Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice An

S S symmetry

Article Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice An

1, 1 2, 3 Nazife Ozdes Koca *, Abeer Al-Siyabi , Mehmet Koca † and Ramazan Koc 1 Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud, 123 Muscat, Oman 2 Department of Physics, Cukurova University, 1380 Adana, Turkey 3 Department of Physics, Gaziantep University, 27310 Gaziantep, Turkey * Correspondence: [email protected] Retired. † Received: 27 July 2019; Accepted: 23 August 2019; Published: 28 August 2019

Abstract: The orthogonal projections of the Voronoi and Delone cells of root lattice An onto the Coxeter plane display various rhombic and triangular prototiles including thick and thin rhombi of Penrose, Amman–Beenker tiles, Robinson triangles, and Danzer triangles to name a few. We point out that the symmetries representing the dihedral subgroup of order 2h involving the Coxeter element of order h = n + 1 of the Coxeter–Weyl group an play a crucial role for h-fold symmetric tilings of the Coxeter plane. After setting the general scheme we give samples of patches with 4-, 5-, 6-, 7-, 8-, and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice A3, 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

1. Introduction Discovery of a fivefold symmetric material [1] has led to growing interest in quasicrystallography. For a review, see for instance [2–4]. 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 [5,6].There have been three major approaches for aperiodic tilings. The first class, perhaps, is the intuitive approach, like Penrose tilings [7,8], which also exhibits the inflation–deflation technique developed later. The second approach is the projection technique of higher dimensional lattices onto lower dimensions, pioneered by de Bruijn [9] by projecting the five-dimensional cubic lattice onto a plane orthogonal to one of the space diagonals of the cube. The cut and project technique works as follows. The lattice is partitioned into two complementing subspaces called E and E . k ⊥ Imagine a cylinder based on the component of the Voronoi cell in the subspace E . Project the ⊥ lattice points lying in the cylinder into the space E . Several authors [10–12] have studied similar k techniques. In particular, Baake et al. [11] exemplified the projection of the A4 lattice. The group theoretical treatments of the projection of n-dimensional cubic lattices have been worked out by the references [13,14] and a general projection technique on the basis of dihedral subgroup of the root lattices has been proposed by Boyle and Steinhardt [15]. In a recent article [16] we pointed out that the two-dimensional facets of the A4 Voronoi cell projects onto thick and thin rhombuses of the Penrose tilings. The cut and project technique has been also worked out by Masáková, Z. et al. [17] by implementing the substitution method. The third method is the model set technique initiated by Meyer [18], followed by Lagarias [19], and developed by Moody [20]. For a detailed treatment, see the reference [5].

Symmetry 2019, 11, 1082; doi:10.3390/sym11091082 www.mdpi.com/journal/symmetry Symmetry 2019, 11, 1082 2 of 18

The Voronoi [21,22] and Delaunay [23–25] cells of the root and weight lattices have been extensively studied in the inspiring book by Conway and Sloane [26] and especially in the reference [27]. The reference [28] contains detailed discussions and information about the Delone and Voronoi polytopes of the root lattices. Higher dimensional lattices have been worked out in [29] and the lattices of the root systems whose point groups are the Coxeter–Weyl groups have been studied extensively in the reference [30]. 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 [31]. In a recent article [32], 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 An and Dn series. A different application of the Voronoi tessellations can be found in the reference [33]. Basic information about the regular polytopes as the orbits of the Coxeter–Weyl groups have been worked out in the references [34,35], the root lattice An are obtained from the Voronoi polytope of the cubic lattice Bn+1 by projection into n-dimensional Euclidean space along one of its space diagonals. It turns out that the facets of the Voronoi polytope of An are rhombohedra in various dimensions such as rhombi in 2D, rhombohedra in 3D, and the generalizations to higher dimensions. Projections of the 2D facets of the Voronoi polytope of An lead to (n + 1)-symmetric tilings of the Coxeter plane by a number of rhombi with different interior angles. Similarly, we show that the 2D facets of the Delone cells of a given lattice An are equilateral triangles; when projected into the Coxeter plane they lead to various triangles, which tile the plane in an aperiodic manner. They include, among many new prototiles and patches, some well-known prototiles such as rhombus tilings by Penrose and Amman–Beenker, and triangle tilings by Robinson and Danzer. The paper displays the lists of the rhombic and triangular prototiles in Tables1 and2, originating from the projections of the Voronoi and Delone cells of the lattice An onto the Coxeter plane, respectively, and illustrates some patches of the aperiodic tilings. Most of the prototiles and the tilings are displayed for the first time in what follows.

Table 1. Rhombic prototiles projected from Voronoi cells of the root lattice A (rhombuses with angles n 2πm , π 2πm , m N). h − h ∈ Root Lattice h # of Prototiles Rhombuses with Pairs of Angles π π A3 4 1 2 , 2 , (square) 2π 3π 4π π A4 5 2 5 , 5 , 5 , 5 , (Penrose’s thick and thin rhombuses) π 2π A5 6 1 3 , 3 2π 5π 4π 3π 6π π A6 7 3 7 , 7 , 7 , 7 , 7 , 7 π 3π π π A7 8 2 4 , 4 , 2 , 2 , (Amman-Beenker tiles) 2π 7π 4π 5π 6π 3π 8π π A8 9 4 9 , 9 , 9 , 9 , 9 , 9 , 9 , 9 2π 3π π 4π A9 10 2 5 , 5 , 5 , 5 , (Penrose’ thick and thin rhombuses) 2π 9π 4π 7π 6π 5π A10 11 3 11 , 11 , 11 , 11 , 11 , 11 π 5π π 2π π π A11 12 3 6 , 6 , 3 , 3 , 2 , 2

We organize the paper as follows. In Section2 we introduce the basics of the root lattice An via its Coxeter–Dynkin diagram, its Coxeter–Weyl group, and projection of its Voronoi cell from that of Bn+1. The projections of the two-dimensional facets of the Voronoi and Delone cells onto the Coxeter plane are studied in Section3. Section4 deals with the examples of periodic and aperiodic patches of tilings. Section5 includes the concluding remarks. In AppendixA we prove that the facets of the Voronoi cell of the 5D cubic lattice projects to those of the Voronoi cell of A4. Symmetry 2019, 11, 1082 3 of 18

n1π n2π n3π Table 2. Triangular prototiles with angles h , h , h projected from Delone cells of the root lattice An.

Triangles Denoted by Triple Natural Numbers Root Lattice h # of Prototiles (n1, n2, n3); n1+ n2+ n3=h

A2 3 1 (1, 1, 1) A3 4 1 (1, 1, 2), (right triangle) A4 5 2 (1, 1, 3), (1, 2, 2), (Robinson triangles) A5 6 3 (1, 1, 4), (1, 2, 3), (2, 2, 2) A6 7 4 (1, 1, 5), (1, 2, 4), (1, 3, 3), (2, 2, 3), (Danzer triangles) A7 8 5 (1, 1, 6), (1, 2, 5), (1, 3, 4), (2, 2, 4), (2, 3, 3) (1, 1, 7), (1, 2, 6), (1, 3, 5), (1, 4, 4), (2, 2, 5), A 9 7 8 (2, 3, 4), (3, 3, 3) (1, 1, 8), (1, 2, 7), (1, 3, 6), (1, 4, 5), (2, 2, 6), A 10 8 9 (2, 3, 5), (2, 4, 4), (3, 3, 4) (1, 1, 9), (1, 2, 8), (1, 3, 7), (1, 4, 6), (1, 5, 5), A 11 10 10 (2, 2, 7), (2, 3, 6), (2, 4, 5), (3, 3, 5), (3, 4, 4) (1, 1, 10), (1, 2, 9), (1, 3, 8), (1, 4, 7), (1, 5, 6), A11 12 12 (2, 2, 8), (2, 3, 7), (2, 4, 6), (2, 5, 5), (3, 3, 6), (3, 4, 5), (4, 4, 4)

2. The Root Lattice An and Its Coxeter–Weyl Group We follow the notations of the reference [27], which can be compared with the standard notations [36], and introduce some new ones when they are needed. The Coxeter–Dynkin diagram of an, describing the Coxeter–Weyl group, and its extended diagram, representing the aSymmetryﬃne Coxeter–Weyl 2019, 11, 1082 group, are shown in Figure1. of 193

𝛼

𝛼 𝛼 𝛼 𝛼

(a) (b)

FigureFigure 1. 1.( (aa)) Coxeter–DynkinCoxeter–Dynkin diagram diagram of ofa 𝑎n,(, b(b)) extended extended Coxeter–Dynkin Coxeter–Dynkin diagram diagram of ofa n𝑎..

TheThe nodesnodes representrepresent thethe lattice-generating lattice-generating vectorsvectors (root(root vectorsvectors inin LieLie algebraalgebra terminology)terminology) α𝛼i ( i(𝑖= 1, = 2,1,... 2,, …n ,) 𝑛) formingforming the Cartan matrix (Gram matrix of the lattice) by the relation (, ) 𝐶 = . (1) (2, αi), αj Cij = . (1) ∗ The fundamental weight vectors 𝜔 , which αgeneratej, αj the dual lattice 𝐴, satisfying the relation 𝛼,𝜔=𝛿, are given by the relations, The fundamental weight vectors ω , which generate the dual lattice A , satisfying the relation i n∗ αi, ωj = δij, are given by the relations,𝜔 =(𝐶 )𝛼 , 𝛼 =𝐶𝜔. (2) X X 1 ω = (C− ) α , α = Cω . (2) The lattice 𝐴 is defined asi the set of ijvectorsj i 𝑝=∑ij j𝑏 𝛼,𝑏 ∈ℤ and the weight lattice ∗ j j 𝐴 consists of the vectors 𝑞=∑ 𝑐 𝜔,𝑐 ∈ℤ. Note that the root lattice is a sublattice of the weight ∗ lattice, 𝐴 ⊂𝐴. Let 𝑟, (𝑖=1,2,…,𝑛) denotes the reflectionPn generator with respect to the hyperplane The lattice An is deﬁned as the set of vectors p = i=1 biαi, bi Z and the weight lattice An∗ orthogonal to the simple rootPn 𝛼, which operates on an arbitrary vector∈ 𝜆 as consists of the vectors q = i=1 ciωi, ci Z. Note that the root lattice is a sublattice of the weight ∈ (,) 𝑟𝜆=𝜆− 𝛼. (3) (,)

The reflection generators generate the Coxeter–Weyl group <𝑟 ,𝑟 ,…,𝑟 |(𝑟 𝑟 ) =1> . Adding another generator, usually denoted by 𝑟 , describing the reflection with respect to the hyperplane bisecting the highest weight vector (𝜔 +𝜔), we obtain the Affine Coxeter group, the infinite discrete group denoted by <𝑟 ,𝑟 ,𝑟 ,…,𝑟 >. The relations between the Voronoi and the ∗ Delone cells of the lattices 𝐴 and 𝐴 have been studied by L. Michel in [37] and [38]. An arbitrary group element of the Coxeter–Weyl group will be denoted by 𝑊(𝑎) and the orbit of an arbitrary ∑ vector 𝑞 will be defined as 𝑊(𝑎)𝑞 =: ( 𝑐 𝜔),𝑐 ∈ℤ. With this notation, the root polytope will

be denoted either by (10 … 01) or simply by (𝜔 +𝜔). The dual polytope of the root polytope [32] is the Voronoi cell 𝑉(0), which is the union of the orbits of the fundamental polytopes ( ) ( ) 𝜔 , 𝑖 = 1, 2, … ,𝑛 , ( ) (𝜔)⋃ 𝜔 …⋃(𝜔). (4)

Each fundamental polytope (𝜔) of 𝐴 is a copy of one of the Delone polytopes. Delone polytopes tile the root lattice in such a way that each polytope centralizes one vertex of the Voronoi

cell. For example, the set of vertices (𝜔) +(𝜔) represent the 2(𝑛 + 1) simplexes centered ( ) ( ) around the vertices (𝜔) and (𝜔) of the Voronoi cell 𝑉 0 . Similarly, 𝜔 +(𝜔) constitutes the vertices of the ambo-simplexes (see the reference [27] for the definition) centralizing ( ) ( ) the vertices 𝜔 and the (𝜔) of the Voronoi cell 𝑉 0 and so on. It is a general practice to work with the set of orthonormal vectors 𝑙, (𝑖=1,2,…,𝑛+1),𝑙,𝑙= 𝛿 to define the root and weight vectors. The root vectors can be defined as 𝛼 = 𝑙 − 𝑙 , (𝑖=1,2,…,𝑛), which are permuted by the generators as 𝑟: 𝑙 ↔ 𝑙, implying that the Coxeter–Weyl group 𝑆 is isomorphic to the symmetric group of order (𝑛+1)!, permuting the 𝑛+ 1 vectors. The fundamental weights then are given as 1 𝜔 = (𝑛 𝑙 − 𝑙 − 𝑙 −⋯− 𝑙 ); (5) 𝑛 + 1 Symmetry 2019, 11, 1082 4 of 18

lattice, An A . Let r , (i = 1, 2, ... , n) denotes the reﬂection generator with respect to the hyperplane ⊂ n∗ i orthogonal to the simple root αi, which operates on an arbitrary vector λ as

2(λ, αi) riλ = λ αi. (3) − (αi, αi)

mij The reflection generators generate the Coxeter–Weyl group < r , r , ... , rn (r r ) = 1 >. 1 2 | i j Adding another generator, usually denoted by r0, describing the reflection with respect to the hyperplane bisecting the highest weight vector (ω1 + ωn), we obtain the Affine Coxeter group, the infinite discrete group denoted by < r0, r1, r2, ... , rn >. The relations between the Voronoi and the Delone cells of the lattices An and An∗ have been studied by L. Michel in [37,38]. An arbitrary group element of the Coxeter–Weyl group will be denoted by W(an) and the orbit of an arbitrary vector q will be defined Pn as W(an)q =: i=1 ciωi , ci Z. With this notation, the root polytope will be denoted either by an ∈ ( ) ( + ) 10 ... 01 an or simply by ω1 ωn an . The dual polytope of the root polytope [32] is the Voronoi cell ( ) ( ) ( = ) V 0 , which is the union of the orbits of the fundamental polytopes ωi an , i 1, 2, ... , n ,

(ω ) (ω ) ... (ωn) . (4) 1 an ∪ 2 an ∪ an ( ) Each fundamental polytope ωi an of An is a copy of one of the Delone polytopes. Delone polytopes tile the root lattice in such a way that each polytope centralizes one vertex of the Voronoi cell. ( ) + ( ) ( + ) For example, the set of vertices ω1 an ωn an represent the 2 n 1 simplexes centered around the ( ) ( ) ( ) ( ) + ( ) vertices ω1 an and ωn an of the Voronoi cell V 0 . Similarly, ω2 an ωn 1 an constitutes the vertices − ( ) of the ambo-simplexes (see the reference [27] for the definition) centralizing the vertices ω2 an and the (ωn 1)a of the Voronoi cell V(0) and so on. − n It is a general practice to work with the set of orthonormal vectors li, (i = 1, 2, ... , n + 1), li, lj = δ to define the root and weight vectors. The root vectors can be defined as α = l l + , (i = 1, 2, ... , n), ij i i − i 1 which are permuted by the generators as r : l l + , implying that the Coxeter–Weyl group S + is i i ↔ i 1 n 1 isomorphic to the symmetric group of order (n + 1)!, permuting the n + 1 vectors. The fundamental weights then are given as

1 ω1 = n+1 (n l1 l2 l3 ln+1); 1 − − − · · · − ω2 = n+1 ((n 1) l1 + (n 1) l2 2 l3 2ln+1); (5) −1 − − − · · · − ωn = ( l + l + + ln n l + ). n+1 1 2 ··· − n 1 It is even more convenient to introduce the non-orthogonal set of vectors

l0 k =: + l , l =: l + l + + l + , k + k + + k + = 0. (6) i −n + 1 i 0 1 2 ··· n 1 1 2 ··· n 1 ( ) The vectors ki are more useful since they represent the vertices of the n-simplex ω1 an and the edges of the Voronoi cell are parallel to the vectors ki. In terms of the fundamental weights they are given by k1 = ω1, k2 = ω2 ω1, k3 = ω3 ω2, ... , kn = ωn ωn 1, kn+1 = ωn. (7) − − − − − Their magnitudes are all the same (k , k ) = n and the angle between any pair is θ = cos 1 1 . i i n+1 − − n They are orthogonal to the vector l , (k , l ) = 0, so that they deﬁne the n dimensional hyperplane 0 i 0 − orthogonal to the vector l0. Following (5) and (6), vertices of the orbits of the fundamental weights

(ω ) = (k + k + + k ) , (i = 1, 2, ... , n) (8) i an 1 2 ··· i an Symmetry 2019, 11, 1082 5 of 18 are obtained as the permutations of the vectors in (8) and each orbit of the fundamental weight represents a copy of the Delone polytope. It is clear that the generators of the Coxeter–Weyl group permute the vectors k as r : k k + . The number of vertices of the Voronoi cell of An is then given by i i i ↔ i 1 n ! X n + 1 = 2n+1 2. (9) i − i=1

The Voronoi cell V(0) of An can also be obtained as the projection along the space diagonal vector l0 of the Voronoi cell of the cubic lattice in n + 1 dimensions with vertices 1 ± ( l l l + ). 2 ± 1 ± 2 ± · · · ± n 1 l0 Substituting li =: n+1 + ki, it is straightforward to show that the vertices of the Voronoi cell of Bn+1 project into the vertices of the Voronoi cell of An, represented by the union of the fundamental polytopes in n-dimensional space given in (4). This follows from the projections

1 2 l0 0, 1 ± → ( l + l + + l + ) k = ω , ± 2 − 1 2 ··· n 1 → ∓ 1 ∓ 1 (10) 1 . ( l l + + l + ) (k + k ) = ω , . ± 2 − 1 − 2 ··· n 1 → ∓ 1 2 ∓ 2 1 ( l l ln + l + ) (k + k + + kn) = (k + ) = ωn. ±2 − 1 − 2 − · · · − n 1 → ∓ 1 2 ··· ± n 1 ∓

Let us consider the root system li lj = ki kj; the hyperplanes bisecting the segment between − − the origin and the root system are represented by the vectors 1 k k . The intersections of these 2 i − j hyperplanes are the vertices of the Voronoi cell V(0) of the lattice A , which are the union of the n orbits of the highest weight vectors given in (4). Actually, the vectors 1 k k are the centers of the 2 i − j n 1 dimensional rhombohedral facets of the Voronoi cell V(0). The 2D facets (squares), 3D (cubes), − and higher dimensional cubic facets of the Voronoi cell of Bn+1 project into the rhombi, rhombohedra, and higher dimensional rhombohedra representing the facets of the Voronoi cell of the V(0) of An. AppendixA discusses further aspects of this projection for two special cases B A and B A . 3 → 2 5 → 4 Two-dimensional faces of the Delone polytopes are equilateral triangles. To give an example, let us take the vertex ω = k . The set of vertices (ω ) = k , i = 1, 2, ... , n + 1 constitute the vertices of the 1 1 1 an { i} n-simplex. One can generate an equilateral triangle by applying the subgroup < r1 , r2 > on ω1, where the edges are given by the root vectors k k , k k , k k . Other faces and the corresponding 1 − 2 2 − 3 3 − 1 edges are generated by applying the group elements W(an), leading to the set of edges k k , a vector i − j ( ) in the root system. The argument is valid for any other Delone polytope ωi an . There is always a group element relating the vector ki to any other vertex kj. To give a further example, let us consider the lattice A . The Delone polytopes (ω ) = (ω ) represent two tetrahedra where the vertices 3 1 a3 − 3 a3 belong to the set {k } or { k }. The six edges of the four triangles of each tetrahedra are represented i − i by the vectors k k , i , j = 1, 2, 3, 4 belonging to the root system. Similarly, the orbit (ω ) is an i − j 2 a3 octahedron with vertices ki + kj, i , j = 1, 2, 3, 4, where the edges of the eight triangular faces are represented by the vectors k k , i , j, the elements of the root system. For general proof, one can i − j take the vertex in (8) and apply the Coxeter Weyl group an to generate the nearest vertices forming − triangles associated with the vertex (8). These arguments imply that the set of edges of an arbitrary two-dimensional face of a Delone polytope can be represented by the root vectors k k , k k , k k , i − j j − l l − i i , j , l = 1, 2, ... , n + 1.

3. Projections of the Faces of the Voronoi and Delone Cells In this section we deﬁne the Coxeter plane and determine the dihedral subgroup of the Coxeter–Weyl group acting in this plane. The Coxeter plane is generally deﬁned by the eigenvector 2 n i2π v = ζ, ζ , ... , ζ , 1 , corresponding to the eigenvalue ζ = e h of the Coxeter element R = r1 r2 r3 ... rn Symmetry 2019, 11, 1082 6 of 18

of the group an, which permutes the vectors k k k + k in the cyclic order. Here, h = n + 1 1 → 2 → · · · n 1 → 1 is the Coxeter number. Going to the real space, the Coxeter element acts on the plane E spanned by 1 i k the unit vectors u1 = (v + v) and u2 = (v v). The components of the vectors kj in the Coxeter √2h √2h − ij2π q h 2 plane are then represented by the complex number (kj) = ce , where j = 1, 2, ... , n + 1, and c = h . In references [14,39] we have introduced an equivalentk deﬁnition of the Coxeter plane through the eigenvalues and eigenvectors of the Cartan matrix of the root system of the lattice An. The eigenvalues and eigenvectors of the Cartan matrix of the group an can be written as

m π + j miπ λ = 2(1 + cos i ); X = ( 1)j 1 sin , i, j = 1, 2, ... , n (11) i h ji − h where mi = 1, 2, ... , n are the Coxeter exponents. We deﬁne the orthonormal set of vectors subject to an arbitrary orthogonal transformation by

n 1 X xˆi = αjXji, xˆi, xˆ j = δij, i = 1, 2, ... , n. (12) √ λi j=1

An equivalent definition of the Coxeter plane E is then given by the pair of vectors xˆ1and xˆn, and k the rest of the vectors in (12) determine the orthogonal space E . With a suitable choice of the simple ⊥ roots αj, the unit vectors xˆ1and xˆn can be associated with the unit vectors u1 and u2, respectively. The Coxeter element R = r1 r2 r3 ... rn can be written as the product of two reflection generators h < R , R (R R ) = 1 > defining a dihedral subgroup of order 2h of the group an [36,40]. 1 2| 1 2 Therefore, the residual symmetry after projection is the dihedral group acting on the Coxeter plane. We will drop the overall factor c as it has no significant meaning in our further discussions. It is (j l)2π − then clear that all possible scalar products ((kj) , (kl) ) = cos h will determine the nature of the k k projected rhombi in the Coxeter plane E . The tiles projected from the Delone cells will be triangles k whose edge lengths are the magnitudes of the vectors (k k ) , (k k ) , (k k ) and, after deleting i − j j − l l − i the common factor c, they will read respectively k k k ! ! ! (i j)π (j l)π (l i)π 2 sin − , 2 sin − , 2 sin − , (13) h h h

Exhausting all possible projections. For each set of integral numbers i , j , l = 1, 2, ... , n + 1 we may assume without loss of generality that the numbers can be ordered as i > j > l and the corresponding (i j)π (j l)π (i l)π angles of the edges in (13) can be represented as − , − , π − . Defining these angles by h h − h n1π n2π n3π , , , ni N, we obtain n1 + n2 + n3 = h. Some edges from the Voronoi cell or Delone cell h h h ∈ overlap after projection, forming a degenerate rhombi or triangle, respectively, which has no effect on tilings of the Coxeter plane. In what follows we will illustrate these results with some examples. The rhombic and triangular prototiles are classified in Tables1 and2, respectively.

4. Examples of Prototiles and Patches of Tilings In what follows, we discuss a few h-fold symmetric tilings with rhombuses and triangles.

4.1. Projection of the Voronoi Cell of A3

The Voronoi cell of the root lattice A3, commonly known as the f.c.c. lattice, is the rhombic dodecahedron (the Wigner–Seitz cell), which tessellates three-dimensional Euclidean space. The vertices of the rhombic dodecahedron can be represented as the union of two tetrahedra and one octahedron:

(ω ) (ω ) (ω ) = ( k , k , k , k )(k + k , k + k , k + k , k + k , k + k , k + k ) . (14) 1 a3 ∪ 2 a3 ∪ 3 a3 { ± 1 ± 2 ± 3 ± 4 1 2 2 3 3 4 4 1 1 3 2 4 Symmetry 2019, 11, 1082 of 197

4. Examples of Prototiles and Patches of Tilings In what follows, we discuss a few h-fold symmetric tilings with rhombuses and triangles.

4.1. Projection of the Voronoi Cell of 𝐴

The Voronoi cell of the root lattice 𝐴 , commonly known as the f.c.c. lattice, is the rhombic dodecahedron (the Wigner–Seitz cell), which tessellates three-dimensional Euclidean space. The vertices of the rhombic dodecahedron can be represented as the union of two tetrahedra and one octahedron: Symmetry 2019, 11, 1082 7 of 18 (𝜔) ⋃(𝜔) ⋃(𝜔) ={(±𝑘,±𝑘,±𝑘,±𝑘) (14) (𝑘 +𝑘, 𝑘 +𝑘, 𝑘 +𝑘, 𝑘 +𝑘, 𝑘 +𝑘,𝑘 +𝑘)}. A typical rhombic face of the rhombic dodecahedron is determined by the set of four vertices A typical rhombic face of the rhombic dodecahedron is determined by the set of four vertices k1 = ω1, k1 + k2 = ω2, k1 + k3 = r2ω2, ω3 = k1 + k2 + k3 = k4. 𝑘 =𝜔 , 𝑘 +𝑘 =𝜔 ,𝑘 +𝑘 =𝑟𝜔 , 𝜔 = 𝑘 +𝑘 +𝑘 =−𝑘− . The ﬁrst four vectors in (14), (ω) = k ,k , k , k represent the vertices of the ﬁrst tetrahedron 1 a3 1 2 3 4 The first four vectors in (14), (𝜔) = {𝑘{ ,𝑘,𝑘,𝑘}} represent the vertices of the first tetrahedron and the permutation group S4 W(a3) is isomorphic to the tetrahedral group of order 24. The set and the permutation group 𝑆 ≅𝑊(𝑎 ) is isomorphic to the tetrahedral group of order 24. The set of of vectors ( k , k , k , k ) represent the second tetrahedron (ω ) . Actually, the Dynkin diagram 1 2 3 4 3 a3 vectors (−𝑘−,−𝑘−,−𝑘−,−𝑘− ) represent the second tetrahedron (𝜔). Actually, the Dynkin diagram symmetry (k1 k4) extends the tetrahedral group to the octahedral group, implying that two symmetry (𝑘 ↔−𝑘↔ − ) 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 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 invariant under the octahedral group of order 48. The last six vectors in (14) represent the vertices of of an octahedron, which split as 4 + 2 under the dihedral subgroup D4 of order 8. The last two an octahedron, which split as 4+2 under the dihedral subgroup 𝐷 of order 8. The last two vectors vectors k1 + k3 and k2 + k4 = (k1 + k3) form a doublet under the dihedral subgroup and they are 𝑘 +𝑘 and 𝑘 +𝑘 =−(𝑘 +𝑘−) form a doublet under the dihedral subgroup and they are perpendicular to the Coxeter plane E , determined by the pair of vectors (xˆ1, xˆ3). The projected perpendicular to the Coxeter plane 𝐸k∥ , determined by the pair of vectors (𝑥,𝑥). The projected components of the vectors ki in the plane E are given by components of the vectors 𝑘 in the plane 𝐸k∥ are given by (k ) = (0, 1), (k ) = ( 1, 0), (k ) = (0, 1), (k ) = (1, 0). (15) (𝑘)1∥ = (0, 1),(𝑘)∥2= (−1, 0),(𝑘)∥ =3 (0, −1),(𝑘)∥ =4 (1, 0). (15) k k − k − k Therefore, projection of the rhombus in Figure2 2 is is aa squaresquare ofof unit unit length, length, determineddetermined byby thethe (𝑘 ) = (−1, 0) and (𝑘 ) = (0, −1) 𝐸 vectors (k2) ∥ = ( 1, 0) and (k3)∥ = (0, 1).. Projection Projection of of the the Voronoi Voronoi cellcell ontoonto thethe planeplane E∥ is shownshown k − k − 𝐷 k in FigureFigure3 3,, which which is is invariant invariant underunder thethe dihedraldihedral groupgroup D4.. We We can can pick up any two vectors in (15) (𝑘 ) (𝑘 ) , 𝐸 𝑞 = 𝑚 (𝑘 ) + say (k1) ∥and and( k2) ,∥then then the the lattice lattice in Ein is∥ a is square a square lattice lattice with awith general a general vector qvector= m1 (k∥1) +m2(k∥2) 𝑚 (𝑘 ) k 𝑚 ,𝑚k ∈ℤ k k k k with m 1∥, withm2 Z. . ∈

𝜔

𝜔 𝜔

𝑟𝜔 Symmetry 2019, 11, 1082 Figure 2. A typical rhombic face of the rhombic dodecahedron (note that edges are represented by the of 198 Figure 2. A typical rhombic face of the rhombic dodecahedron (note that edges are represented by the vectors k2 and k3 with an angle 109.5◦). vectors 𝑘 and 𝑘 with an angle 109.5° ).

Figure 3. The Wigner–Seitz cell (rhombic dodecahedron) onto the Coxeter plane E . k Figure 3. The Wigner–Seitz cell (rhombic dodecahedron) onto the Coxeter plane 𝐸∥. 4.2. Projection of the Delone Cells of the Root Lattice A3 4.2. Projection of the Delone Cells of the Root Lattice 𝐴 P3 P3 The lattice vector of the lattice A3 can be written as p = biαi = niki with bi, ni Z such P i=1 i=1 ∈ that 3 n = even. This shows that the root lattice is a sublattice of the weight lattice. Equilateral Thei=1 latticei vector of the lattice 𝐴 can be written as 𝑝=∑ 𝑏 𝛼 = ∑ 𝑛 𝑘 with 𝑏,𝑛 ∈ℤ such triangles of the Delone cells project as right triangles, any two constitute a square of length √2. One can that ∑ 𝑛 = even. This shows that the root lattice is a sublattice of the weight lattice. Equilateral check that the projected Delone cells form a square lattice with a general vector p = m1(k1) + m√2(k2) , triangles of the Delone cells project as right triangles, any two constitute a squarek of length . One with m + m = even integer as shown in Figure4. k k can check1 that2 the projected Delone cells form a square lattice with a general vector 𝑝∥ = 𝑚(𝑘)∥ + 𝑚(𝑘)∥, with 𝑚 +𝑚 = even integer as shown in Figure 4.

Figure 4. Projection of the root lattice as a square lattice made of right triangles (not shown).

It is clear that it is a sublattice of the square lattice obtained from the weight lattice and it is rotated by 45° 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. This is expected because the dihedral group is a crystallographic group. Two overlapping square lattices have been depicted in Figure 5.

∗ Figure 5. Projections of the lattices 𝐴 and 𝐴 , leading to two overlapping square lattices, 𝐴 is the sublattice of the other.

4.3. Projection of the Voronoi Cell of the Root Lattice 𝐴 We would like to discuss this projection leading to the Penrose tiling, which can be obtained from the cut and project technique or by the substitution technique directly. The Voronoi cell of the Symmetry 2019, 11, 1082 Symmetry 2019, 11, of 1082 198 of 198

Figure 3. The Wigner–Seitz cell (rhombic dodecahedron) onto the Coxeter plane 𝐸∥.

Figure 3. The Wigner–Seitz cell (rhombic dodecahedron) onto the Coxeter plane 𝐸∥. 4.2. Projection of the Delone Cells of the Root Lattice 𝐴 4.2. Projection of the Delone Cells of the Root Lattice 𝐴 The lattice vector of the lattice 𝐴 can be written as 𝑝=∑ 𝑏 𝛼 = ∑ 𝑛 𝑘 with 𝑏,𝑛 ∈ℤ such that ∑The 𝑛lattice = even. vector This of theshows lattice that 𝐴 the can root be writtenlattice isas a 𝑝= sublattice∑ 𝑏 𝛼of =the∑ weight𝑛 𝑘 withlattice. 𝑏,𝑛 Equilateral ∈ℤ such trianglesthat ∑ 𝑛of =the even. Delone This cells shows project that as the right root tr iangles,lattice is any a sublattice two constitute of the a weight square lattice.of length Equilateral √2. One cantriangles check of that the the Delone projected cells projectDelone ascells right form triangles, a square any lattice two constitutewith a general a square vector of length𝑝∥ = 𝑚 √(𝑘2. One)∥ + Symmetry𝑚can(𝑘 check)∥2019, with that, 11 ,𝑚 1082the +𝑚 projected = even Delone integer cells as shown form a in square Figure lattice 4. with a general vector 𝑝∥ = 𝑚(𝑘8 of)∥ 18+ 𝑚(𝑘)∥, with 𝑚 +𝑚 = even integer as shown in Figure 4.

Figure 4. Projection of the root lattice as a square lattice made of right triangles (not shown). FigureFigure 4.4. Projection of the rootroot latticelattice asas aa squaresquare latticelattice mademade ofof rightright trianglestriangles (not(not shown).shown). It is clear that it is a sublattice of the square lattice obtained from the weight lattice and it is rotatedItIt is isby clear clear 45° that that with it isit respect ais sublattice a sublattice to the of theprojected of square the square square lattice lattice obtainedlattice obtained of from the Voronoi the from weight thecell. lattice weight With and this lattice it example, is rotated and it we by is

45haverotated◦ with obtained by respect 45° ato withsublattice the respect projected whose to squarethe unit projected latticecell is ofinvariant square the Voronoi lattice under cell.of the the With dihedral Voronoi this example, groupcell. With 𝐷 we. this This have example, is obtained expected we a sublatticebecausehave obtained the whose dihedral a unitsublattice cell group is invariantwhose is a crystallographic unit under cell theis invariant dihedral group. groupunder Two Dthe overlapping4. Thisdihedral is expected group square because𝐷 lattices. This the is have expected dihedral been groupdepictedbecause is athe in crystallographic Figure dihedral 5. group group. is a crystallographic Two overlapping gr squareoup. Two lattices overlapping have been depictedsquare lattices in Figure have5. been depicted in Figure 5.

∗ FigureFigure 5.5. Projections of the lattices A𝐴3∗ and A𝐴3 ,, leadingleading toto twotwo overlappingoverlapping square lattices, lattices, A𝐴3 isis thethe ∗ sublatticesublatticeFigure 5. Projections ofof thethe other.other. of the lattices 𝐴 and 𝐴 , leading to two overlapping square lattices, 𝐴 is the sublattice of the other. 4.3. Projection of the Voronoi Cell of the Root Lattice A4 4.3. Projection of the Voronoi Cell of the Root Lattice 𝐴 4.3. Projection of the Voronoi Cell of the Root Lattice 𝐴 WeWe wouldwould likelike toto discussdiscuss thisthis projectionprojection leadingleading toto thethe PenrosePenrose tiling,tiling, whichwhich cancan bebe obtainedobtained fromfrom We the wouldcut and and like project project to discuss technique technique this orprojection or by by the the substitution substitutionleading to techniquethe technique Penrose directly. tiling, directly. Thewhich The Voronoi can Voronoi be cell obtained cellof the of thefrom root the lattice cut and tiles project the four-dimensional technique or by Euclideanthe substitution reciprocal technique space, directly. which is The the Voronoi weight latticecell of Athe4∗ P4 P5 P4 represented by a general vector q = ciωi = miki = niki with ci, mi, ni Z. Union of the i=1 i=1 i=1 ∈ Delone polytopes (ω ) (ω2) (ω3) (ω ) (16) 1 a4 ∪ a4 ∪ a4 ∪ 4 a4 constitutes the Voronoi cell V(0) with 30 = 5 + 10 + 10 + 5 vertices. It is a polytope comprised of rhombohedral facets. A typical three-facet of the Voronoi cell is a rhombohedron with six rhombic faces [32], as depicted in Figure6. Its vertices are given by

(k , k + k , k + k , k + k , k , ( k + k ), ( k + k ), ( k + k )) 1 1 2 1 3 1 4 − 5 − 2 5 − 3 5 − 4 5 and its center is at 1 (k k ). 2 1 − 5 The Voronoi cell is formed by 20 such rhombohedra, whose rhombic two faces are generated by the pairs of vectors satisfying the scalar product (k , k ) = 1 , i , j = 1, ... , 5, and (k , k ) = 4 . It is perhaps i j − 5 i i 5 useful to discuss some of the technicalities of the projection with this example. The components of the ijπ P j 5 5 j vectors in the plane E (kj) = (lj) ζ = e , j = 1, ... , 5 satisfying i=1 ζ = 0 form a pentagon, k ⇒ k k 4π while the vectors (kj) form a pentagram since the angle between two successive vectors (kj) is 5 . ⊥ ⊥ Symmetry 2019, 11, 1082 Symmetry 2019, 11, 1082 9of of 19 189 root lattice tiles the four-dimensional Euclidean reciprocal space, which is the weight lattice ∗ A matrix representation of the Coxeter element R in four-dimensional space is given by 𝐴 represented by a general vector 𝑞=∑ 𝑐 𝜔 = ∑ 𝑚 𝑘 = ∑ 𝑛 𝑘 with 𝑐,𝑚,𝑛 ∈ℤ. Union of the Delone polytopes  cos 2π sin 2π 0 0   5 − 5  ( 𝜔 ) ⋃2π( 𝜔 ) ⋃(2π 𝜔 ) ⋃( 𝜔 )   sin 5 cos 5 0 0  (16) R =  π π  (17)  0 0 cos 4 sin 4  constitutes the Voronoi cell 𝑉(0) with 30 = 5 + 10 + 10 + 5 vertices.− 5It is a polytope comprised of  0 0 sin 4π cos 4π  rhombohedral facets. A typical three-facet of the Voronoi cell5 is a rhombohedron5 with six rhombic faces [32], as depicted in Figure 6. Its vertices are given by P5 A general vector of the weight lattice A4∗ projects as q = i=1 ni(ki) . Note that the rhombic k k 2-faces of the𝑘 Voronoi, 𝑘 +𝑘 cell, 𝑘 project+𝑘, 𝑘 onto +𝑘 the,−𝑘 plane,−(E 𝑘as+ 𝑘 two),− Penrose( 𝑘+ 𝑘 rhombuses,),−( 𝑘+ 𝑘thick) and thin, for the k ij2π acute angles between any pair of the vectors (k ) e 5 are either 72 or 36 . and its center is at (𝑘−𝑘). j ◦ ◦ k ⇒

Figure 6.6. AA 3-dimensional3-dimensional rhombohedron, rhombohedron, the rhombohedralthe rhombohedral facet offacet the Voronoiof the Voronoi cell of the cell root of lattice the rootA4 . lattice 𝐴 . De Bruijn [9] proved that the projection of the five-dimensional cubic lattice onto E leads to the k PenroseThe rhombusVoronoi cell tiling is offormed the plane by 20 with such thick rhombohedra, and thin rhombuses. whose rhombic The same two faces technique are generated is extended by theto an pairs arbitrary of vectors cubic satisfying lattice [13 ].the It isscalar not aproduct surprise (𝑘 that, 𝑘 we)=− obtain,𝑖 the ≠𝑗=1,…,5 same tiling, and from (𝑘 , the 𝑘 )=A4 lattice. It is perhapsbecause Voronoiuseful to cell discuss of A4 is some obtained of the as thetechnicali projectionties ofof the the Voronoi projection cell ofwith the cubicthis example. lattice B5 The[14] 5 W(b ) (see also AppendixA for further details). The Coxeter–Weyl group 5 is of order 2 5! and its 𝐸 (𝑘 ) =(𝑙) ⇒𝜁 =𝑒 ,𝑗 =1,…,5 ×∑ 𝜁 =0 componentsCoxeter number of the is h vectors= 10. Normally, in the plane one expects∥ ∥ a 10-fold ∥ symmetric tiling from satisfying the projection of the (𝑘 ) formfive-dimensional a pentagon, cubicwhile lattice. the vectors The Voronoi form cell a of pentagramB has 32 vertices since the1 ( anglel betweenl l twol successivel ), which 5 2 1 2 3 4 5 vectors (𝑘 ) is . ± ± ± ± ± decompose under the Coxeter–Weyl group W(a4) as A matrix representation of the Coxeter element 𝑅 in four-dimensional space is given by 32 = 1 + 1 + 5 + 10 + 10 + 5. (18) 2π 2π cos −sin The first two terms in (18) are represented5 5 by the vector00s 1 l 0 in four-space and the ⎛ ± 2⎞0 → 2π 2π 00+ + + remaining vertices of the five-dimensional⎜sin cubecos decompose as 5 10 10⎟ 5, representing the vertices 5 5 of the Voronoi cell (ω1) (ω2𝑅=) ⎜(ω3) (ω4) . In terms of the vectors⎟ li, the vectors ki can also be a4 ∪ a4 ∪ a4 ∪ a4 4π 4π (17) written as (compare with (6)) ⎜ cos −sin ⎟ ⎜ 00 5 5 ⎟ 3 001 4π 4π k1 10 l0 = 2 ( l1 l2 sinl3 l4 cosl5), ⎝−3 1 − − 5− − 5 ⎠ k2 10 l0 = 2 ( l1 + l2 l3 l4 l5), − 3 1 − − − − k3 l0 = ∗( l1 l2 + l3 l4 l5), (19) A general vector of the weight lattice10 𝐴2 projects as 𝑞∥ = ∑ 𝑛(𝑘)∥ . Note that the rhombic 2- − 3 1 − − − − k4 l0 = ( l1 l2 l3 + l4 l5), faces of the Voronoi cell project onto− 10 the plane2 − 𝐸∥− as two− Penrose− rhombuses, thick and thin, for the 3 = 1 ( + ) k5 10 l0 2 l1 l2 l3 l4 l5 , ° ° acute angles between any pair of the− vectors (𝑘−)∥−⇒𝑒− are− either 72 or 36 . DeThis Bruijn implies [9] thatproved the that set ofthe vertices projection of theof the five-dimensional five-dimensional cube cubic on lattice the right onto of 𝐸 Equation∥ leads to (19)the Penroserepresent rhombus a four-simplex; tiling of the any plane three withki represent thick and thethin vertices rhombuses. of an The equilateral same technique triangle, is extended any four toconstitute an arbitrary a tetrahedron. cubic lattice The [13]. four-simplex It is not ais surprise one of the that Delone we obtain polytopes. the same Similarly, tiling from pairwise the and𝐴 lattice triple becausecombinations Voronoi of thecell vectorsof 𝐴 isk obtainedi constitute as the the projection Delone cells of the composed Voronoi of cell octahedra of the cubic and lattice tetrahedra 𝐵 [14] as (see also Appendix A for further details). The Coxeter–Weyl group 𝑊(𝑏 ) is of orderP 25 ×5! and its three-dimensional facets. De Bruijn proved that the integers ni in the expression q = i=1 ni(ki) take CoxeterP number5 is ℎ=10. Normally, one expects a 10-fold symmetric tiling fromk the projectionk of values ni 1, 2, 3, 4 . The vectors in the direction ( ki) count positive and negative depending i=1 the five-dimensional∈ { cubic} lattice. The Voronoi cell of± 𝐵 khas 32 vertices (± 𝑙 ± 𝑙 ± 𝑙 ± 𝑙 ± on its sign. A patch of Penrose tiling with the numbering of vertices is shown in Figure 7 follow from 𝑙), which decompose under the Coxeter–Weyl group 𝑊(𝑎) as 32=1+1+5+10+10+5. (18) Symmetry 2019, 11, 1082 of 1910

The first two terms in (18) are represented by the vector 𝑠 ± 𝑙 →0 in four-space and the remaining vertices of the five-dimensional cube decompose as 5+10+10+5, representing the ( ) ( ) ( ) vertices of the Voronoi cell (𝜔)⋃ 𝜔 ⋃ 𝜔 ⋃ 𝜔 . In terms of the vectors 𝑙, the vectors 𝑘 can also be written as (compare with (6)) 3 1 𝑘 − 𝑙 = ( 𝑙 − 𝑙 − 𝑙 − 𝑙 − 𝑙 ), 10 2 3 1 𝑘 − 𝑙 = (− 𝑙 + 𝑙 − 𝑙 − 𝑙 − 𝑙 ), 10 2 3 1 𝑘 − 𝑙 = (−𝑙 − 𝑙 + 𝑙 − 𝑙 − 𝑙 ), (19) 10 2 3 1 𝑘 − 𝑙 = (− 𝑙 − 𝑙 − 𝑙 + 𝑙 − 𝑙 ), 10 2 𝑘 − 𝑙 = (−𝑙 − 𝑙 − 𝑙 − 𝑙 + 𝑙 ), This implies that the set of vertices of the five-dimensional cube on the right of Equation (19) represent a four-simplex; any three 𝑘 represent the vertices of an equilateral triangle, any four constitute a tetrahedron. The four-simplex is one of the Delone polytopes. Similarly, pairwise and triple combinations of the vectors 𝑘 constitute the Delone cells composed of octahedra and Symmetrytetrahedra2019 as, 11 three-dimensional, 1082 facets. De Bruijn proved that the integers 𝑛 in the expression10 𝑞 of∥ 18= ∑ 𝑛(𝑘)∥ take values ∑ 𝑛 ∈ {1, 2, 3, 4}. The vectors in the direction (±𝑘)∥ count positive and negative depending on its sign. A patch of Penrose tiling with the numbering of vertices is shown in the cut and project technique. Note that the projection of the Voronoi cell onto E form four intersecting Figure 7 follow from the cut and project technique. Note that the projection ofk the Voronoi cell onto pentagons deﬁned by de Bruijn and denoted in our notation by 𝐸∥ form four intersecting pentagons defined by de Bruijn and denoted in our notation by

𝑉 =V1 (= 𝜔 (()ω1) =−𝑉) = =V4 −=( 𝜔(()ω4) , )𝑉, =V2(= 𝜔 (()ω2) =−𝑉) = =−V3 =( 𝜔(()ω3) ) ∥a4 ∥a4 ∥a4 ∥a4 k − − k k − − k

FigureFigure 7.7. Patch of the Penrose rhombicrhombic tilingtiling obtainedobtained byby thethe cutcut andand projectproject methodmethod fromfrom thethe latticelattice ∗ A 𝐴4∗ .. TheThe fourfour typestypes ofof verticesvertices areare distinguisheddistinguished byby numbersnumbers asas statedstated inin thethe text.text.

4.4. Projection of the Root Lattice A4 by Delone Cells 4.4. Projection of the Root Lattice 𝐴 by Delone Cells The Delone polytopes (ω ) = (ω ) , (ω2) = (ω3) , tile the root lattice such that the centers 1 a4 − 4 a4 a4 − a4 of the Delone cells correspond to the vertices of the Voronoi cells. Take, for example, the four-simplex (ω ) = k , k2, k3, k , k5 and (ω ) = k , k2, k3, k , k5 . Adding vertices of these two 1 a4 { 1 4 } 4 a4 {− 1 − − − 4 − } simplexes, one obtains 10 simplexes of Delone cells centered around 10 vertices of the Voronoi cell V(0). If we add the vector k to the vertices of the Delone cell (ω ) , we obtain the vertices of the 1 4 a4 Delone cell 0, k k , k k , k k , k k whose center is at the vertex k of the Voronoi cell V(0). { 1 − 2 1 − 3 1 − 4 1 − 5n} o 1 When the vertices of the polytope (ω ) = k + k , i j = 1, 2, ... , 5 are added to the vertices of 2 a4 i j , (ω ) , we obtain 20 Delone cells centered around the vertices of (ω ) and (ω ) . For example when, 3 a4 2 a4 3 a4 we add the vector k + k to the vertices of (ω ) we obtain 10 vertices of the Delone cell centered 1 2 3 a4 around the vertex k1 + k2 of the Voronoi cell V(0) as 0, k k , k k + k k , k k + k k , k k , k k , k k , k k + k k , k k , k k . (20) { 1 − 3 1 − 3 2 − 4 1 − 4 2 − 5 2 − 5 2 − 3 1 − 4 1 − 3 2 − 5 2 − 4 1 − 5 } Note that the vertices of the Delone cells are either the root vectors or their linear combinations, as expected. Using (13) for n = 4 and j , k , l = 1, 2, ... , 5, we obtain two isosceles Robinson triangles with π π 2π 2π 2π 4π edges (2 sin 5 , 2 sin 5 , 2 sin 5 ) and (2 sin 5 , 2 sin 5 , 2 sin 5 ). The darts and kites obtained from the Robinson triangles are shown in Figure8. A patch of Penrose tiling by darts and kites is shown Figure9. 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 reference [5]. Symmetry 2019, 11, 1082 of 1911

( ) The Delone polytopes (휔1)푎4 = −(휔4)푎4, 휔2 푎4 = −(휔3)푎4, tile the root lattice such that the centers of the Delone cells correspond to the vertices of the Voronoi cells. Take, forSymmetry example 2019,, the 11, 1082four- of 1911 simplex (휔1)푎4 = {푘1, 푘2, 푘3, 푘4, 푘5} and (휔4)푎4 = {− 푘1, − 푘2, − 푘3, − 푘4, − 푘5}. Adding vertices of these two simplexes, one obtains 10 simplexes( of) Delone cells centered around 10 vertices of the The Delone polytopes (𝜔) =−(𝜔), 𝜔 =−(𝜔), tile the root lattice such that the Voronoi cell 푉(0). If we add the vector 푘 to the vertices of the Delone cell (휔 ) , we obtain the centers of the Delone cells correspond to the 1vertices of the Voronoi cells. Take, for example,4 푎4 the four- vertices of the Delone cell {0, 푘1 − 푘2, 푘1 − 푘3, 푘1 − 푘4, 푘1 − 푘5} whose center is at the vertex 푘1 of simplex (𝜔) ={𝑘, 𝑘, 𝑘, 𝑘, 𝑘} and (𝜔) ={− 𝑘,− 𝑘,− 𝑘,− 𝑘,− 𝑘}. Adding vertices of the Voronoi cell 푉(0). When the vertices of the polytope (휔 ) = {푘 + 푘 , 푖 ≠ 푗 = 1, 2, … , 5} are these two simplexes, one obtains 10 simplexes of Delone cells centered2 푎4 around푖 푗 10 vertices of the ( ) ( ) added to the vertices of 휔3 푎4, we obtain 20 Delone cells centered around the vertices of 휔2 푎4 and Voronoi cell 𝑉(0). If we add the vector 𝑘 to the vertices of the Delone cell (𝜔), we obtain the (휔3)푎 . For example when, we add the vector 푘1 + 푘2 to the vertices of (휔3)푎 we obtain 10 vertices vertices4 of the Delone cell {0, 𝑘 −𝑘, 𝑘 −𝑘, 𝑘 − 𝑘, 𝑘 −𝑘} whose center is4 at the vertex 𝑘 of of the Delone cell centered around the vertex 푘1 + 푘2 of the( Voronoi) cell 푉(0) as the Voronoi cell 𝑉(0). When the vertices of the polytope 𝜔 ={𝑘 + 𝑘,𝑖 ≠𝑗=1,2,…,5} are ( ) ( ) added{0 to, 푘 the1 − vertices푘3, 푘1 − of 푘 3𝜔+ 푘2, −we푘4 obtain, 푘1 − 푘204 +Delone푘2 − 푘 cells5, 푘2 ce−ntered푘5, 푘2 −around푘3, 푘1 the− 푘 vertices4, 푘1 − 푘of3 +𝜔 and ( ) ( ) (20) 𝜔 . For example when, we add 푘 the2 − vector푘5, 푘2 −𝑘푘+4, 𝑘푘1 −to 푘the5 } .vertices of 𝜔 we obtain 10 vertices of the Delone cell centered around the vertex 𝑘 + 𝑘 of the Voronoi cell 𝑉(0) as Note that the vertices of the Delone cells are either the root vectors or their linear combinations, as {expected.0, 𝑘 −𝑘 , 𝑘 − 𝑘 + 𝑘 −𝑘 , 𝑘 −𝑘 +𝑘 −𝑘 , 𝑘 −𝑘 , 𝑘 −𝑘 , 𝑘 −𝑘 , 𝑘 − 𝑘 + (20) Using (13) for 푛 = 4 and 푗 ≠ 𝑘푘−𝑘≠ 푙=, 𝑘1,−𝑘2, …,5 𝑘, we−𝑘 obtain }. two isosceles Robinson triangles with 휋 휋 2휋 2휋 2휋 4휋 edges (2sin , 2sin , 2sin ) and (2sin , 2sin , 2sin ). The darts and kites obtained from the Note that5 the vertices5 of5 the Delone cells5 are ei5ther the5 root vectors or their linear combinations, asRobinson expected. triangles are shown in Figure 8. Using (13) for 𝑛 = 4 and 𝑗 ≠ 𝑘 ≠ 𝑙 = 1, 2,…,5, we obtain two isosceles Robinson triangles with edges (2sin ,2sin ,2sin ) and (2sin ,2sin ,2sin ). The darts and kites obtained from the RobinsonSymmetry 2019 triangles, 11, 1082 are shown in Figure 8. 11 of 18

Figure 8. Darts and kites obtained from Robinson triangles.

A patch of Penrose tiling by darts and kites is shown Figure 9. FigureFigure 8. 8. DartsDarts and and kites kites obtained obtained from from Robinson Robinson triangles. triangles.

A patch of Penrose tiling by darts and kites is shown Figure 9.

Figure 9. A 5-fold symmetric patch of Penrose tiling with darts and kites. Figure 9. A 5-fold symmetric patch of Penrose tiling with darts and kites. In AppendixA, we have displayed the detailed relations between the Voronoi cells of A4 and a Symmetry 2019, 11, 1082 five-dimensionalThere are two cubic inflation lattice. techniques using the Robinson triangles for aperiodic tiling. One version of 1912 is the Penrose–Robinson tiling (PRT) and the other is the Tubingen triangle tiling (TTT). For further Figure 9. A 5-fold symmetric patch of Penrose tiling with darts and kites. 4.5. Projection of the Voronoi Cell of the Root Lattice A5 discussionsHere, the see Coxeter reference number [5]. is ℎ=6 and the dihedral subgroup 𝐷 of the Coxeter–Weyl group ( In) appendix A, we have displayed= the detailed relations between the Voronoi cells( ) of (퐴4 )and a 𝑊 𝑎ThereHere, is ofare the order two Coxeter inflation 12. The number techniquesDelone is h poly using6topesand the the are Robinson dihedral three different triangles subgroup orbits forD aperiodic6 of(𝜔 the) Coxeter–Weyl=− tiling.𝜔 One, version𝜔 group = W−five((𝜔a5-)dimensional) is, ( of𝜔 order) . The cubic 12. Voronoi The lattice. Delone cell 𝑉(0) polytopes of the lattice are three 𝐴 is di aff erentpolytope orbits in (five-dimensionalω ) = (ω5) ,space(ω2) and= is the Penrose–Robinson tiling (PRT) and the other is the Tubingen triangle 1tilinga5 (TTT).− Fora5 furthera5 is( ωthe4) disjoint, (ω3) . unionThe Voronoi of the cell aboveV(0 )Deloneof the lattice polytopes.A5 is aIts polytope four-dimensional in five-dimensional facets are space the andfour- is discussions− a5 seea5 reference [5]. 4.5. Projection of the Voronoi Cell of the Root Lattice 퐴5 thedimensional disjointIn appendix union rhombohedra, A, of we the have above implyingdisplayed Delone polytopes.that the detathe two-dimensionaliled Its relations four-dimensional between facets facetsthe are Voronoi the are rhombuses the cells four-dimensional of 𝐴generated and a five-dimensionalrhombohedra,by the pair of vectors implying cubic lattice.(𝑘 that, 𝑘 )=− the two-dimensional,𝑖 ≠𝑗=1,2,…,6 facets. In the are Coxeter the rhombuses plane the generated scalar product by the pairwould of vectors (k , k ) = 1 , i , j = 1, 2, ... , 6. In the Coxeter plane the scalar product would read read i j − 6 4.5. Projection of the Voronoi Cell of the Root Lattice 𝐴 (( ) ) ! ( 𝑘)∥,( 𝑘)∥ = cos 2π j. i (21) ((ki) , (kj) ) = cos − . (21) k k 6 All possible projected rhombuses turn out to be the one with interior angles (60°, 120°) . The tilingAll with possible this rhombus projected isrhombuses known in the turn literature out to be as the the one rhombille with interior tiling, angleswhich (60is used◦, 120 for◦). Thethe study tiling withof spin this structures rhombus in is knowndiatomic in molecules the literature based as theon rhombillethe Ising tiling,models. which The isrhombille used for thetiling study is depicted of spin structuresin Figure 10. in diatomic molecules based on the Ising models. The rhombille tiling is depicted in Figure 10.

Figure 10. The rhombille tiling with a D 𝐷6 symmetry.

4.6. Projection of the Delone Cells of the Root Lattice A 4.6. Projection of the Delone Cells of the Root Lattice 𝐴5 The prototiles obtained from the two-dimensional Delone faces are of three types of triangles The prototiles obtained from the two-dimensional Delone faces are of three types of triangles with angles π, π ,π , π ,π, 2π , π, π, π , represented by dark blue, yellow, and green triangles, with angles 3 , 3, 3, 6 , 6, 3, 6, ,3 ,2 represented by dark blue, yellow, and green triangles, respectively. Some patches of six-fold symmetric tilings by three triangles are shown in Figure 11, which follow from the substitution technique.

(a) Symmetry 2019, 11, 1082 of 1912

Here, the Coxeter number is ℎ=6 and the dihedral subgroup 𝐷 of the Coxeter–Weyl group ( ) ( ) ( ) 𝑊 𝑎 is of order 12. The Delone polytopes are three different orbits (𝜔) =− 𝜔 , 𝜔 = ( ) ( ) − 𝜔 , 𝜔 . The Voronoi cell 𝑉(0) of the lattice 𝐴 is a polytope in five-dimensional space and is the disjoint union of the above Delone polytopes. Its four-dimensional facets are the four- dimensional rhombohedra, implying that the two-dimensional facets are the rhombuses generated by the pair of vectors (𝑘 , 𝑘 )=− ,𝑖 ≠𝑗=1,2,…,6. In the Coxeter plane the scalar product would read

() ( 𝑘 ) ,( 𝑘 ) = cos . (21) ∥ ∥ All possible projected rhombuses turn out to be the one with interior angles (60°, 120°) . 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 Figure 10.

Figure 10. The rhombille tiling with a 𝐷 symmetry.

4.6. Projection of the Delone Cells of the Root Lattice 𝐴 Symmetry 2019, 11, 1082 12 of 18 The prototiles obtained from the two-dimensional Delone faces are of three types of triangles with angles , , , , , , , , , represented by dark blue, yellow, and green triangles, respectively. Some patches of six-fold symmetric tilings by three triangles are shown in Figure 11, respectively. Some patches of six-fold symmetric tilings by three triangles are shown in Figure 11, which follow from the substitution technique. which follow from the substitution technique.

Symmetry 2019, 11, 1082 (a) of 1913

(b)

(c)

Figure 11. Some patches (a-b-c) from three triangular prototiles from Delone cells of the root lattice Figure 11. Some patches (a-b-c) from three triangular prototiles from Delone cells of the root lattice A5, obtained𝐴 , obtained by substitution by substitution rule. rule.

4.7.4.7. Prototiles Prototiles from from the the Projection Projection of of the the Voronoi Voronoi Cell 𝑉 V((00)) ofof the the Root Root Lattice Lattice A𝐴6 The rhombic prototiles from the Voronoi cell 𝑉V((00)) can be obtained from the formula The rhombic prototiles from the Voronoi cell can be obtained from the formula 2π((ji)) ((( k𝑘i))∥,,((k 𝑘j)))∥ = =cos 7− ,𝑖, i , ≠𝑗j = 1, =1,2,…,7 2, .... ,This 7. Thiswould would lead to lead three to types three of types rhombi of rhombi with angles with k k , ,π ,6π ,2π, 5π. There3π are4π numerous discussions on the seven-fold symmetric rhombic tiling angles 7, 7 , 7 , 7 , 7 , 7 . There are numerous discussions on the seven-fold symmetric withrhombic three tiling rhombi. with One three particular rhombi. construction One particular is based construction on the cut is and based projection on the cut technique and projection from a seven-dimensional cubic lattice [13]. See also the website of Professor Gerard t’ Hooft. A patch obtained by substitution technique is illustrated in Figure 12.

Figure 12. A patch of 7-fold symmetric rhombic tiling.

4.8. Projection of the Delone Cells of the Root Lattice 𝐴 Symmetry 2019, 11, 1082 of 1913

(b)

(c)

Figure 11. Some patches (a-b-c) from three triangular prototiles from Delone cells of the root lattice

𝐴 , obtained by substitution rule.

4.7. Prototiles from the Projection of the Voronoi Cell 𝑉(0) of the Root Lattice 𝐴 The rhombic prototiles from the Voronoi cell 𝑉(0) can be obtained from the formula () ( 𝑘)∥,( 𝑘)∥ = cos ,𝑖 ≠𝑗 =1,2,…,7. This would lead to three types of rhombi with angles Symmetry 2019, 11, 1082 13 of 18 , , , , , . There are numerous discussions on the seven-fold symmetric rhombic tiling with three rhombi. One particular construction is based on the cut and projection technique from a technique from a seven-dimensional cubic lattice [13]. See also the website of Professor Gerard t’ Hooft. seven-dimensional cubic lattice [13]. See also the website of Professor Gerard t’ Hooft. A patch A patch obtained by substitution technique is illustrated in Figure 12. obtained by substitution technique is illustrated in Figure 12.

FigureFigure 12. 12.A A patch patch of of 7-fold 7-fold symmetric symmetric rhombic rhombic tiling. tiling. Symmetry 2019, 11, 1082 4.8.4.8. Projection Projection of of the the Delone Delone Cells Cells of of the the Root Root Lattice Lattice A 𝐴6 of 1914 The prototiles obtained from two-dimensional Delone faces are of four types of triangles with The π prototilesπ 5π 2obtainedπ 2π 3π from 3π two-dimensional3π π π 2 πDelone4π faces are of four types of triangles with angles 7 , 7 ,7 , 7 ,7 , 7 , 7, 7, 7 , and 7 , 7 , 7 . The substitution tilings by the last three angles , , , , , , , , ,and , , . The substitution tilings by the last three triangles are widely known as Danzer tiling. The number of prototiles obtained by projection here is trianglesthe same are as the widely tiles known obtained as from Danzer the tiling. tangents The of number the deltoid of prototiles [41]. They obtained have illustrated by projection the patches here is theinvolving same as only the threetiles obtained of the triangles. from the In tangents Figure 13 of, wethe illustratedeltoid [41]. a patch They of have sevenfold illustrated symmetric the patches tiling involvingcomprising only ofall three four of triangles. the triangles. In Figure 13, we illustrate a patch of sevenfold symmetric tiling comprising of all four triangles.

(a) (b)

Figure 13. TwoTwo 7-fold symmetric patches (a-b) obtained from the projection of the Delone cells of the root lattice 𝐴A6 ..

4.9. Prototiles from the Projection of the Voronoi CELL V( ) of the Root Lattice A 4.9. Prototiles from the Projection of the Voronoi CELL 𝑉(00) of the Root Lattice 𝐴7 The famous eightfold symmetric rhombic tiling by two prototiles is known as Amman–Beenker tiling [[6].6]. Interestingly,Interestingly, enoughenough wewe obtain obtain the the same same prototiles prototiles from from the the projection projection of theof the Voronoi Voronoi cell V( ) A D cell0 𝑉(of0) the of the root root lattice lattice7. The𝐴 . pointThe point group group of the of aperiodic the aperiodic tiling tiling is the is dihedral the dihedral group group8 of 𝐷 order of order16 and 16 the and prototiles the prototiles are generated are generated by the pair by of the vectors pair satisfying of vectors the satisfying scalar product the scalar((ki) ,product(kj) ) = k k 2π(j i) () (𝑘 ) ,(𝑘− ) = cos , 𝑖 ≠ 𝑗 = 1, 2, …,8 . We obtained two Amman–Beenker prototiles, one cos ∥ 8 ∥, i , j = 1, 2, ... , 8. We obtained two Amman–Beenker prototiles, one rhombus with angles rhombusπ 3π with angles π , π and a square , . A patch of aperiodic tiling is illustrated in Figure 14. 4 , 4 and a square 2, 2 . A patch of aperiodic tiling is illustrated in Figure 14.

Figure 14. The patch obtained from the projection of the Voronoi cell of the root lattice 𝐴 .

4.10. Projection of the Delone Cells of the Root Lattice 𝐴 Substituting 𝑛=7in (13), we obtain five different triangular prototiles from the projection of the

Delone cells of the root lattice 𝐴 . Three of the prototiles are the isosceles triangles with angles , , , , , , , , and the other two are the scalene triangles with angles , , , , , . So far as we know, no one has studied the tilings of the plane with these prototiles. Two patches of eightfold symmetric tilings with these prototiles are depicted in Figure 15. Symmetry 2019, 11, 1082 of 1914

The prototiles obtained from two-dimensional Delone faces are of four types of triangles with angles , , , , , , , , ,and , , . The substitution 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 [41]. They have illustrated the patches involving only three of the triangles. In Figure 13, we illustrate a patch of sevenfold symmetric tiling comprising of all four triangles.

(a) (b)

Figure 13. Two 7-fold symmetric patches (a-b) obtained from the projection of the Delone cells of the

root lattice 𝐴 .

4.9. Prototiles from the Projection of the Voronoi CELL 𝑉(0) of the Root Lattice 𝐴 The famous eightfold symmetric rhombic tiling by two prototiles is known as Amman–Beenker tiling [6]. Interestingly, enough we obtain the same prototiles from the projection of the Voronoi

cell 𝑉(0) of the root lattice 𝐴 . The point group of the aperiodic tiling is the dihedral group 𝐷 of order 16 and the prototiles are generated by the pair of vectors satisfying the scalar product () (𝑘 ) ,(𝑘 ) = cos , 𝑖 ≠ 𝑗 = 1, 2, …,8 . We obtained two Amman–Beenker prototiles, one Symmetry 2019∥ , 11 ,∥ 1082 14 of 18 rhombus with angles , and a square , . A patch of aperiodic tiling is illustrated in Figure 14.

FigureFigure 14. 14.The The patch patch obtained obtained fromfrom the projection of of the the Voronoi Voronoi cell cell of the of theroot root lattice lattice 𝐴 . A7.

4.10.4.10. Projection Projection of the of the Delone Delone Cells Cells of of the the Root Root LatticeLattice A𝐴7 SubstitutingSubstitutingn = 𝑛=77 in (13),in (13), we we obtain obtain ﬁve five didifferentﬀerent triangul triangularar prototiles prototiles from from the theprojection projection of the of the DeloneDelone cells ofcells the of root the lattice root Alattice. Three 𝐴 of . theThree prototiles of the areprototiles the isosceles are the triangles isosceles with triangles angles withπ , π , 6π , 7 8 8 8 angles , , , , , , , , and the other two are the scalene triangles with angles 2π , 2π , 4π , 3π , 3π , 2π and the other two are the scalene triangles with angles π , 2π , 5π , π , 3π , 4π . 8 8 8 8 8 8 8 8 8 8 8 8 , , , , , . So far as we know, no one has studied the tilings of the plane with these So far as we know, no one has studied the tilings of the plane with these prototiles. Two patches of prototiles. Two patches of eightfold symmetric tilings with these prototiles are depictedSymmetry in 2019, Figure 11, 108215. eightfold symmetric tilings with these prototiles are depicted in Figure 15. Symmetry 2019, 11, 1082 of 1915 of 1915

(a) (b) (a) (b)

Figure 15. Patches (a-b) of prototiles from Delone cells of the root lattice 𝐴 with 8-fold symmetry. FigureFigure 15. Patches15. Patches (a-b) (a-b) of of prototiles prototiles from from DeloneDelone cells cells of of the the root root lattice lattice 𝐴A with7 with 8-fold 8-fold symmetry. symmetry.

4.11. Prototiles from Projection of the Voronoi Cell of the Root Lattice 𝐴 4.11.4.11. Prototiles Prototiles from from Projection Projection of of the the Voronoi Voronoi Cell Cell ofof thethe Root Lattice Lattice 𝐴 A11 The 12-fold symmetric rhombic tiling has 3 prototiles of rhombuses with interior angles , π, 5π The 12-foldThe 12-fold symmetric symmetric rhombic rhombic tiling tiling has has 33 prototilesprototiles of of rhombuses rhombuses with with interior interior angles angles , ,, , 6 6 π 2π, , and π π,. A patch of 12-fold symmetric tiling obtained by substitution technique is depicted , , and , and, , . A. A patch patch of of 12-fold12-fold symmetric tiling tiling obtain obtaineded by bysubstitution substitution technique technique is depicted is depicted 3 3 2 2 in Figure 16. in Figurein Figure 16. 16.

Figure 16. Three rhombuses illustrated with different colors obtained from the Voronoi cell of 𝐴 FigureFigure 16. Three 16. Three rhombuses rhombuses illustrated illustrated with with di differentﬀerent colors colors obtained obtained fromfrom the Voronoi cell cell of of 𝐴A and and a patch is depicted. 11 a patchand is a depicted.patch is depicted.

4.12. Prototiles from the Projection of the Delone Cells of the Root Lattice 𝐴 4.12. Prototiles from the Projection of the Delone Cells of the Root Lattice 𝐴 The number of prototiles in this case is 12. As the rank of the Coxeter–Weyl group is increasing, 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 the number of the triangular prototiles are also increasing. The 12 triangles are given as the set of integers (𝑛,𝑛,𝑛) defined in Section 3 as integers (𝑛,𝑛,𝑛) defined in Section 3 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). (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 Figure 17. A patch of 12-fold symmetric tiling is depicted in Figure 17. Symmetry 2019, 11, 1082 15 of 18

4.12. Prototiles from the Projection of the Delone Cells of the Root Lattice A11 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 (n1, n2, n3) deﬁned in Section3 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).

Symmetry 2019, 11, 1082 A patch of 12-fold symmetric tiling is depicted in Figure 17. of 1916

Figure 17.17. A 12-fold symmetric patch obtained by substitutionsubstitution from triangular prototiles originating from Delone cells of the root lattice 𝐴 . from Delone cells of the root lattice A11.

5. Concluding Remarks

Rhombic prototiles usually arise from projecti projectionon of of the the higher higher dimensional dimensional cubic cubic lattices lattices 𝐵Bn+1 , because two-dimensional square faces project onto rhombuses and represent the local 2(n𝑛+1+ 1))-fold-fold symmetric rhombic aperiodicaperiodic tilings ofof thethe CoxeterCoxeter plane.plane. Depending on the shift of thethe CoxeterCoxeter plane,plane, oneone maymay reduce reduce the the tilings tilings to to(n (+𝑛+11)-fold)-fold symmetric symmetric aperiodic aperiodic tilings. tilings. The The rhombic rhombic(n + (𝑛+11)-fold)- symmetricfold symmetric aperiodic aperiodic tilings tilings can also canbe also obtained be obtained from thefrom lattice the latticeAn, for 𝐴 the, for latter the latticelatter lattice is a sub-lattice is a sub- oflattice the lattice of theB latticen+1. One 𝐵 may . One visualize may visualize that the lattice that theBn +lattice1 projects 𝐵 onto projects the lattice ontoA then at lattice first stage, 𝐴 at as first the Voronoistage, as cell the of Voronoi the cubic cell lattice of th projectse cubic lattice onto the projects Voronoi onto cell the of theVoronoi root lattice cell ofA then. So, root the lattice classification 𝐴 . So, ofthe the classification rhombic aperiodic of the rhombic tilings of aperiodic both lattices tilings are the of same.both lattices An advantage are the ofsame. the root An latticeadvantageAn is thatof the it canroot be lattice tiled by 𝐴 the is Delonethat it cellscan be with tiled triangular by thetwo-dimensional Delone cells with faces. triangular Projections two-dimensional of the Delone cells faces. of theProjections root lattice of Athen allow Delone the classificationscells of the root of the lattice triangular 𝐴 allow aperiodic the tilings.classifications A systematic of the study triangular of the projectionsaperiodic tilings. of the DeloneA systematic and Voronoi study cellsof the of projecti the rootons and of weight the Delone lattices and of theVoronoi simply cells laced of ADEthe root Lie algebrasand weight may lattices lead to of more the simply interesting laced prototiles ADE Lie and algebras aperiodic may tilings. lead to The more present interesting paper wasprototiles concerned and onlyaperiodic with thetilings. aperiodic The present rhombic paper and triangularwas concerned tilings only of the with root the lattice aperiodic exemplifying rhombic the and 5-fold, triangular 8-fold andtilings 12-fold of the symmetric root lattice aperiodic exemplifying tilings the asthey 5-fold, represent 8-fold someand 12-fold quasicrystallographic symmetric aperiodic structures. tilings as they represent some quasicrystallographic structures. Author Contributions: N.O.K.: Project administration, investigation, methodology, formal analysis, supervision, writing-review&editing;Author Contributions: A.A.-S.: N. O. Investigation, K.: Project software, administration, visualization; invest M.K.:igation, Project methodology, administration, conceptualization,formal analysis, investigation,supervision, methodology,writing-review&editing; original draft, writing-reviewA.A.-S.: Investigation, and editing; software, R.K.: Software, visualization; visualization. M. K.: Project Funding:administration,This research conceptualization, was funded investigation, by Sultan Qaboos methodolo University,gy, original Oman. draft, writing-review and editing; R. K.: Software, visualization. Conflicts of Interest: The authors declare no conflict of interest. Funding: This research was funded by Sultan Qaboos University, Oman.

Conflicts of Interest: The authors declare no conflict of interest.

Appendix A

Appendix A.1. Three-Dimensional Cube and the Voronoi Cell 𝑉(0) of 𝐴 Vertices of the Voronoi cell of a three-dimensional cubic lattice are given by the vectors (±𝑙 ± 𝑙 ± 𝑙 ). Since the vector representing the space diagonal projects to the origin ± 𝑙 →0, the images of the six remaining vertices of the cube can be represented by the vectors ±𝑘 ,(𝑖=1,2,3) in 2D space. Each set of vectors 𝑘 and −𝑘 represents an equilateral triangle. The union of two sets is the hexagon representing the Voronoi cell 𝑉(0) of the root lattice 𝐴. In particle physics, it is used to define the color and anti-color degrees of freedom of the quarks. Vertices of the six Delone cells (equilateral triangles) centralizing the vertices of the Voronoi cell 𝑉(0) consists of the vectors 0, ± ( 𝑘 −𝑘). Symmetry 2019, 11, 1082 16 of 18

Appendix A.

Appendix A.1. Three-Dimensional Cube and the Voronoi Cell V(0) of A2 Vertices of the Voronoi cell of a three-dimensional cubic lattice are given by the vectors 1 ( l l l ). Since the vector representing the space diagonal projects to the origin 1 l 0, 2 ± 1 ± 2 ± 3 ± 2 0 → the images of the six remaining vertices of the cube can be represented by the vectors k , (i = 1, 2, 3) ± i in 2D space. Each set of vectors k and k represents an equilateral triangle. The union of two sets is the i − i hexagon representing the Voronoi cell V(0) of the root lattice A2. In particle physics, it is used to deﬁne the color and anti-color degrees of freedom of the quarks. Vertices of the six Delone cells (equilateral triangles) centralizing the vertices of the Voronoi cell V(0) consists of the vectors 0, (k k ). ± i − j

Appendix A.2. Five-Dimensional Cube and the Voronoi Cell V(0) of A4 Vertices of the Voronoi cell of a five-dimensional cubic lattice are given by the vectors 1 ( l l l l l ). The cube is cut by hyperplanes orthogonal to the vector l at four 2 ± 1 ± 2 ± 3 ± 4 ± 5 0 levels, which splits the cube into four Delone polytopes of A4, the union of which represents the Voronoi cell of A4. When projected into four-dimensional space they can be represented as (ω ) = (ω ) , (ω2) = (ω ) . Vertices of a typical two-dimensional face of five 5-dimensional 1 a4 − 4 a4 a4 − 4 a4 cube can be taken as 1 ( l + l + l + l + l ), 1 ( l + l + l + l + l ), 1 ( l l + l + l + l ), 2 1 2 3 4 5 2 − 1 2 3 4 5 2 1 − 2 3 4 5 1 ( l l + l + l + l ). The edges of the square are represented by the vectors parallel to l 2 − 1 − 2 3 4 5 1 and l . In four-dimensional space, l projects into the origin by (6) and therefore l and l are 2 0 1 2 replaced by the vectors k and k , constituting a rhombus with obtuse angle θ = cos 1 1 104.5 . 1 2 − − 4 ◦ Each three-dimensional cubic facet of a five-dimensional cube projects into a rhombohedral facet of the Voronoi cell V(0), as it is shown in the following example. The set of vertices 1 ( l l l l l ) is a three-dimensional cube centered at the point 2 1 ± 2 ± 3 ± 4 − 5 1 ( l l ). This cube projects into a three-dimensional rhombohedral facet of the Voronoi cell V(0) of 2 1 − 5 A4, as shown in Figure6 where the vertices of the rhombohedron are given by

1 (l l l l l ) k 2 1 − 2 − 3 − 4 − 5 → 1 1 (l + l + l + l l ) k , 2 1 2 3 4 − 5 → − 5 1 (l + l l l l ) k + k , 2 1 2 − 3 − 4 − 5 → 1 2 1 (l l + l l l ) k + k , 2 1 − 2 3 − 4 − 5 → 1 3 (A1) 1 (l l l + l l ) k + k , 2 1 − 2 − 3 4 − 5 → 1 4 1 (l + l + l l l ) (k + k ), 2 1 2 3 − 4 − 5 → − 4 5 1 (l + l l + l l ) (k + k ), 2 1 2 − 3 4 − 5 → − 3 5 1 (l l + l + l l ) (k + k ). 2 1 − 2 3 4 − 5 → − 2 5 All the rhombohedral facets of the Voronoi cell V(0) are obtained by permutations. Centers of the rhombohedral facets are the halves of the roots of the root system. This also explains why the Voronoi cell of A4 is the dual polytope of the root polytope. In brief, the square face of the ﬁve-dimensional cube projects onto the rhombic face of the Voronoi cell V(0) in four-dimensional space and all three-dimensional cubic facets project onto the three-dimensional rhombohedra. Of course, the results of both projections lead to the same Penrose rhombic aperiodic tilings. This is true for all B + An projections, the only diﬀerence is that the n 1 → aperiodic tiles from the projections of Bn+1 exhibit 2(n + 1)-fold symmetry, while An projections display (n + 1)-fold symmetry only.

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