = [ 3, 3, 5 ]. To see this we work in the f i basis. Any element of G occurs as the top left corner of some matrix g g. If u # there is a vector v Ψ = π X (Ω) such that (u,v) E. Then (u,v) g g = (ug, vg ) E. But Ω has the same symmetries as E, so vg Ψ, and hence ug #. (ii) # is closed under multiplication by τ. Proof. Consider the map from V to V defined by the matrix

9 - - S = H tr I H in the e i basis, and by S = Φ tr SΦ = diag{τ, τ, τ, τ, σ, σ, σ, σ} in the f i basis. This map preserves E, S commutes with the elements g of G in the 4-D representation defined in 2, and S, S have determinant and satisfy the equation Z 2 Z I =. The map acts as an inflation on X (increasing lengths) and as a contraction on X (decreasing lengths). In particular, working in the f i basis, suppose u # with (u,v) E. Then the map sends (u,v) to (τ u, σ v) E, and so τ u #. (iii) # is a discrete set of points. In fact if t,u are distinct points of # then t u /( τ 4 c 2 ) = Proof. Let t,u be the projections of distinct points P,Q E, and let R = P Q = Σ i = r i e i, R = π X (R), R 2 = π X (R). Then R (since P Q), and R, R 2 (since no point of E except lies in X or X ). We write α = (r,..., r 4 ), β = (r 5,..., r ), obtain R as the left part of (α, β) Φ, and find R. R = c 2 {2 α. α + 2 β. β + σ( 2 (α β) H (α + β) tr + (α β). (α β) ) }. R 2. R 2 is the algebraic conjugate of R. R. Using the fact that inner products in E are integers it follows that R. R R 2. R 2 = c 2 c 2 n, for some positive integer n. Therefore R. R R 2. R 2 c 2 c 2 = /2. Furthermore, π X (P) and π X (Q) Ψ, so R 2. R 2 ( 2 τ 2 c ) 2, and the result follows.

10 - 9 - No lattice has properties (i) or (ii), and (i)-(iii) show that # is a discrete nonperiodic set of points. (iv) Finding which points are in #. It is not easy to apply the definition of # directly, since it is hard to tell if a point is in Ψ. We know from 3 that Ψ contains a sphere of radius 2 3/2 τ 4 c =. 9..., and is contained in a sphere of radius τ 2 c = It follows that π X (P) is certainly in # if π X (P) <. 9..., and is certainly not in # if π X (P) > This test is enough to show for example that exactly 2 of the 24 minimal vectors of E project into #, producing points c (, ±, ± σ, ± τ), etc., forming a copy of {3, 3, 5}. Similarly exactly 2 of the 26 vectors in E of length 2 project into #, producing points τ c (, ±, ± σ, ± τ), etc., forming a slightly larger {3, 3, 5} concentric with the first. (Of course we know from property (i) that # is highly symmetric when viewed from the origin. The neighborhoods of other points are not so symmetric.) There is however a simple algorithm for determining precisely which points are in #. To decide whether a point P = Σ i = v i e i E projects into #, we first express P in terms of the f i as P = Σ i = v i f i, where (v,..., v ) = (v,..., v ) Φ. Let β = (v 5,..., v ). We must test whether β Ψ, or equivalently whether there is an 6,, 3 α = (α, α 2, α 3, α 4 ) such that (α, β) is in the Voronoi cell Ω. It is known that Ω is the set of points P V satisfying P. T (k) (k =,..., 24 ), where the T (k) are the 24 minimal vectors of E. Writing T (k) = (k) Σ i = t i f i,

11 - - γ (k) = (t (k),..., t 4 (k) ), δ (k) = (t 5 (k),..., t (k) ), we conclude that a necessary and sufficient condition for π X (P) to be in # is that there exists a real vector α = (α,..., α 4 ) such that α. γ (k) β. δ (k), for k =,..., 24. This is a question of the existence of a feasible solution to a system of 24 linear inequalities in four variables, a standard (and easy) problem in linear programming. (v) # has a cross-section which is a 3-D quasicrystal with icosahedral symmetry. Proof. We write elements of G with respect to the f i basis. The top left corner matrices (the matrices g, in the notation of 2) corresponding to the elements L i R i and L ω R ω are respectively a = diag{,,, } and b = 2 τ σ τ σ σ τ. + 2 Then a 2 = b 3 = (a b) 5 = I, so a, b generate a subgroup of G that is isomorphic to the alternating group A 5 and fixes the fourth (or f 4 ) coordinate. This 4-D representation of A 5 is the sum of 3-D and -D representations. A cross-section of # in which the coefficient of f 4 is constant therefore has the desired properties. Acknowledgements We thank J. H. Conway for some helpful discussions.

12 - - List of Table Captions Table I. Identification of icosians and E vectors. Table II. Vertices of Ω.

13 - 2 - Table I Identification of icosians and E vectors (,,, ) e + e 5, (,,, ) e 2 + e 6, (,,, ) e 3 + e 7, (,,, ) e 4 + e, (σ,,, ) 2 ( e + e 2 + e 3 + e 4 + e 5 e 6 e 7 e ), (, σ,, ) 2 ( e e 2 + e 3 e 4 + e 5 + e 6 e 7 + e ), (,, σ, ) 2 ( e e 2 e 3 + e 4 + e 5 + e 6 + e 7 e ), (,,, σ) 2 ( e + e 2 e 3 e 4 + e 5 e 6 + e 7 + e ).

14 - 3 - Table II Vertices of Ω Components Signs Number (±, 7 ) all 6 (± 4, 4 )/2 all 2 ( 3, ± 7 )/4 odd 24 (, ± 7 )/3 odd 2 (±2, ± 4, 3 )/3 all 96 ( 3 3, ± 5 )/6 odd 76 (±5, ± 7 )/6 even

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