Prototiles and Tilings from Voronoi and Delone cells of the Root Lattice A_n

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.


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(Penrose, & 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 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 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(Voronoi, , 1909 and Delaunay [Delaunay, 1929[Delaunay, , 1938a 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 and series.We pointed out that the 2-dimensional faces of the Voronoi cells of the root lattice are identical rhombuses and the root lattice 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 via its Coxeter-Dynkin diagram by introducing + 1 -dimensional orthonormal vectors ( = 1,2, … , + 1). We identify its dihedral subgroup of order 2( + 1) generated by two reflection generators whose products define the Coxeter element of order ℎ = + 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 by introducing the non-orthogonal vectors with 1 + 2 + ⋯ + +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 = − +1 . Sec. 4 deals with the projections of the Voronoi and Delone 2-faces and give classifications of the prototiles induced by some lattices and Sec. 5 discusses periodic and aperiodic tilings with examples. Sec. 6 includes the concluding remarks.

The Root Lattice and its Coxeter-Weyl Group
We briefly introduce the root lattice . The Coxeter-Dynkin diagram describing the Coxeter-Weyl group and its extended diagram representing the affine Coxeter group are shown in  The nodes represent the simple roots ( = 1, 2, … , ) of the associated Lie algebra ( + 1) of rank n where the norm of the roots are given by ( , ) = 2 and the angle between adjacent roots is 120 0 and non-adjacent roots are orthogonal to each other. These properties define the Cartan matrix (Gram matrix in the lattice terminology) by the relation = 2( , ) . (1) The fundamental weight vectors are defined by the relation ( , ) = where is the Kronecker-delta and they are related to each other by the relations where the scalar product of the fundamental weights define the matrix elements of the inverse Cartan matrix ( , ) = ( −1 ) . The root lattice is defined as the set of vectors = ∑ =1 , ∈ ℤ . Among many other tessellations the lattice is tiled with the parallelotope generated by the simple roots which, in an orthogonal base, constitute the rows of the generator matrix M where the Cartan matrix can be written as = . This implies that the volume of the parallelotope is √ 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 * is the inverse Cartan matrix in (1) and the lattice is generated by the , ∈ ℤ with the volume of the parallelotope is 1/√ . Note that the root lattice is a sublattice of the weight lattice, ⊂ * . Let , ( = 1, 2, … , ) denotes the reflection generator with respect to the hyperplane orthogonal to the simple root i  which operates on an arbitrary vector as = − 2( , ) . ( It transforms a fundamental weight vector as = − . The reflection generators generate the Coxeter group ( ) =< 1 , 2 , … , |( ) = 1 > which is also called the Coxeter-Weyl group for the crystallographic Coxeter group. Adding another generator, usually denoted by 0 , describing the reflection with respect to the hyperplane bisecting the highest weight vector ( 1 + ) we obtain the Affine Coxeter group ( ), the infinite discrete group denoted by ( ) =< 0 , 1 , 2 , … , > . The generator 0 acts as a translation which translates zero vector (0) to the highest weight ( 1 + ) of the adjoint representation of the associated Lie algebra. The Coxeter-Weyl group ( ) is the normal subgroup of the affine Weyl group ( ) which is the full symmetry of the root lattice . The relations between the Voronoi and the Delone cells of the root and weight lattices of have been studied by L. Michel (Michel, 1995(Michel, , 1997). An arbitrary polytope of the point group will be the orbit ( ) =: (∑ =1 ) , ∈ ℤ. With this notation the root polytope generated by ( ) will be denoted either by (10 … 01) or simply by ( 1 + ) . The dual polytope of the root polytope (Koca et al., 2018b) is the Voronoi cell (0) which is the union of the orbits of the fundamental polytopes ( ) , ( = 1, 2, … , ), ( 1 ) ⋃( 2 ) … ⋃( ) .
One can prove that the Voronoi cell of tiles the weight lattice * . Each fundamental polytope ( ) 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 ( 1 ) + ( ) represent the 2( + 1) simplexes centered around the vertices ( 1 ) and ( ) of the Voronoi cell (0). Similarly, ( 2 ) + ( −1 ) constitutes the vertices of the ambo-simplexes centralizing the vertices ( 2 ) and the ( −1 ) of the Voronoi cell (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.

An alternative technique for the Delone and Voronoi Cells of
It is a general practice to work with the set of orthonormal vectors , ( = 1, 2, … , + 1), ( , ) = . Then the simple roots of the associated Lie algebra, in other words, the basis vectors of the lattice can be defined as = − +1 , ( = 1, 2, … , ) which are permuted by the generators as : ↔ +1 implying that the Coxeter-Weyl group ( ) ≃ +1 is isomorphic to the the symmetric group of order ( + 1)! permuting the + 1 vectors. The fundamental weights are given as Let us introduce the vectors which represent the vertices of the n-simplex ( 1 ) and denote the edges of the Voronoi cell. In terms of the fundamental weights they are given by Their magnitudes are all the same ( , ) = +1 and the angle between any pair is = cos −1 (− 1 ). They are orthogonal to the vector 0 , ( , 0 ) = 0 and therefore the vectors represent the dimensional space orthogonal to the vector 0 . The orbits of the fundamental weights can be easily written as Note that this number is 2 less than the number of vertices 2 +1 of a cube in + 1 dimensions. The vertices of the Voronoi cell of the + 1dimensional cubic lattice +1 is given by Projection of this cube along its diagonal ± 1 2 ( 1 + 2 + ⋯ + +1 ) into dimensional vector space leads to the Voronoi cell (0) of the root lattice . Therefore, the projection of the root lattice via cut and project technique where the Voronoi cell (0) of the lattice plays the role of window is equivalent to the projection of the + 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 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.

6
If we proceed to study the emergence of the Voronoi cell (0) of from the Voronoi cell of the + 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 which follows from the equation (6) as ( , ) = − 1 +1 , ≠ ). One of the − 1 dimensional facet of the Voronoi cell (0) of the root lattice is a − 1 dimensional rhombohedron generated by the edges ( = 2, … , ) whose center is represented by half the highest weight vector ( 1 ± 2 ± ⋯ ± − +1 ) of the Voronoi cell of the cubic lattice. Applying the Coxeter-Weyl group ( ) on these vectors one generates all ( − 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 generate the identical rhombuses. Any facet of the Voronoi cell (0) of the root lattice can be obtained as the projections of the facets of the cube whose vertices are given by (10).
Each polytope ( ) 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 1 = 1 . One can generate an equilateral triangle by applying the group < 1 , 2 > whose edges are represented by the vectors 1 − 2 , 2 − 3 , 3 − 1 . Other faces and the corresponding edges are generated by applying the group elements ( ) leading to the set of edges − , a vector in the root system. The argument is valid for any other Delone polytope ( ) . Therefore the set of edges of an arbitrary two dimensional face (an equilateral triangle) of the Delone cells can be simply given by − , − , − , ≠ ≠ = 1, 2, … , + 1.

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 ( ) 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 ( ) can be written as where = 1, 2, … , are the Coxeter exponents and ℎ = + 1 is the Coxeter number. Earlier, (Koca, et al., 2014(Koca, et al., , 2015 we have defined a set of orthonormal set of vectors subject to an arbitrary orthogonal transformation We define the Coxeter plane ∥ determined by the pair of vectors ̂1and ̂ and the rest as the orthogonal space ⊥ . A dihedral subgroup of order 2ℎ of the group ( ) can be defined, up to a conjugation, by two generators < 1 , 2 |( 1 2 ) ℎ = 1 > where 1 = 1 3 5 … , 2 = 2 4 6 … and each generator consists of the products of the set of commuting generators of the Coxeter-Weyl group ( ) (Carter, 1972;Humphreys, 1990). The Coxeter element can be defined up to a conjugation by the product = 1 2 satisfying ℎ = 1. The Coxeter element permutes in some order. Since all the Coxeter elements are conjugates of each other it is always possible to obtain the Coxeter element ′ = 1 2 3 … = −1 , ∈ ( ) which permutes in the cyclic order ′ : 1 → 2 → ⋯ → +1 → 1 , accordingly, the simple roots in (12) must be replaced by ′ = . With this modification and using the freedom of defining ̂ up to a further orthogonal transformation the components of the vectors in the parallel plane ∥ can be obtained as a complex number ( ) ∥ = 2 ℎ , where = 1,2, … , + 1 and = √ 2 ℎ . We will drop the factor as it is an overall factor and has no significant meaning in our further discussions. It is then clear that the scalar product (( ) ∥ , ( ) ∥ ) = cos ( ( − )2 ℎ ) will determine the nature of the projected rhombus onto the plane ∥ . The projected rhombuses will determine the Voronoi tiling of the plane ∥ . The tiles projected from the Delone cells will be the triangles whose edge lengths are the magnitudes of the vectors ( − ) ∥ , ( − ) ∥ , ( − ) ∥ and, after deleting the common factor , they will turn out to be respectively 2 sin ( Note the triangles obtained from (13)  satisfying 1 + 2 + 3 = ℎ.
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.

Examples of prototiles and tiling schemes
In what follows we discuss a few h-fold symmetric tilings with rhombuses and the triangles.

Figure 3
A typical rhombic face of the rhombic dodecahedron (note that edges are represented by the vectors 2 and 3 with an angle 109.5 0 ).
The first four vectors ( 1 ) 3 = { 1 , 2 , 3 , 4 } represent the vertices of the first tetrahedron and the permutation group 4 ≅ ( 3 ) is the tetrahedral group of order 24. The set of vectors (− 1 , − 2 , − 3 , − 4 ) represent the second tetrahedron ( 3 ) 3 . Actually the Dynkin diagram symmetry ( 1 ↔ − 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) Therefore projection of the rhombus in Fig. 3 is a square of unit length determined by the vectors ( 2 ) ∥ = (−1, 0) and ( 3 ) ∥ = (0, −1). Actual lengths of the projected vectors are = 1 √2 if we had rescaled the length. Projection of the Voronoi cell onto the plane ∥ is shown in Fig. 4 which is invariant under the dihedral group 4 .We can pick up any two vectors in (15) say ( 1 ) ∥ and ( 2 ) ∥ then the lattice in ∥ is a square lattice with a general vector ∥ = 1 ( 1 ) ∥ + 2 ( 2 ) ∥ with 1 , 2 ∈ ℤ . By projecting the weight lattice 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 3 * tiled by the rhombic dodecahedron.

5.1b. Projection of the Delone cells of the root lattice
The lattice vector of the root lattice can be written as = ∑ 3 =1 = ∑ 3 =1 with , ∈ ℤ such that ∑ 3 =1 = 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 ∥ = 1 ( 1 ) ∥ + 2 ( 2 ) ∥ , with 1 + 2 =even integer as shown in Fig. 6.

Figure 6
Projection of the root lattice as a square lattice made of right triangles.
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 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.
The first two terms in (18) are represented by the vector ± 1 2 0 → 0 in 4-space and the remaining vertices of 5-dimensional cube decompose as 5 + 10 + 10 + 5 representing the vertices of the Voronoi cell ( 1 ) 4 ⋃( 2 ) 4 ⋃( 3 ) 4 ⋃( 4 ) 4 . In terms of the vectors , the vectors can also be written as (compare with (6) This implies that the set of vertices of the 5-dimensional cube on the right of equation ( . The vectors in the direction (± ) ∥ 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 ∥ form four intersecting pentagons defined by de Bruijn as

Figure 9
A patch of the Penrose rhombic tiling by projection of the lattice 4 * . The four types of vertices are distinguished by numbers as stated in the text.

Figure 10
Darts and kites obtained from Robinson triangles.
A patch of Penrose tiling by darts and kites is shown Fig. 11.

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 4 .
In appendix A we have displayed the detailed relations between the Voronoi cells of 4 and 5-dimensional cubic lattice. In brief, the square face of 5-dimensional cube projects onto rhombic face of the Voronoi cell (0) in 4-dimensional space and all 3dimensional 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 4 by the Delone cells lead to triangular tilings which are absent in the 5dimensional cubic lattice projection.

5.3a. Projection of the Voronoi cell of the root lattice
Here the Coxeter number is ℎ = 6 and the dihedral subgroup of the Coxeter-Weyl group ( 5 ) is 6 of order 12. The Delone ploytopes centered around the origin are ( 1 ) 5 = −( 5 ) 5 , ( 2 ) 5 = −( 4 ) 5 , ( 3 ) 5 whose 2-dimensional facets are equilateral triangles. The Voronoi cell (0) of the lattice 5 is a polytope in 5dimensional space and is the the disjoint union of the above Delone cells. Its 4dimensional facets are the 4-dimensional rhombohedra implying that the 2dimensional facets are the rhombuses generated by the pair of vectors ( , ) = − 1 6 , ≠ = 1, 2, … ,6. In the Coxeter plane the scalar product would read All possible rhombuses turn out to be the one with interior angles (60 0 , 120 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. 12.

Figure 12
The rhombille tiling with a 6 symmetry.

5.3b. Projection of the Delone cells of the root lattice
The prototiles obtained from the 2-dimensional Delone faces are of three types of triangles with angles ( 3 , 3 , 3 ), ( 6 , 6 , 2 3 ) , ( 6 , 3 , 2 ) . 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 5 and some patches made by these prototiles.

5.5a. Prototiles from the projection of the Voronoi cell (0) of the root lattice
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 ( ) and a square ( 2 , 2 ). A patch of aperiodic tiling is illustrated in Fig. 16.

Figure 16
The prototiles from the projection of the Voronoi cell of the root lattice 7 and an 8-fold symmetric patch of the tiling.

5.5b. Projection of the Delone cells of the root lattice
Following the formula (13) and substituting = 7 we obtain five different triangular prototiles from the projection of the Delone cells of the root lattice 7 . Three of the prototiles are the isosceles triangles with angles ( 8 , 8 , ). 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.17.

Figure 17
Prototiles from the Delone cells of the root lattice 7 and two patches 8-fold symmetric tilings with all prototiles are included.

Figure 18
The four rhombuses illustrated with different colors obtained from the Voronoi cell of 11 and a patch with 12-fold symmetry at the center.

Figure 19
The triangular prototiles from the Delone cells of the root lattice 11 .

Concluding Remarks
Rhombic prototiles usually arise from projection of the higher dimensional cubic lattices +1 because 2-dimensional square faces project onto rhombuses and represent the local 2( + 1)-fold symmetric rhombic aperiodic tilings of the Coxeter plane. Depending on the shift of the Coxeter plane one may reduce the tilings to ( + 1)-fold symmetric aperiodic tilings. The rhombic ( + 1)-fold symmetric aperiodic tilings can also be obtained from the lattice for the latter lattice is a sublattice of the lattice +1 . One may visualize that the lattice +1 projects onto the lattice at first stage as the Voronoi cell of the cubic lattice projects onto the Voronoi cell of the root lattice . So the classification of the rhombic aperiodic tilings of both lattices are the same. An advantage of the root lattice is that it can be tiled by the Delone cells with triangular 2-dimensional faces. Projections of the Delone cells of the root lattice 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. 22 angle = − cos −1 ( 1 4 ) ≅ 104.5 0 . Each 3-dimensional cubic facet of 5-dimensional cube projects into a rhombohedral facet of the Voronoi cell (0) as shown in the following example. The set of vertices