Prototiles and Tilings from Voronoi and Delone cells of the Root

Transcription

1 Prototiles and Tilings from Voronoi and Delone cells of the Root Lattice A n Nazife Ozdes Koca a), Abeer Al-Siyabi b) Department of Physics, College of Science, Sultan Qaboos University P.O. Box 36, Al-Khoud, 123 Muscat, Sultanate of Oman, Mehmet Koca c) Department of Physics, Cukurova University, Adana, Turkey, and Ramazan Koc d) Department of Physics, Gaziantep University, 27310, Gaziantep, Turkey ABSTRACT We exploit the fact that two-dimensional facets of the Voronoi and Delone cells of the root lattice A n in n-dimensional space are the identical rhombuses and equilateral triangles respectively.the prototiles obtained from orthogonal projections of the Voronoi and Delaunay (Delone) cells of the root lattice of the Coxeter-Weyl group W(a n ) are classified. Orthogonal projections lead to various rhombuses and several triangles respectively some of which have been extensively discussed in the literature in different contexts. For example, rhombuses of the Voronoi cell of the root lattice A 4 projects onto only two prototiles: thick and thin rhombuses of the Penrose tilings. Similarly the Delone cells tiling the same root lattice projects onto two isosceles Robinson triangles which also lead to Penrose tilings with kites and darts. We point out that the Coxeter element of order h = n + 1 and the dihedral subgroup of order 2n plays a crucial role for h-fold symmetric aperiodic tilings of the Coxeter plane. After setting the general scheme we give examples leading to tilings with 4-fold, 5- fold, 6-fold,7-fold, 8-fold and 12-fold symmetries with rhombic and triangular tilings of the plane which are useful in modelling the quasicrystallography with 5-fold, 8-fold and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice A 3 whose Wigner-Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an h = 4 fold symmetry. Keywords: Lattices, Coxeter-Weyl groups, Voronoi and Delone cells, tilings by rhombi and triangles a) electronic-mail: nazife@squ.edu.om b) electronic-mail: s21168@student.squ.edu.om c) electronic-mail: mehmetkocaphysics@gmail.com ; retired professor d) electronic-mail: koc@gantep.edu.tr 1

2 1. Introduction Discovery of a 5-fold symmetric material (Shectman et al., 1984) has lead to growing interest in quasicrystallography. For a review see for instance (DiVincenzo & Steinhardt, 1991; Janot, 1993; Senechal, 2009). Aperiodic tilings of the plane with dihedral point symmetries or icosahedral symmetry in three dimensions have been the central research area of mathematicians and mathematical physicists to explain the quasicrystallography. For an excellent review see for instance (Baake & Grimm, 2013) and (Grunbaum & Shephard, 1987). There have been three major approaches for aperiodic tilings. The first class, perhaps, is the intuitive approach like Penrose tilings (Penrose, 1974 & 1978) which also exibits the inflation technique developed later. The second approach is the projection technique of higher dimensional lattices onto lower dimensions pioneered by de Bruijn (de Bruijn, 1981) by projecting the 5-dimensional cubic lattice onto a plane orthogonal to one of the major diagonal of the cube. The others (Duneau & Katz, 1985; Baake, Joseph, Kramer & Schlottmann, 1990; Chen, et al., 1998) studied similar techniques. In particular, Baake et al. (Baake et al., 1990) exemplified the projection of the A 4 lattice. The group theoretical treatment of the projection of n-dimensional cubic lattices has been worked out by the references (Whittaker & Whittaker, 1987; Koca et al., 2015) and a general projection technique on the basis of dihedral subgroup of the root lattices has been proposed by Boyle & Steinhardt (Boyle & Steinhardt, 2016). In a recent article (Koca et al, 2108a) we pointed out that the 2-dimensional facets of the A 4 Voronoi cell projects onto thick and thin rhombuses of the Penrose tilings. The third one is the the model set technique initiated by Meyer (Meyer, 1972) and followed by Lagarias (Lagarias, 1996) and developed by Moody (Moody, 1997). For a detailed treatment see the reference (Baake & Grimm, 2013). The Voronoi (Voronoi, 1908, 1909) and Delaunay [Delaunay, 1929, 1938a,b) cells of the root and weight lattices have been extensively studied in the inspiring book by Conway and Sloane (Conway & Sloane, 1988; chapter 21) and especially in the reference (Conway and Sloane, 1991). The reference (Deza & Grishukhin, 2004) contains detailed discussions and informations about the Delone and Voronoi polytopes of the root lattices. Higher dimensional lattices have been worked out in (Engel, 1986) and the lattices of the root systems whose point grups are the Coxeter- Weyl groups have been studied extensively in the reference (Engel et al., 1994). Numbers of facets of the Voronoi and Delone cells of the root lattice have been also determined by a technique of decorated Coxeter-Dynkin diagrams (Moody & Patera, 1992). In (Koca et al., 2108b) we have worked out the detailed structures of the facets of the Voronoi and the Delone cells of the root and weight lattices of the A n and D n series.we pointed out that the 2-dimensional faces of the Voronoi cells of the root lattice A n are identical rhombuses and the root lattice A n is tiled with Delone cells with 2-dimensional faces of equilateral triangles. Basic information about the regular polytopes as the orbits of the Coxeter-Weyl groups have been worked out in the references (Coxeter, 1973) and (Grunbaum, 1967). The Lie algebras derived from the Coxeter-Weyl groups have been studied extensively in the references (Bourbaki, 1968), (Carter, 1971) and (Humphreys, 1992). We organize the paper as follows. In Sec. 2 we introduce the root lattice A n via its Coxeter-Dynkin diagram by introducing n + 1 -dimensional orthonormal vectors 2

3 l i (i = 1,2,, n + 1). We identify its dihedral subgroup of order 2(n + 1) generated by two reflection generators whose products define the Coxeter element of order h = n + 1, where h is called the Coxeter number. We discuss the projection technique employing the eigenvalues and eigenvectors of the Cartan matrix. Sec. 3 reviews the Voronoi and Delone cells of the lattice A n by introducing the non-orthogonal vectors k i with k 1 + k k n+1 = 0 representing the edges of the Voronoi cells. We also prove that the edges of the Delone cells are represented by the root vectors α i = k i k i+1. Sec. 4 deals with the projections of the Voronoi and Delone 2-faces and give classifications of the prototiles induced by some A n lattices and Sec. 5 discusses periodic and aperiodic tilings with examples. Sec. 6 includes the concluding remarks. 2. The Root Lattice A n and its Coxeter-Weyl Group We briefly introduce the root lattice A n. The Coxeter-Dynkin diagram describing the Coxeter-Weyl group and its extended diagram representing the affine Coxeter group are shown in Fig.1. α 0 α 1 α 2 α n 1 α n α 1 α 2 α n 1 α n (a) (b) Figure 1 (a) Coxeter-Dynkin diagram of a n, (b) Extended Coxeter Dynkin diagram of a n. The nodes represent the simple roots α i (i = 1, 2,, n) of the associated Lie algebra SU(n + 1) of rank n where the norm of the roots are given by (α i, α i ) = 2 and the angle between adjacent roots is and non-adjacent roots are orthogonal to each other. These properties define the Cartan matrix (Gram matrix in the lattice terminology) by the relation C ij = 2(α i, α j ) (α j, α j ). (1) The fundamental weight vectors ω i are defined by the relation (α i, ω j ) = δ ij where δ ij is the Kronecker-delta and they are related to each other by the relations ω i = (C 1 j ) ij α j, α i = j C ij ω j (2) where the scalar product of the fundamental weights define the matrix elements of the inverse Cartan matrix (ω i, ω j ) = (C 1 ) ij. The root lattice A n is defined as the set of vectors p = n i=1 b i α i, b i Z. Among many other tessellations the lattice is tiled with the parallelotope generated by the simple roots α i which, in an orthogonal base, constitute the rows of the generator matrix M where the Cartan matrix can be written as C = MM T. This implies that the volume of the parallelotope is detc which is called the volume of the lattice and also representing the volume of the Voronoi cell. The Gram matrix of the the weight lattice A n is the inverse Cartan matrix in (1) and the lattice is generated by the 3

4 fundamental weights ω i and an arbitrary lattice vector can be written as the linear n combination of the weight vectors q = i=1 c i ω i, c i Z with the volume of the parallelotope is 1/ detc. Note that the root lattice is a sublattice of the weight lattice, A n A n. Let r i, (i = 1, 2,, n) denotes the reflection generator with respect to the hyperplane orthogonal to the simple root i which operates on an arbitrary vector λ as r i λ = λ 2(λ,α i) α (α i,α i ) i. (3) It transforms a fundamental weight vector ω j as r i ω j = ω j α i δ ij. The reflection generators generate the Coxeter group W(a n ) =< r 1, r 2,, r n (r i r j ) m ij = 1 > which is also called the Coxeter-Weyl group for the crystallographic Coxeter group. Adding another generator, usually denoted by r 0, describing the reflection with respect to the hyperplane bisecting the highest weight vector (ω 1 + ω n ) we obtain the Affine Coxeter group W(A n ), the infinite discrete group denoted by W(A n ) =< r 0, r 1, r 2,, r n >. The generator r 0 acts as a translation which translates zero vector (0) to the highest weight (ω 1 + ω n ) of the adjoint representation of the associated Lie algebra. The Coxeter-Weyl group W(a n ) is the normal subgroup of the affine Weyl group W( A n ) which is the full symmetry of the root lattice A n. The relations between the Voronoi and the Delone cells of the root and weight lattices of A n have been studied by L. Michel (Michel, 1995, 1997). An arbitrary polytope of the point group n will be the orbit W(a n )q =: ( i=1 c i ω i ) an, c i Z. With this notation the root polytope generated by W(a n ) will be denoted either by (10 01) an or simply by (ω 1 + ω n ) an. The dual polytope of the root polytope (Koca et al., 2018b) is the Voronoi cell V(0) which is the union of the orbits of the fundamental polytopes (ω i ) an, (i = 1, 2,, n), (ω 1 ) an (ω 2 ) an (ω n ) an. (4) One can prove that the Voronoi cell of A n tiles the weight lattice A n. Each fundamental polytope (ω i ) an is a Delone cell centered around the origin. Delone cells tile the root lattice such that each polytope centralizes one vertex of the Voronoi cell. For example, the set of vertices (ω 1 ) an + (ω n ) an represent the 2(n + 1) simplexes centered around the vertices (ω 1 ) an and (ω n ) an of the Voronoi cell V(0). Similarly, (ω 2 ) an + (ω n 1 ) an constitutes the vertices of the ambo-simplexes centralizing the vertices (ω 2 ) an and the (ω n 1 ) an of the Voronoi cell V(0) and so on. The root lattice is tiled with the Delone cells by translation.we will elaborate this point in the following sections by giving examples. 3. An alternative technique for the Delone and Voronoi Cells of A n It is a general practice to work with the set of orthonormal vectors l i, (i = 1, 2,, n + 1), ( l i, l j ) = δ ij. Then the simple roots of the associated Lie algebra, in other words, the basis vectors of the lattice can be defined as α i = l i l i+1, (i = 1, 2,, n) which are permuted by the generators as r i : l i l i+1 implying that the Coxeter-Weyl group W(a n ) S n+1 is isomorphic to the the symmetric group of order (n + 1)! permuting the n + 1 vectors. The fundamental weights are given as ω 1 = 1 n + 1 (n l 1 l 2 l 3 l n+1 ); 4

5 ω 2 = 1 n+1 ((n 1) l 1 + (n 1) l 2 2 l 3 2 l n+1 ); ω n = 1 n+1 ( l 1 + l 2 + +l n n l n+1 ). (5) Let us introduce the vectors k i = l 0 n+1 + l i, l 0 =: l 1 + l l n+1, k 1 + k k n+1 = 0, (6) which represent the vertices of the n-simplex (ω 1 ) an and denote the edges of the Voronoi cell. In terms of the fundamental weights they are given by k 1 = ω 1, k 2 = ω 2 ω 1, k 3 = ω 3 ω 2,, k n = ω n ω n 1, k n+1 = ω n. (7) Their magnitudes are all the same (k i, k i ) = n n+1 and the angle between any pair is θ = cos 1 ( 1 n ). They are orthogonal to the vector l 0, (k i, l 0 ) = 0 and therefore the vectors k i represent the n dimensional space orthogonal to the vector l 0. The orbits of the fundamental weights can be easily written as (ω 1 ) an = (k 1 ) an ; (ω 2 ) an = (k 1 + k 2 ) an ; (ω 3 ) an = (k 1 + k 2 + k 3 ) an ; (ω n ) an = (k 1 + k k n ) an = ( ω 1 ) an (8) each of which represents the vertices of the Delone cells centered around the origin. It is clear that the generators of the Coxeter-Weyl group permute the vectors k i as r i : k i k i+1. The list of vectors representing each orbit can be easily determined by permutations of the vectors k i. The union of all these orbits represent the vertices of the Voronoi cell of the root lattice A n. The number of vertices of the Voronoi cell of A n is given by n i=1 ( n+1 = 2 n+1 2. (9) i ) Note that this number is 2 less than the number of vertices 2 n+1 of a cube in n + 1 dimensions. The vertices of the Voronoi cell of the n + 1dimensional cubic lattice B n+1 is given by 1 2 (± l 1 ± l 2 ± ± l n+1 ). (10) Projection of this cube along its diagonal ± 1 2 ( l 1 + l l n+1 ) into n dimensional vector space leads to the Voronoi cell V(0) of the root lattice A n. Therefore, the projection of the root lattice A n via cut and project technique where the Voronoi cell V(0) of the lattice A n plays the role of window is equivalent to the projection of the n + 1 dimensional cubic lattice. As we will discuss later it will lead to the classification of the rhombic prototiles. The advantage of working with the lattice A n is that the root lattice is tiled with the Delone cells and their projections will lead to the classification of the triangular prototiles such as the Robinson triangles. 5

6 If we proceed to study the emergence of the Voronoi cell V(0) of A n from the Voronoi cell of the n + 1 dimensional cubic lattice we have a few additional remarks. First of all we note that the 2-dimensional square face of the cubic lattice projects into the rhombic 2-dimensional face of the Voronoi cell of A n which follows from the equation (6) as (k i, k j ) = 1, i j). One of the n 1 dimensional facet of the Voronoi n+1 cell V(0) of the root lattice A n is a n 1 dimensional rhombohedron generated by the edges k i (i = 2,, n) whose center is represented by half the highest weight vector 1 (ω ω n ) = 1 (k 2 1 k n+1 ) = 1 (l 2 1 l n+1 ). It can be shown explicitly that the (n 1)-dimensional rhombohedron centered at 1 (l 2 1 l n+1 ) is the projection of the cubic facet whose vertices are given by 1 2 ( l 1 ± l 2 ± ± l n l n+1 ) of the Voronoi cell of the cubic lattice. Applying the Coxeter-Weyl group W(a n ) on these vectors one generates all (n 1)-dimensional rhombohedra whose edges are represented by the set of vectors of (6). Therefore all two-dimensional faces of the Voronoi cell are identical as any pair of the vectors k i generate the identical rhombuses. Any facet of the Voronoi cell V(0) of the root lattice A n can be obtained as the projections of the facets of the cube whose vertices are given by (10). Each polytope (ω i ) an is a Delone cell centered around the origin as we stated earlier. Two dimensional faces of the Delone cells are equilateral triangles. To give an example let us take ω 1 = k 1. One can generate an equilateral triangle by applying the group < r 1, r 2 > whose edges are represented by the vectors k 1 k 2, k 2 k 3, k 3 k 1. Other faces and the corresponding edges are generated by applying the group elements W(a n ) leading to the set of edges k i k j, a vector in the root system. The argument is valid for any other Delone polytope (ω i ) an. Therefore the set of edges of an arbitrary two dimensional face (an equilateral triangle) of the Delone cells can be simply given by k i k j, k j k l, k l k i, i j l = 1, 2,, n The Projections of the faces of the Voronoi and Delone cells We use eigenvalues and eigenvectors of the Cartan matrix of the Coxeter-Weyl group W(a n ) to define a set of orthonormal vectors consisting of n vectors. This is useful to define the Coxeter plane for the projection. Eigenvales and the eigenvectors of the Cartan matrix of the group W(a n ) can be written as λ i = 2(1 + cos m iπ ); X h ji = ( 1) j+1 sin j m iπ, j = 1, 2,, n, (11) h where m i = 1, 2,, n are the Coxeter exponents and h = n + 1 is the Coxeter number. Earlier, (Koca, et al., 2014, 2015) we have defined a set of orthonormal set of vectors subject to an arbitrary orthogonal transformation x i = 1 n α λ j=1 j X ji, (x i, x j) = δ ij, i = 1, 2,, n. (12) i We define the Coxeter plane E determined by the pair of vectors x 1and x n and the rest as the orthogonal space E. A dihedral subgroup of order 2h of the group W(a n ) can be defined, up to a conjugation, by two generators < R 1, R 2 (R 1 R 2 ) h = 1 > where R 1 = r 1 r 3 r 5 6

7 , R 2 = r 2 r 4 r 6 and each generator consists of the products of the set of commuting generators of the Coxeter-Weyl group W(a n ) (Carter, 1972; Humphreys, 1990). The Coxeter element can be defined up to a conjugation by the product R = R 1 R 2 satisfying R h = 1. The Coxeter element R permutes k i in some order. Since all the Coxeter elements are conjugates of each other it is always possible to obtain the Coxeter element R = r 1 r 2 r 3 r n = grg 1, g W(a n ) which permutes k i in the cyclic order R : k 1 k 2 k n+1 k 1, accordingly, the simple roots in (12) must be replaced by α j = gα j. With this modification and using the freedom of defining x i up to a further orthogonal transformation the components of the vectors k i in the parallel plane E can be obtained as a complex number (k j ) = ce ij2π h, where j = 1,2,, n + 1 and c = 2. We will drop the factor c as it is an h overall factor and has no significant meaning in our further discussions. It is then clear that the scalar product ((k j ), (k l ) ) = cos ( (j l)2π ) will determine the nature of the projected rhombus onto the plane E. The projected rhombuses will determine the Voronoi tiling of the plane E. The tiles projected from the Delone cells will be the triangles whose edge lengths are the magnitudes of the vectors (k i k j ), (k j k l ), (k l k i ) and, after deleting the common factor c, they will turn out to be respectively 2 sin ( (i j)π ), 2 sin ( (j l)π ), 2 sin ( (l i)π ). (13) h h h h Note the triangles obtained from (13) has the angles satisfying m 1 + m 2 + m 3 = h. ( m 1π h, m 2π h, m 3π ), m h i Z As it is clear from this discussion that the prototiles of tilings from projection of the Voronoi cells will be the rhombuses and the prototiles from projection of the Delone cells will be various triangles. In what follows we will illustrate this fact with some examples.the rhombic and triangular prototiles are classified in Table 1 and Table 2 respectively. 7

8 Table 1 Classification of the rhombic prototiles projected from Voronoi cells of the root lattice A n (rhombuses with angles ( 2πm, π 2πm ), m N). h h Root h # of prototiles Rhombuses with pairs of angles Lattice A ( π, π ), (square) 2 2 A ( 2π, 3π ), 5 5 (4π, π ), (thick and thin rhombuses 5 5 A A of Penrose) ( π 3, 2π 3 ) ( 2π 7, 5π 7 ), (4π 7, 3π 7 ), (6π 7, π 7 ) A ( π, 3π ), 4 4 (π, π ), (Amman-Beenker tiles) 2 2 A ( 2π 9, 7π 9 ), (4π 9, 5π 9 ), (6π 9, 3π 9 ), (8π 9, π 9 ) A ( 2π, 3π ), 5 5 (π, 4π ), (thick and thin rhombuses 5 5 A A of Penrose) ( 2π 11, 9π 11 ), (4π 11, 7π 11 ), (6π 11, 5π 11 ) ( π 6, 5π 6 ), (π 3, 2π 3 ), (π 2, π 2 ) 8

9 Table 2 Classification of the triangular prototiles projected from Delone cells of the root lattice A n. Root Lattice h # of prototiles Triangles denoted by triple natural numbers (m 1, m 2, m 3 ) A (1, 1, 1) A (1, 1, 2) A (1, 1, 3), (1, 2, 2), (Robinson triangles) A (1, 1, 4), (1, 2, 3), (2, 2, 2) A (1, 1, 5), (1, 2, 4), (1, 3, 3), (2, 2, 3), (Danzer tiles) A (1, 1, 6), (1, 2, 5), (1, 3, 4), (2, 2, 4), (2, 3, 3) A (1, 1, 7), (1, 2, 6), (1, 3, 5), (1, 4, 4), (2, 2, 5), (2, 3, 4), (3, 3, 3) A (1, 1, 8), (1, 2, 7), (1, 3, 6), (1, 4, 5), (2, 2, 6), (2, 3, 5), (2, 4, 4), (3, 3, 4) A (1, 1, 9), (1, 2, 8), (1, 3, 7), (1, 4, 6), (1, 5, 5), (2, 2, 7), (2, 3, 6), (2, 4, 5), (3, 3, 5), (3, 4, 4) A (1, 1, 10), (1, 2, 9), (1, 3, 8), (1, 4, 7), (1, 5, 6), (2, 2, 8), (2, 3, 7), (2, 4, 6), (2, 5, 5), (3, 3, 6), (3, 4, 5), (4, 4, 4) 5. Examples of prototiles and tiling schemes In what follows we discuss a few h-fold symmetric tilings with rhombuses and the triangles. 5.1a. Projection of the Voronoi cell of A 3 The Voronoi cell of the root lattice A 3 commonly known as the the f.c.c. lattice is the rhombicdodecahedron which tiles the 3-dimensional Euclidean space. The vertices of the rhombic dodecahedron can be represented as the union of two tetrahedra and an octahedron (see Fig. 2): (ω 1 ) a3 (ω 2 ) a3 (ω 3 ) a3 = {(±k 1, ±k 2, ±k 3, ±k 4 ); (k 1 + k 2, k 2 + k 3, k 3 + k 4, k 4 + k 1, k 1 + k 3, k 2 + k 4 )}. (14) 9

10 Figure 2 The rhombic dodecahedron (Voronoi cell of the root lattice A 3 ) A typical rhombic face of the rhombic dodecahedron is determined by the set of four vertices k 1 = ω 1, ω 3 = k 4, k 1 + k 2 = ω 2, k 1 + k 3 = r 2 ω 2. ω 2 ω 1 ω 3 r 2 ω 2 Figure 3 A typical rhombic face of the rhombic dodecahedron (note that edges are represented by the vectors k 2 and k 3 with an angle ). The first four vectors (ω 1 ) a3 = {k 1, k 2, k 3, k 4 } represent the vertices of the first tetrahedron and the permutation group S 4 W(a 3 ) is the tetrahedral group of order 24. The set of vectors ( k 1, k 2, k 3, k 4 ) represent the second tetrahedron (ω 3 ) a3. Actually the Dynkin diagram symmetry (k 1 k 4 ) extends the tetrahedral group to the octahedral group implying that two tetrahedra form a cube. Since the Voronoi cell is dual to the root polytope it is face transitive and invariant under the octahedral group of order 48. The last six vectors in (14) represent the vertices of an octahedron which split as under the dihedral subgroup D 4 of order 8. The last two vectors k 1 + k 3 and k 2 + k 4 = (k 1 + k 3 ) form a doublet under the dihedral subgroup and they are perpendicular to the Coxeter plane E determined by the pair of vectors (x 1, x 3). The projected components of the vectors k i in the plane E are given by (k 1 ) = (0, 1), (k 2 ) = ( 1, 0), (k 3 ) = (0, 1), (k 4 ) = (1, 0). (15) Therefore projection of the rhombus in Fig. 3 is a square of unit length determined by the vectors (k 2 ) = ( 1, 0) and (k 3 ) = (0, 1). Actual lengths of the projected vectors are c = 1 if we had rescaled the length. Projection of the Voronoi cell onto 2 the plane E is shown in Fig. 4 which is invariant under the dihedral group D 4.We can pick up any two vectors in (15) say (k 1 ) and (k 2 ) then the lattice in E is a square lattice with a general vector q = m 1 (k 1 ) + m 2 (k 2 ) with m 1, m 2 Z. 10

11 Figure 4 Projection of the rhombic dodecahedron onto the Coxeter plane E. By projecting the weight lattice A 3 tiled by the unit cell rhombic dodecahedron we obtain the square lattice illustrated in Fig. 5. Figure 5 Square lattice obtained as the projection of the lattice A 3 tiled by the rhombic dodecahedron. 5.1b. Projection of the Delone cells of the root lattice A 3 The lattice vector of the root lattice can be written as p = i=1 b i α i = 3 3 i=1 n i k i with b i, n i Z such that i=1 n i = even. This shows that the root lattice is a sublattice of the weight lattice. Equilateral triangles of the Delone cells (remember, tetrahedra and octahedra tiling the root lattice have identical 2-faces as equilateral triangles) project as right triangles, any two constitute a square of length 2. One can check that the projected Delone cells form a square lattice with a general vector p = m 1 (k 1 ) + m 2 (k 2 ), with m 1 + m 2 =even integer as shown in Fig Figure 6 Projection of the root lattice as a square lattice made of right triangles. 11

12 It is clear that it is a sublattice of the square lattice obtained from the weight lattice and it is rotated by 45 0 with respect to the projected square lattice of the Voronoi cell. With this example we have obtained a sublattice whose unit cell is invariant under the dihedral group D 4. This is expected because the dihedral group is a crystallographic group.two overlapping square lattices have been depicted as in Fig. 7. Figure 7 Projections of two square lattices. 5.2a. Projection of the Voronoi cell of the root lattice A 4 The Voronoi cell of the root lattice tiles the 4-dimensional Euclidean reciprocal space which is the weight lattice A 4 4 represented by a general vector q = i=1 c i ω i = 5 4 i=1 m i k i = i=1 n i k i with c i, m i, n i Z. Remember that the sum of the vectors 5 = 0.The union of the Delone polytopes centered around the origin i=1 k i ( ω 1 ) a4 ( ω 2 ) a4 ( ω 3 ) a4 ( ω 4 ) a4 (16) constitutes the Voronoi cell with 30 = vertices. It is a polytope comprised of the rhombohedral facets. A typical 3-facet of the Voronoi cell is a rhombohedron with six rhombic faces (Koca et al., 2018b) as depicted in Fig. 8. Its vertices are given by (k 1, k 1 + k 2, k 1 + k 3, k 1 + k 4, k 5, ( k 2 + k 5 ), ( k 3 + k 5 ), ( k 4 + k 5 )) and centered at 1 2 (k 1 k 5 ). Figure 8 A 3-dimensional rhombohedron, the rhombohedral facet of the Voronoi cell of the root lattice A 4. The Voronoi cell is formed by 20 such rhombohedra whose rhombic 2-faces are generated by the pairs of vectors satisfying the scalar product (k i, k j ) = 1, i j = 5 1,,5 and (k i, k i ) = 4. It is perhaps useful to discuss some of the technicalities of 5 12

13 the projection with this example. We choose the Coxeter plane E generated by the unit vectors x 1and x 4 determined from x i = 1 4 α λ j=1 j X ji with α j = gα j where g = i r 1 r 2. The main purpose of choosing this set of orthonormal vectors is that the Coxeter element R = r 1 r 2 r 3 r 4 permutes the vectors k i in the cyclic order. The dihedral group D 5 =< R 1, R 2 (R 1 R 2 ) 5 = 1 > of order 10 is a non-crystallographic group where the generators are given by R 1 = r 1 r 2 r 1 r 3 r 2 r 1, R 2 = r 1 r 2 r 1 r 4. Since the unit vectors x i are defined up to an orthogonal transformation we can choose the components of the vectors in the plane E as (k j ) = (l j ) = ξ j = e ijπ 5, j = 1,,5 5 i=1 satisfying ξ j = 0. The vectors ( k j ) form a pentagon while the vectors (k j ) form a pentagram since the angle between two successive vectors (k j ) is 4π 5. A matrix representation of the Coxeter element R in 4-dimensional space is given by R = ( cos 2π 5 sin 2π sin 2π 5 cos 2π 5 cos 4π 5 sin 4π sin 4π 5 cos 4π 5 ). (17) A general vector of the weight lattice A 4 projects as q = i=1 m i (k i ). We have dropped the scale vector c = 2 since it is irrelevant in what follows. Note that the 5 rhombic 2-faces of the Voronoi cell project onto the plane E as two Penrose rhombuses, thick and thin, for the acute angles between any pair of the vectors (k j ) = e ij2π 5 are either 72 0 or De Bruijn (de Bruijn, 1981) proved that the projection of the 5-dimensional cubic lattice onto E leads to the Penrose rhombus tiling of the plane with the thick and tin rhombuses as prototiles. The same technique is extended to an arbitrary cubic lattice (Whittaker & Whittaker, 1987). It is not a surprise that we obtain the same tiling from the A 4 lattice because it is a sublattice of the cubic lattice B 5 (Koca et al., 2015). The Coxeter- Weyl group W(b 5 ) is of order ! and its Coxeter number is h = 10. Normally one expects a 10-fold symmetric tiling from the projection of the 5- dimensional cubic lattice. The Voroni cell of B 5 has 32 vertices 1 2 (± l 1 ± l 2 ± l 3 ± l 4 ± l 5 ) which decompose under the Coxeter-Weyl group W(a 4 ) as 32 = (18) The first two terms in (18) are represented by the vectors ± 1 2 l 0 0 in 4-space and the remaining vertices of 5-dimensional cube decompose as representing the vertices of the Voronoi cell (ω 1 ) a4 (ω 2 ) a4 (ω 3 ) a4 (ω 4 ) a4. In terms of the vectors l i, the vectors k i can also be written as (compare with (6)) 5 13

14 k 1 = 3 10 l ( l 1 l 2 l 3 l 4 l 5 ) k 2 = 3 10 l ( l 1 + l 2 l 3 l 4 l 5 ) k 3 = 3 10 l ( l 1 l 2 + l 3 l 4 l 5 ) k 4 = 3 10 l ( l 1 l 2 l 3 + l 4 l 5 ) k 5 = 3 10 l ( l 1 l 2 l 3 l 4 + l 5 ). (19) This implies that the set of vertices of the 5-dimensional cube on the right of equation (19) represent a 4-simplex with 5 vertices in 4-dimensional space; any three k i represent the vertices of an equilateral triangle, any four constitute a tetrahedron.the 4-simplex is one of the Delone cells centered at the origin. Similarly pairwise and triple combinations of the vectors k i constitute the Delone cells composed of octahedra and tetrahedra as 3-dimensional facets. De Bruijn proved that the integers n i in the 5 5 expression p = i=1 n i (k i ) take values i=1 n i {1, 2, 3, 4}. The vectors in the direction (±k i ) count positive and negative depending on its sign. A patch of Penrose tiling with the numbering of vertices is shown in Fig. 9. Note that the projection of the Voronoi cell onto E form four intersecting pentagons defined by de Bruijn as V 1 = (( ω 1 ) a4 ) = V 4 = (( ω 4 ) a4 ), V 2 = (( ω 2 ) a4 ) = V 3 = (( ω 3 ) a4 ). Figure 9 A patch of the Penrose rhombic tiling by projection of the lattice A 4. The four types of vertices are distinguished by numbers as stated in the text. 14

15 5.2b. Projection of the root lattice A 4 by Delone cells The Delone polytopes (ω 1 ) a4 = (ω 4 ) a4, (ω 2 ) a4 = (ω 3 ) a4, tile the root lattice such that the centers of the Delone cells correspond to the vertices of the Voronoi cells. Take, for example, the 4-simplex (ω 1 ) a4 = {k 1, k 2, k 3, k 4, k 5 } and (ω 4 ) a4 = { k 1, k 2, k 3, k 4, k 5 }. Adding vertices of these two simplexes one obtains 10 simplexes of Delone cells centered around 10 vertices of the Voronoi cell V(0) consisting of 30 vertices altogether of (ω 1 ) a4 (ω 2 ) a4 (ω 3 ) a4 (ω 4 ) a4. If we add the vector k 1 to the vertices of the Delone cell (ω 4 ) a4 we obtain the vertices of the Delone cell {0, k 1 k 2, k 1 k 3, k 1 k 4, k 1 k 5 } whose center is at the vertex k 1 of the Voronoi cell V(0). When the vertices of the polytope (ω 2 ) a4 = {k i + k j, i j = 1, 2,, 5} are added to the vertices of (ω 3 ) a4 we obtain 20 Delone cells centered arount the vertices of (ω 2 ) a4 and (ω 3 ) a4. For example when we add the vector k 1 + k 2 to the vertices of (ω 3 ) a4 we obtain the Delone cell with 10 vertices centered around the the vertex k 1 + k 2 of the Voronoi cell V(0) {0, k 1 k 3, k 1 k 3 + k 2 k 4, k 1 k 4 + k 2 k 5, k 2 k 5, k 2 k 3, k 1 k 4, k 1 k 3 + k 2 k 5, k 2 k 4, k 1 k 5 }. (20) Note that the vertices of the Delone cells are either the root vectors or their linear combinations as expected. One can repeat the procedure to complete the construction that all Delone cells centered around the vertices of the Voronoi cell V(0) tile the root lattice by lattice translation. Using (13) for n = 4 and j k l = 1, 2,,5 we obtain two isosceles Robinson triangles with edges (2sin π, 2sin π, 2sin 2π ) and (2sin 2π, 2sin 2π, 2sin 4π ). The darts and kites obtained from the Robinson triangles are shown in Fig. 10. Figure 10 Darts and kites obtained from Robinson triangles. A patch of Penrose tiling by darts and kites is shown Fig

16 Figure 11 A patch of Penrose tiling with darts and kites. There are two inflation techniques using the Robinson triangles for aperiodic tiling. One version is the Penrose-Robinson tiling (PRT) and the other is the Tubingen triangle tiling (TTT). For further discussions see (Baake & Grimm, 2013). On the other hand one can construct darts and kites of Penrose tiling by folding each triangle along one of its equal edges. One can find various tilings by darts and kites in the literature other than the original paper. The advantage of our approach is that both the triangular and rhombic tilings of the plane follows from the projections of the Delone and Voronoi cells of the same root lattice A 4. In appendix A we have displayed the detailed relations between the Voronoi cells of A 4 and 5-dimensional cubic lattice. In brief, the square face of 5-dimensional cube projects onto rhombic face of the Voronoi cell V(0) in 4-dimensional space and all 3- dimensional cubic facets project onto the 3-dimensional rhombohedra. Of course the result of both projections lead to the same tilings. However tessellations of the root lattice A 4 by the Delone cells lead to triangular tilings which are absent in the 5- dimensional cubic lattice projection. 5.3a. Projection of the Voronoi cell of the root lattice A 5 Here the Coxeter number is h = 6 and the dihedral subgroup of the Coxeter-Weyl group W(a 5 ) is D 6 of order 12. The Delone ploytopes centered around the origin are (ω 1 ) a5 = (ω 5 ) a5, (ω 2 ) a5 = (ω 4 ) a5, (ω 3 ) a5 whose 2-dimensional facets are equilateral triangles. The Voronoi cell V(0) of the lattice A 5 is a polytope in 5- dimensional space and is the the disjoint union of the above Delone cells. Its 4- dimensional facets are the 4-dimensional rhombohedra implying that the 2- dimensional facets are the rhombuses generated by the pair of vectors (k i, k j ) = 1 6, i j = 1, 2,,6. In the Coxeter plane the scalar product would read (( k i ), ( k j ) ) = cos ( 2π(j i) ). (21) 6 All possible rhombuses turn out to be the one with interior angles (60 0, ). The tiling with this rhombus is known in the literature as the rhombille tiling which is used for the study of spin structures in diatomic molecules based on the Ising models. The rhombille tiling is depicted in Fig

17 Figure 12 The rhombille tiling with a D 6 symmetry. 5.3b. Projection of the Delone cells of the root lattice A 5 The prototiles obtained from the 2-dimensional Delone faces are of three types of triangles with angles ( π, π, π ), (π, π, 2π ), (π, π, π ). By employing the inflation technique one can embed the smaller prototiles into the inflated ones. See for example (Nischke & Danzer 1996). A patch of 6-fold symmetric tiling by three triangles is shown in Fig. 13. Figure 13 Three triangular prototiles from Delone cells of the root lattice A 5 and some patches made by these prototiles. 17

18 5.4a. Prototiles from the projection of the Voronoi cell V(0) of the root lattice A 6 Since the dihedral group is D 7 of order 14 one expects 7-fold symmetric patches of the tilings. The rhombic prototiles from the Voronoi cell V(0) can be obtained from the formula (( k i ), ( k j ) ) = cos ( 2π(j i) ), i j = 1, 2,,7. This would lead to three 7 types of rhombuses with angles ( π, 6π ), 7 7 (2π, 5π ), 7 7 (3π, 4π ). There are numerous 7 7 discussions on the 7- fold syymetric rhombic tiling with three rhombuses. One particular construction is based on the projection technique from 7-dimensional cubic lattice (Whittaker& Whittaker, 1988). See also the website of professor Gerard t Hooft. A patch of this tiling is illustrated in Fig. 14. Figure 14 7-fold symmetric rhombic tiling. 5.4b. Projection of the Delone cells of the root lattice A 6 The prototiles obtained from 2-dimensional Delone faces are of four types of triangles with angles ( π, π, 5π ), (2π, 2π, 3π ), (3π, 3π, π ) and (π, 2π, 4π ). The inflation tilings by the last three triangles are widely known as Danzer tiling. The number of prototiles obtained by projection here is the same as the tiles obtained from the tangents of the deltoid (Nischke & Danzer, 1996). They have illustrated the inflations involving only three of the triangles. In Fig.15 we illustrate a patch of 7- fold symmetric tiling comprising of all four triangles. Figure 15 Two 7-fold symmetric patches obtained from the projection of the Delone cells of the root lattice A 6. 18

19 5.5a. Prototiles from the projection of the Voronoi cell V(0) of the root lattice A 7 The famous 8-fold symmetric rhombic tiling by two prototiles is known as Amman- Beenker tiling (Grunbaum & Shephard, 1987). Interestingly enough we obtain the same prototiles from the projection of the Voronoi cell V(0) of the root lattice A 7. The point group of the aperiodic tiling is the dihedral group D 8 of order 16 and the prototiles are generated by the pair of vectors satisfying the scalar product ((k i ), (k j ) ) = cos ( 2π(j i) ), i j = 1, 2,,8. We obtain two Amman-Beenker 8 prototiles one rhombus with angles ( π, 3π ) and a square 4 4 (π, π ). A patch of aperiodic 2 2 tiling is illustrated in Fig. 16. Figure 16 The prototiles from the projection of the Voronoi cell of the root lattice A 7 and an 8-fold symmetric patch of the tiling. 5.5b. Projection of the Delone cells of the root lattice A 7 Following the formula (13) and substituting n = 7 we obtain five different triangular prototiles from the projection of the Delone cells of the root lattice A 7. Three of the prototiles are the isosceles triangles with angles ( π 8, π 8, 6π 8 ), (2π 8, 2π 8, 4π 8 ), (3π 8, 3π 8, 2π 8 ) and the other two are the triangles with angles ( π, 2π, 5π ), (π, 3π, 4π ). So far as we know no one has studied the tilings of the plane with these prototiles. Two patches of 8-fold symmetric tilings with these prototiles are depicted in Fig

20 Figure 17 Prototiles from the Delone cells of the root lattice A 7 and two patches 8-fold symmetric tilings with all prototiles are included. 5.6a. Prototiles from projection of the Voronoi cell of the root lattice A 11 The 12-fold symmetric rhombic tiling has 3 prototiles of rhombuses with interior angles ( π, 5π ), 6 6 (π, 2π ), and 3 3 (π, π ). A patch of 12-fold symmetric tiling by rhombuses is depicted in Fig.18. Figure 18 The four rhombuses illustrated with different colors obtained from the Voronoi cell of A 11 and a patch with 12-fold symmetry at the center. 5.6b. Prototiles from the projection of the Delone cells of the root lattice A 11 The number of prototiles in this case is 12. As the rank of the Coxeter-Weyl group is increasing the number of the triangular prototiles are also increasing. The 12 triangles are given as the set of integers (m 1, m 2, m 3 ) defined in Sec. 4 as (1, 1, 10), (1, 2, 9), (1, 3, 8), (1, 4, 7), (1, 5, 6), (2, 2, 8), (2, 3, 7), (2, 4, 6), (2, 5, 5), (3, 3, 6), (3, 4, 7), (4, 4, 4). A patch of 12-fold symmetric tiling is depicted in Fig

21 Figure 19 The triangular prototiles from the Delone cells of the root lattice A Concluding Remarks Rhombic prototiles usually arise from projection of the higher dimensional cubic lattices B n+1 because 2-dimensional square faces project onto rhombuses and represent the local 2(n + 1)-fold symmetric rhombic aperiodic tilings of the Coxeter plane. Depending on the shift of the Coxeter plane one may reduce the tilings to (n + 1)-fold symmetric aperiodic tilings. The rhombic (n + 1)-fold symmetric aperiodic tilings can also be obtained from the lattice A n for the latter lattice is a sublattice of the lattice B n+1. One may visualize that the lattice B n+1 projects onto the lattice A n at first stage as the Voronoi cell of the cubic lattice projects onto the Voronoi cell of the root lattice A n. So the classification of the rhombic aperiodic tilings of both lattices are the same. An advantage of the root lattice A n is that it can be tiled by the Delone cells with triangular 2-dimensional faces. Projections of the Delone cells of the root lattice A n allow the classifications of the triangular aperiodic tilings. A systematic study of the projections of the Delone and Voronoi cells of the root and weight lattices of the simply laced ADE Lie algebras may lead to more interesting prototiles and aperiodic tilings. The present paper was concerned only with the aperiodic rhombic and triangular tilings of the root lattice exemplifying the 5-fold, 8-fold and 12-fold symmetric aperiodic tilings as they represent some quasicrystallographic structures. Appendix A: 5-dimensional cube and the Voronoi cell V(0) of A 4 Vertices of the Voronoi cell of 5-dimensional cubic lattice are given by the vectors 1 2 ( ±l 1 ± l 2 ± l 3 ± l 4 ± l 5 ). The cube is cut by hyperplanes orthogonal to the vector l 0 at four levels which siplit the cube into four Delone cells of A 4 centered around the origin. Using (8) they can be grouped as (ω 1 ) a4 = (ω 4 ) a4, (ω 2 ) a4 = (ω 3 ) a4. Vertices of a typical 2-dimensional face of the 5-dimensional cube can be taken as 1 2 ( l 1 + l 2 + l 3 + l 4 + l 5 ), 1 2 ( l 1 + l 2 + l 3 + l 4 + l 5 ), 1 2 ( l 1 l 2 + l 3 + l 4 + l 5 ), 1 2 ( l 1 l 2 + l 3 + l 4 + l 5 ). The edges of the square are represented by l 1 and l 2. In 4-dimensional space l 0 projects into the origin by (12) and therefore l 1 and l 2 are replaced by the vectors k 1 and k 2 constituting a rhombus with obtuse 21

22 angle θ = cos 1 ( 1 4 ) Each 3-dimensional cubic facet of 5-dimensional cube projects into a rhombohedral facet of the Voronoi cell V(0) as shown in the following example. The set of vertices 1 2 ( l 1 ± l 2 ± l 3 ± l 4 l 5 ) is a 3-dimensional cube centered at the point 1 2 ( l 1 l 5 ). This cube projects into the 3-dimensional rhombohedral facet of the Voronoi cell V(0) of A 4 as shown in Fig. 8 where the vertices of the rhombohedron are given by k 1 = 3 10 l ( l 1 l 2 l 3 l 4 l 5 ), k 5 = 3 10 l ( l 1 + l 2 + l 3 + l 4 l 5 ), k 1 + k 2 = 1 10 l ( l 1 + l 2 l 3 l 4 l 5 ), k 1 + k 3 = 1 10 l ( l 1 l 2 + l 3 l 4 l 5 ), k 1 + k 4 = 1 10 l ( l 1 l 2 l 3 + l 4 l 5 ), k 1 + k 2 + k 3 = ( k 4 + k 5 ) = 1 10 l ( l 1 + l 2 + l 3 l 4 l 5 ), k 1 + k 2 + k 4 = ( k 3 + k 5 ) = 1 10 l ( l 1 + l 2 l 3 + l 4 l 5 ), k 1 + k 3 + k 4 = ( k 2 + k 5 ) = 1 l ( l 2 1 l 2 + l 3 + l 4 l 5 ). (A.1) All the rhombohedral facets of the Voronoi cell V(0) are obtained by permutations. Centers of the rhombohedral facets are the halfs of the roots of the root system. This also explains why the Voronoi cell is the dual polytope of the root polytope. References Baake, M., Joseph, D., Kramer, P. & Schlottmann, M. (1990). J. Phys. A: Math. & Gen. 23, L1037-L1041. Baake, M. & Grimm, U. (2013). Aperiodic Order, Volume 1: A Mathematical Invitation, Cambridge University Press, Cambridge. Boyle, L. & Steinhardt, P. J. (2016). Coxeter Pairs, Amman Patterns and Penrose-like Tilings. arxiv: Bourbaki, N. (1968). Groupes et Algèbres de Lie. Chap. IV-VI, Actualités Scientifiques et Industrielles (Paris: Hermann) 288; English translation: Lie Groups and Lie algebras (Springer 2 nd printing, 2008). Carter, R. W. (1971). Simple Groups of Lie Type. John Wiley & Sons, New York. Chen, L., Moody, R. V. & Patera, J. (1998). Quasicrystals and Discrete Geometry, edited by J. Patera, pp Fields Institute Monographs, Vol. 10. Providence, RI: American Mathematical Society. Conway, J. H. & Sloane, N. J. A. (1988). Sphere Packings, Lattices and Groups. Springer-Verlag New York Inc. Conway, J. H. & Sloane, N. J. A. (1991). The cell structures of certain lattices. Miscellanea Mathematica, ed. Hilton, P., Hirzebruch, F. & Remmert, R.( New York: Springer). Coxeter, H. S. M. (1973). Regular Polytopes, Third Edition, Dover Publications. 22

23 de Bruijn, N. G. (1981). Algebraic theory of Penrose's non-periodic tilings of the plane. Nederl. Akad. Wetensch. Proceedings Ser. A 84 (=Indagationes Math. 43), 38. Delaunay, N. B. (1929). Sur la partition reguilere de l espace a 4-dimensions, Izv. Akad. Nauk SSSR Otdel. Fiz.-Mat. Nauk. pp and Delaunay, N. B. (1938a). Geometry of positive quadratic forms, Usp. Mat. Nauk. 3, Delaunay, N. B. (1938b). Usp. Mat. Nauk. 4, Deza, M. & Grishukhin,V. (2004). Non-rigidity degrees of root lattices and their duals, Geometriae Dedicata 104, 15-24, Kluwer. Di Vincenzo, D. & Steinhardt, P. J. (1991). Quasicrystals: the state of the art, World Scientific Publishers, Singapore. Duneau, M. & Katz, A. (1985). Phys. Rev. Lett. 54, Engel, P. (1986). Geometric Crystallography: an Axiomatic Introduction to Crystallography. Dordrecht: Springer. Engel, P., Michel, L. & Senechal, M. (1994). Lattice Geometry, preprint IHES /P/04/45. Grunbaum, B. (1967). Convex Polytopes, Wiley, N.Y. Grunbaum, B. & Shephard, G. C. (1987). Tilings and Patterns, Freeman, New York. Humphreys, J. E. (1992). Reflection Groups and Coxeter Groups. Cambridge University Press. Janot, C.(1993). Quasicrystals: a primer, Oxford, Oxford University Press. Koca, N. O., Koca, M. & Al-Siyabi, A. (2018a). Int. J. Geom. Methods Mod. Phys.15 (4), Koca, M., Ozdes Koca, N., Al-Siyabi, A. & Koc, R. (2108b). Acta Cryst. A74, Koca, M., Koca, N. O. & Koc, R. (2014). Int. J. Geom. Meth. Mod. Phys. 11, Koca, M., Koca, N. & Koc, R. (2015). Acta Cryst A71, Lagarias, J. C. (1996). Commun. Math. Phys. 179, Meyer, Y. (1972). Algebraic Numers and Harmonic Analysis, North Holland, Amsterdam. Michel, L. (1995). Bravais classes, Voronoi cells, Delone cells, Symmetry and Structural Properties of Condensed Matter, ed. Lulek, T., Florek, W. & Walcerz, S., Zajaczkowo, Worl Sci., Singapore, p Michel, L. (1997). Complete description of the Voronoi cell of the Lie algebra A n weight lattice. On the bounds for the number of d-faces of the n-dimensional Voronoi cells, IHES/P/97/53. Moody, R.V. & Patera, J. (1992). J. Phys. A: Math. Gen. 25, Moody, R. V. (1997). Meyer Sets and their duals, in The Mathematics of Long Range Aperiodic Order (ed. R. Moody), NATO ASI Series C 489, Kluwer, Dordrecht, pp Nischke, K. P. & Danzer, L. (1996). Discrete Comput. Geom 15, Penrose, R. (1974). The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10, 266. Penrose, R. (1978). Pentaplexity, Eureka 39, 16. Shechtman, D., Blech, I., Gratias, D. & Cahn, J.W. (1984). Phys. Rev. Lett. 53, Senechal, M. (1995). Quasicrystals an Geometry,Cambridge University Press, Cambridge. Voronoi, G. (1908). Recherches sur les paralleloedres primitifs I. Proprietes generals des paralleloedres. J. reine angew. Math.133,

24 Voronoi, G. (1909). Recherches sur les paralleloedres primitifs II. Domaines de formes quadratiques corresondant aux different types de Nouvelles applications des parame etres continus `a l` a th eorie des formes quadratiques primitifs. J. reine angew. Math.136, Whittaker, E. J. W. & Whittaker, R. M. (1987). Acta Cryst. A44,