Icosahedral Polyhedra from 6 Lattice and Danzer's ABCK Tiling

Home , Icosahedron, Rhombic dodecahedron, Rhombic icosahedron, Rhombic triacontahedron

Icosahedral Polyhedra from 퐷6 lattice and Danzer’s ABCK tiling

Abeer Al-Siyabi, a Nazife Ozdes Koca, a* and Mehmet Kocab aDepartment of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud, 123 Muscat, Sultanate of Oman, bDepartment of Physics, Cukurova University, Adana, Turkey, retired, *Correspondence e-mail: [email protected]

ABSTRACT

It is well known that the point group of the root lattice 퐷6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group 퐻3 , its roots and weights are determined in terms of those of 퐷6 . Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of 퐷6 determined by a pair of integers (푚1, 푚2 ) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of 퐻3 and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers which are linear combinations of the integers (푚1, 푚2 ) with coefficients from Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the < 퐴퐵퐶퐾 > octahedral tilings in 3D space with icosahedral symmetry 퐻3 and those related transformations in 6D space with 퐷6 symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in 퐷6 .The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron” and “C-polyhedron” generated by the icosahedral group have been discussed.

Keywords: Lattices, Coxeter-Weyl groups, icosahedral group, projections of polytopes, polyhedra, aperiodic tilings, quasicrystals

1. Introduction

Quasicrystallography as an emerging science attracts the interests of many scientists varying from the fields of material science, chemistry and physics. For a review see for instance the references (Di Vincenzo & Steinhardt, 1991; Janot, 1993; Senechal, 1995). It is mathematically intriguing as it requires the aperiodic tiling of the space by some prototiles. There have been several approaches to describe the aperiodicity of the quasicrystallographic space such as the set theoretic techniques, cut-and-project scheme of the higher dimensional lattices and the intuitive approaches such as the Penrose-like tilings of the space. For a review of these techniques, we propose the reference (Baake & Grimm, 2013). There have been two major approaches for the aperiodic tiling of the 3D space with local icosahedral symmetry. One of them is the Socolar-Steinhardt tiles (Socolar & Steinhardt, 1986) consisting of acute rhombohedron with golden rhombic faces, Bilinski rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, the latter three are constructed with two Ammann tiles of acute and obtuse rhombohedra. Later it was proved that (Danzer, Papadopolos & Talis, 1993), (Roth, 1993) they can be constructed by the Danzer’s ABCK tetrahedral tiles (Danzer, 1989). Katz (Katz, 1989) and recently Hann-Socolar-Steinhardt (Hann, Socolar & Steinhardt, 2018) proposed a model of tiling scheme with decorated Ammann tiles. A detailed account of the Danzer ABCK tetrahedral tilings can be found in “math.uni- bielefeld.de/icosahedral tilings in ℝ3: the ABCK tilings” and in page 231 of the reference (Baake & Grimm, 2013) where the substitution matrix, its eigenvalues and the corresponding eigenvectors are studied. The right and left eigenvectors of the substitution matrix corresponding to the Perron-Frobenius eigenvalue are well known and will not be repeated here. Ammann rhombohedral and Danzer ABCK tetrahedral tilings are intimately related with the projection of six-dimensional cubic lattice and the root and weight lattices of 퐷6, the point symmetry group of which is of order 256!. See for a review the paper “Modelling of quasicrystals” by Kramer (Kramer, 1993) and references therein. Similar work has also been carried out in the reference (Koca, Koca & Koc, 2015). Kramer and Andrle (Kramer & Andrle, 2004) have investigated Danzer tiles from the wavelet point of view and their relations with the lattice 퐷6. In what follows we point out that the icosahedral symmetry requires a subset of the root lattice 퐷6 characterized by a pair of integers (푚1, 푚2) with 푚1 + 푚2 = 푒푣푒푛 which are the coefficients of the orthogonal set of vectors 푙𝑖, (푖 = 1, 2, … ,6). Our approach is different than the cut and project scheme of lattice 퐷6 as will be seen in the sequel. The paper consists of two major parts; first part deals with the determination of Platonic and Archimedean icosahedral polyhedra by projection of the fundamental weights of the root lattice 퐷6 into 3D space and the second part employs the technique to determine the images of the Danzer tiles in 퐷6. Inflation of the Danzer tiles are related to a redefinition of the pair of integers (푚1, 푚2) by the Fibonacci sequence. Embeddings of basic tiles in the inflated ones require translations and rotations in 3D space where the corresponding transformations in 6D space can be easily determined by the technique we have introduced. This technique which restricts the lattice 퐷6 to its subset has not been discussed elsewhere. The paper is organized as follows. In Sec. 2, we introduce the root lattice 퐷6, its icosahedral subgroup, decomposition of its weights in terms of the weights of the icosahedral group 퐻3 leading to the Archimedean polyhedra projected from the 퐷6 lattice. It turns out that the lattice vectors to be projected are determined by a pair of integers (푚1 , 푚2 ). In Sec. 3 we introduce the ABCK tetrahedral Danzer tiles in terms of the fundamental weights of the icosahedral 1+ 5 group 퐻 . Tiling by inflation with 휏푛 where 휏 = √ and 푛 ∈ ℤ is studied in 퐻 space by 3 2 3 prescribing the appropriate rotation and translation operators. The corresponding group elements of 퐷6 are determined noting that the pair of integers (푚1 , 푚2 ) can be expressed as the linear

2 combinations of similar integers with coefficients from Fibonacci sequence. Sec. 4 is devoted for conclusive remarks.

2. Projection of 푫ퟔ lattice under the icosahedral group 푯ퟑ and the Archimedian polyhedra

We will use the Coxeter diagrams of 퐷6 and 퐻3 to introduce the basic concepts of the root systems, weights and the projection technique. A vector of the 퐷6 lattice can be written as a linear combination of the simple roots with integer coefficients:

6 6 6 휆 = ∑𝑖=1 푛𝑖훼𝑖 = ∑𝑖=1 푚𝑖푙𝑖, ∑𝑖=1 푚𝑖 = 푒푣푒푛, 푛𝑖, 푚𝑖 ∈ ℤ. (1)

Here 훼𝑖 are the simple roots of 퐷6 defined in terms of the orthonormal set of vectors as 훼𝑖 = 푙𝑖 − 푙𝑖+1, 푖 = 1, … ,5 and 훼6 = 푙5 + 푙6 and the reflection generators of 퐷6 act as 푟𝑖: 푙𝑖 ⟷ 푙𝑖+1 and 푟6: 푙5 ⟷ −푙6. The generators of 퐻3 can be defined as (Koca, Koca & Koc, 2015) 푅1 = 푟1푟5, 푅2 = 푟2푟4 and 푅3 = 푟3푟6 where the Coxeter element, for example, can be taken as 푅 = 푅1푅2푅3. The weights of 퐷6 are given by

ω1 = 푙1, ω2 = 푙1 + 푙2, ω3 = 푙1 + 푙2 + 푙3, ω4 = 푙1 + 푙2 + 푙3 + 푙4, 1 1 ω = (푙 + 푙 + 푙 + 푙 + 푙 − 푙 ) , ω = (푙 + 푙 + 푙 + 푙 + 푙 + 푙 ). (2) 5 2 1 2 3 4 5 6 6 2 1 2 3 4 5 6

The Voronoi cell of the root lattice 퐷6 is the dual polytope of the root polytope of ω2 (Koca et. al, 2018) and determined as the union of the orbits of the weights ω1, ω5 and ω6 which correspond to the holes of the root lattice (Conway & Sloane, 1999). If the roots and weights of ́ 퐻3 are defined in two complementary 3D spaces 퐸∥ and 퐸⊥as 훽𝑖(훽𝑖) and 푣𝑖(푣́𝑖), (푖 = 1, 2, 3) respectively then they can be expressed in terms of the roots and weights of 퐷6 as

1 1 훽 = (훼 + 휏훼 ), 푣 = (ω + 휏ω ), 1 √2+휏 1 5 1 √2+휏 1 5 1 1 훽 = (훼 + 휏훼 ), 푣 = (ω + 휏ω ), (3) 2 √2+휏 2 4 2 √2+휏 2 4 1 1 훽 = (훼 + 휏훼 ), 푣 = (ω + 휏ω ). 3 √2+휏 6 3 3 √2+휏 6 3

1+ 5 For the complementary 3D space replace 훽 by 훽́ , 푣 by 푣́ in (3) and 휏 = √ by its algebraic 𝑖 𝑖 𝑖 𝑖 2 1− 5 conjugate 휎 = √ = −휏−1. Consequently, Cartan matrix of 퐷 (Gram matrix in lattice 2 6 terminology) and its inverse block diagonalize as

2 −1 0 0 0 0 2 −1 0 0 0 0 −1 2 −1 0 0 0 −1 2 −휏 0 0 0

0 −1 2 −1 0 0 0 −휏 2 0 0 0 퐶 = → 0 0 −1 2 −1 −1 0 0 0 2 −1 0 0 0 0 −1 2 0 0 0 0 −1 2 −휎 [ 0 0 0 −1 0 2] [ 0 0 0 0 −휎 2] (4)

2 2 2 2 1 1 2 + 휏 2휏2 휏3 0 0 0

2 4 4 4 2 2 2휏2 4휏2 2휏3 0 0 0

1 1 3 3 2 퐶−1 = 2 4 6 6 3 3 ⟶ 휏 2휏 3휏 0 0 0 . 2 2 4 6 8 4 4 2 0 0 0 2 + 휎 2휎2 휎3 1 2 3 4 3 2 0 0 0 2휎2 4휎2 2휎3 [1 2 3 4 2 3] [ 0 0 0 휎3 2휎3 3휎2 ]

Fig.1 shows how one can illustrate the decomposition of the 6D space as the direct sum of two complementary 3D spaces where the corresponding nodes 훼𝑖, (푖 = 1, 2, . . . ,6), 훽𝑖, (푖 = 1, 2, 3) ́ represent the corresponding simple roots and 훽𝑖 stands for 훽𝑖 in the complementary space.

Figure 1 Coxeter-Dynkin diagrams of 퐷6 and 퐻3 illustrating the symbolic projection.

1 For a choice of the simple roots of 퐻 as 훽 = (√2, 0, 0), 훽 = − (1, 휎, 휏), 훽 = (0, 0, √2) 3 1 2 √2 3 and the similar expressions for the roots in the complementary space we can express the components of the set of vectors 푙𝑖 푖 = 1, 2, … ,6) as

푙1 1 휏 0 휏 −1 0

푙2 −1 휏 0 −휏 −1 0

푙 1 0 1 휏 0 휏 −1 3 = . (5) 푙4 √2(2+휏) 0 1 −휏 0 휏 1 푙5 휏 0 1 −1 0 휏 [ ] [푙6] −휏 0 1 1 0 휏

First three and last three components project the vectors 푙𝑖 into 퐸∥ and 퐸⊥ spaces respectively. On 1 the other hand the weights are represented by the vectors 푣 = (1, 휏, 0), 푣 = √2(0, 휏, 0), 푣 = 1 √2 2 3 1 (0, 휏2, 1). The generators of 퐻 in the space 퐸 read √2 3 ∥

−1 0 0 1 −휎 −휏 1 0 0 1 푅 = [ 0 1 0], 푅 = [−휎 휏 1], 푅 = [0 1 0], (6) 1 2 2 3 0 0 1 −휏 1 휎 0 0 −1 satisfying the generating relations

2 2 2 2 3 5 푅1 = 푅2 = 푅3 = (푅1푅3) = (푅1푅2) = (푅2푅3) = 1. (7)

Their representations in the 퐸⊥ space follows from (6) by algebraic conjugation. The orbits of the weights 푣𝑖 under the icosahedral group 퐻3 can be obtained by applying the group elements on the weight vectors as (푣1)ℎ 1 3 = {(±1, ±휏, 0), (±휏, 0, ±1), (0, ±1, ±휏)}, √2 2

(휏−1 푣 ) 1 1 1 1 2 ℎ3 = {(±1,0,0), (0,0, ±1), (0, ±1,0), (±1, ±휎, ±휏), (±휎, ±휏, ±1), (±휏, ±1, ±휎)}, (8) 2√2 2 2 2 2

(휏−1푣 ) 1 3 ℎ3 = {(±1, ±1, ±1), (0, ±휏, ±휎), (±휏, ±휎, 0), (±휎, 0, ±휏)}, √2 2

where the notation (푣𝑖)ℎ3is introduced for the set of vectors generated by the action of the icosahedral group on the weight 푣𝑖 . The sets of vectors in (8) represent the vertices of an icosahedron, an icosidodecahedron and a dodecahedron respectively in the 퐸∥ space. The weights 푣1, 푣2 and 푣3 denote also the 5-fold, 2- (푣 ) fold and 3-fold symmetry axes of the icosahedral group. The union of the orbits of 1 ℎ3 and √2 (휏−1푣 ) 3 ℎ3 constitute the vertices of a rhombic triacontahedron. √2 It is obvious that the Coxeter group 퐷6 is symmetric under the algebraic conjugation and this is more apparent in the characteristic equation of the Coxeter element of 퐷6 given by

(휆3 + 휎휆2 + 휎휆 + 1)(휆3 + 휏휆2 + 휏휆 + 1) = 0, (9) whose eigenvalues lead to the Coxeter exponents of 퐷6 . The first bracket is the characteristic polynomial of the Coxeter element of the matrices in (6) describing it in the 퐸∥ space and the second bracket describes it in the 퐸⊥ space (Koca, Koc, Al-Barwani, 2001). Therefore, projection of 퐷6 into either space is the violation of the algebraic conjugation. It would be interesting to discuss the projections of the fundamental polytopes of 퐷6 into 3D space possessing the icosahedral symmetry. It is beyond the scope of the present paper however we may discuss a few interesting cases. The orbit generated by the weight ω1 is a polytope with 12 vertices called cross polytope and represents an icosahedron when projected into either space. The orbit of weight ω2 constitutes the “root polytope” of 퐷6 with 60 vertices which projects into 3D space as two icosidodecahedra with 30 vertices each, the ratio of radii of the circumspheres is 휏. The dual of the root polytope is the union of the three polytopes generated by ω1, ω5 and ω6 which constitute the Voronoi cell of the lattice 퐷6 as mentioned earlier and projects into an icosahedron and two rhombic triacontahedra. Actually, they consist of three icosahedra with the ratio of radii 1, 휏, 휏2 and two dodecahedra with the radii in proportion to 휏. The orbit generated by the weight vector ω3 is a ∗ polytope with 160 vertices and constitutes the Voronoi cell of the weight lattice 퐷6 . It projects into two dodecahedra and two polyhedra with 60 vertices each. Voronoi cells can be used as windows for the cut and projects scheme however we prefer the direct projection of the root lattice as described in what follows. A general root vector can be decomposed in terms of the weights 푣𝑖(푣́𝑖) as

1 푚 푙 + 푚 푙 + 푚 푙 + 푚 푙 + 푚 푙 + 푚 푙 = [(푚 − 푚 + 휏푚 − 휏푚 ) 푣 + 1 1 2 2 3 3 4 4 5 5 6 6 √2+휏 1 2 5 6 1 (푚2 − 푚3 + 휏푚4 − 휏푚5) 푣2 + (푚5 + 푚6 + 휏푚3 − 휏푚4) 푣3] + ( 푣𝑖 ⟶ 푣́𝑖, 휏 ⟶ 휎). (10)

Projection of an arbitrary vector of 퐷6 into the space 퐸∥, or more thoroughly, onto a particular weight vector, for example, onto the weight 푣1 is given by

푣 [푚 푙 + 푚 (푙 + 푙 + 푙 + 푙 − 푙 )] ≡ [(푚 − 푚 ) ω + 2푚 ω ] = 푐( 1 ) (11) 1 1 2 2 3 4 5 6 ∥ 1 2 1 2 5 ∥ √2

2 and represents an icosahedron where 푐 ≡ 푐(푚 , 푚 ) = √ (푚 − 푚 + 2푚 휏) is an overall 1 2 2+휏 1 2 2 scale factor. The subscript ∥ means the projection into the space 퐸∥. The expression in (11) shows that 푚1 + 푚2 = even implying that the pair of integers (푚1, 푚2) are either both even or both odd. We will see that not only icosahedron but also dodecahedron and the icosahedral Archimedean polyhedra can be obtained by relations similar to (11). The Platonic and Archimedean polyhedra with icosahedral symmetry are listed in Table 1 as projections of 퐷6 lattice vectors determined by the pair of integers (푚1, 푚2). Table 1 shows that only a certain subset of vectors of 퐷6 project onto regular icosahedral polyhedra. We will see in the next section that the Danzer tiling also restricts the 퐷6 lattice in a domain where the vectors are determined by the pair of integers (푚1, 푚2).These quantities studied here are potentially applicable to many graph indices (Alhevaz, Baghipur, Shang, 2019)

Table1 Platonic and Archimedean icosahedral polyhedra projected from 퐷6 (vertices are orbits under the icosahedral group).

Vector of the Polyhedron in 푯ퟑ Corresponding Vector in 푫ퟔ 2 푐 ≡ √ (푚 − 푚 + 2푚 휏) 푚 , 푚 ∈ ℤ 2 + 휏 1 2 2 1 2

Icosahedron 푚1푙1 + 푚2(푙2 + 푙3 + 푙4 + 푙5 − 푙6) = (푚1 − 푚2)휔1 + 2푚2휔5 푣1 푐 푐 = (1, 휏, 0) 푚 , 푚 ∈ 2ℤ or 2ℤ + 1 √2 2 1 2

Dodecahedron 1 [(푚 + 3푚 )(푙 + 푙 + 푙 ) 푣 푐 1 2 1 2 3 푐 3 = (0, 휏2, 1) 2 √2 2 + (푚1 − 푚2)(푙4 + 푙5 + 푙6)] = (푚1 − 푚2)휔6 + 2푚2휔3

푚1, 푚2 ∈ 2ℤ or 2ℤ + 1

Icosidodecahedron (푚1 + 푚2)(푙1 + 푙2) + 2푚2(푙3 + 푙4) 푣2 = (푚1 − 푚2)휔2 + 2푚2휔4 푐 = 푐(0, 휏, 0) √2 푚1, 푚2 ∈ 2ℤ or 2ℤ + 1

Truncated Icosahedron (2푚1 + 푚2)푙1 + (푚1 + 2푚2)푙2 +푚 (3푙 + 3푙 + 푙 − 푙 ) (푣1 + 푣2) 푐 2 3 4 5 6 푐 = (1, 3휏, 0) = (푚 − 푚 )(휔 + 휔 ) + 2푚 (휔 + 휔 ) 2 1 2 1 2 2 4 5 √2 푚1, 푚2 ∈ 2ℤ or 2ℤ + 1 Small Rhombicosidodecahedron 1 [(3푚1 + 3푚2)푙1 + (푚1 + 5푚2)(푙2 + 푙3) (푣1 + 푣3) 푐 2 푐 = (1, 2휏 + 1, 1) + (푚1 + 푚2)(푙4 + 푙5) √2 2 + (푚1 − 3푚2)푙6] = (푚1 − 푚2)(휔1 + 휔6) + 2푚2(휔3 + 휔5)

Truncated Dodecahedron 1 [(3푚1 + 5푚2)(푙1 + 푙2) + (푚1 + 7푚2)푙3 (푣2 + 푣3) 푐 2 푐 = (0, 3휏 + 1, 1) + (푚1 + 3푚2)푙4 √2 2 + (푚1 − 푚2)(푙5+푙6)] ( ) = 푚1 − 푚2 (휔2 + 휔6) + 2푚2(휔3 + 휔4) Great Rhombicosidodecahedron 1 [5(푚1 + 푚2)푙1 + (3푚1 + 7푚2)푙2 + (푚1 + 9푚2)푙3 (푣1 + 푣2 + 푣3) 푐 2 푐 = (1, 4휏 + 1, 1) + (푚1 + 5푚2)푙4 + (푚1 + 푚2)푙5 √2 2 + (푚1 − 3푚2)푙6] = (푚1 − 푚2)(휔1 + 휔2 + 휔6) + 2푚2(휔3 + 휔4 + 휔5)

3. Danzer’s ABCK tiles and 푫ퟔ lattice

We introduce the ABCK tiles with their coordinates in Fig. 2 as well as their images in lattice 퐷6. For a fixed pair of integers (푚1, 푚2) ≠ (0, 0) let us define the image of K by vertices 퐷1(푚1, 푚2), 퐷2(푚1, 푚2), 퐷3(푚1, 푚2) and 퐷0 (0,0) in lattice 퐷6 and its projection in 3D space where 퐷0 (0,0) represents the origin. They are given as

푐푣 퐷 (푚 , 푚 ) = 1 = [푚 푙 + 푚 (푙 + 푙 + 푙 + 푙 − 푙 )] = [(푚 − 푚 )휔 + 2푚 휔 ] , 1 1 2 √2 1 1 2 2 3 4 5 6 ∥ 1 2 1 2 5 ∥

푐푣 1 1 퐷 (푚 , 푚 ) = 2 = [(푚 + 푚 )(푙 + 푙 ) + 2푚 (푙 + 푙 )] = [(푚 − 푚 )휔 + 2푚 휔 ] , (12) 2 1 2 2√2 2 1 2 1 2 2 3 4 ∥ 2 1 2 2 2 4 ∥

푐휏−1푣 1 퐷 (푚 , 푚 ) = 3 = [(푚 + 푚 )(푙 + 푙 + 푙 ) + (−푚 + 3푚 )(푙 + 푙 + 푙 )] 3 1 2 √2 2 1 2 1 2 3 1 2 4 5 6 ∥

= [2푚2휔3 + (−푚1 + 3푚2)휔6]∥, which follows from Table 1 with redefinitions of the pair of integers (푚1, 푚2). With removal of the notation ∥ they represent the vectors in 6D space. A set of scaled vertices defined by 퐷́ 𝑖 ≡ 퐷 (푚 , 푚 ) 𝑖 1 2 are used for the vertices of the Danzer’s tetrahedra in Fig. 2. 푐

A-tetrahedron B-tetrahedron

C-tetrahedron K-tetrahedron Figure 2 Danzer’s tetrahedra.

It is instructive to give a brief introduction to Danzer’s tetrahedra before we go into details. The edge lengths of ABC tetrahedra are related to the weights of the icosahedral group 퐻3 whose 2+휏 ‖ 푣 ‖ 3 ‖ 휏−1푣 ‖ ‖ 휏−1푣 ‖ edge lengths are given by 푎 = √ = 1 , 푏 = √ = 3 , 1 = 2 and their multiples 2 √2 2 √2 √2 by 휏 and 휏−1, where 푎, 푏, 1 are the original edge lengths introduced by Danzer. However, the 1 휏 휏−1 tetrahedron K has edge lengths also involving , , . As we noted in Sec. 2 the vertices of a 2 2 2 (푣 ) (휏−1푣 ) rhombic triacontahedron consist of the union of the orbits 1 ℎ3 and 3 ℎ3. One of its cells is √2 √2 a pyramid based on a golden rhombus with vertices

푣 1 휏−1푣 1 푣 1 휏−1푣 1 1 = (1, 휏, 0), 3 = (0, 휏, −휎), 푅 1 = (−1, 휏, 0), 푅 3 = (0, 휏, 휎), (13) √2 2 √2 2 1 √2 2 3 √2 2 and the apex is at the origin. The coordinate of the intersection of the diagonals of the rhombus is 푣 1 the vector 2 = (0, 휏, 0) and its magnitude is the in-radius of the rhombic triacontahedron. 2√2 2 푣 휏−1푣 푣 Therefore, the weights 1 , 3 , 2 and the origin can be taken as the vertices of the tetrahedron √2 √2 2√2 K. As such, it is the fundamental region of the icosahedral group from which the rhombic triacontahedron is generated (Coxeter, 1973). The octahedra generated by these tetrahedra denoted by < 퐵 >, < 퐶 > and < 퐴 > comprise 4 copies of each obtained by a group of order 4 generated by two commuting generators 푅1and 푅3 or their conjugate groups. The octahedron

< 퐾 > consists of 8K generated by a group of order 8 consisting of three commuting generators 푅1, 푅3 as mentioned earlier and the third generator 푅0 is an affine reflection (Humphrey,1992) with respect to the golden rhombic face induced by the affine Coxeter group 퐷̃6. One can dissect the octahedron < 퐾 > into three non-equivalent pyramids with rhombic bases by cutting along the lines orthogonal to three planes 푣1 − 푣2, 푣2 − 푣3, 푣3 − 푣1. One of the pyramids is generated by the tetrahedron K upon the actions of the group generated by the reflections 푅1 and 푅3. If we call 푅1퐾 = 퐾́ as the mirror image of K the others can be taken as the 0 tetrahedra obtained from K and 푅1퐾 by a rotation of 180 around the axis 푣2. A mirror image of 4퐾 = (2퐾 + 2퐾́) with respect to the rhombic plane (corresponding to an affine reflection) complements it up to an octahedron of 8K as we mentioned above. In addition to the vertices in (13) the octahedron 8K also includes the vertices (0, 0, 0) and (0, 휏, 0). Octahedral tiles are depicted in Fig. 3. Dissected pyramids constituting the octahedron < 퐾 > as depicted in Fig. 4 have bases, two of which with bases of golden rhombuses with edge lengths (heights) 휏 휏−1 1 휏−1푎 ( ) , 푎 ( ) and the third is the one with 푏( ). The rhombic triacontahedron consists of 2 2 2 60퐾 + 60퐾́ = 30(2퐾 + 2퐾́ ) where (2퐾 + 2퐾́ ) form a cell of pyramid based on a golden 휏 rhombus of edge 휏−1푎 and height . Faces of the rhombic triacontahedron are the rhombuses 2 orthogonal to one of 30 vertices of the icosidodecahedron given in (8).

A-octahedron B-octahedron C-octahedron K-octahedron

Figure 3 The octahedra generated by the ABCK tetrahedra.

(a) (b) (c) Figure 4 The three pyramids composed of 4K tetrahedra.

When the 4K of Fig. 4 (a) is rotated by the icosahedral group, it generates the rhombic triacontahedron as shown in Fig. 5.

Figure 5 A view of rhombic triacontahedron.

The octahedron < 퐵 > is a non-convex polyhedron whose vertices can be taken as 휏퐷́ 1, −1 휏푅1퐷́ 1, 휏퐷́ 3, 휏푅3퐷́ 3, 2휏 퐷́ 2 and the sixth vertex is the origin 퐷́ 0. It consists of 4 triangular faces with edges 푎, 휏푎, 휏푏 and 4 triangular faces with edges 휏−1푎, 푎, 푏. Full action of the icosahedral group on the tetrahedron 퐵 would generate the “B-polyhedron” consisting of 60퐵 + (휏푣 ) (휏−1푣 ) 60퐵́ where 퐵́ = 푅 퐵 is the mirror image of 퐵. It has 62 vertices (12 like 1 ℎ3 , 30 like 2 ℎ3, 1 √2 √2 (푣 ) 20 like 3 ℎ3), 180 edges and 120 faces consisting of faces with triangles of √2 edges with 휏−1푎, 푎, 푏. The face transitive “B-polyhedron” is depicted in Fig. 6 showing 3-fold and 5-fold axes simultaneously.

Figure 6 The “B-polyhedron”.

The octahedron < 퐶 > is a convex polyhedron which can be represented by the vertices

1 1 1 1 1 (휏, 0,1), (휏, 휏, 휏), (1, 휏, 0), (휏2, 1,0), (휏2, 휏, 1), (0,0,0). (14) 2 2 2 2 2

This is obtained from that of Fig. 2 by a translation and inversion. It consists of 4 triangular faces with edges 푎, 푏, 휏푏 and 4 triangular faces with edges 휏−1푎, 푎 and 푏. (푣 ) (푣 ) The “C-polyhedron” is a non-convex polyhedron with 62 vertices (12 like 1 ℎ3, 30 like 2 ℎ3, √2 √2 (푣 ) 20 like 3 ℎ3), 180 edges and 120 faces consisting of only one type of triangular face with edge √2 lengths 휏−1푎, 푎 and 푏 as can be seen in Fig. 7 (see also the reference Baake & Grimm, 2013, page 234). It is also face transitive same as “B-polyhedron”.

Figure 7 The “C-polyhedron”.

The octahedron < 퐴 > is a non-convex polyhedron which can be represented by the vertices

1 1 1 1 1 (−휏, 0,1), (1, −휏, 0), (1,1,1), (휎, 0, −휏), (휎, −휏, 1), (0,0,0), (15) 2 2 2 2 2 as shown in Fig. 3. Now we discuss how the tiles are generated by inflation with an inflation factor 휏. First of all, let us recall that the projection of subset of the 퐷6 vectors specified by the pair of integers (푚1, 푚2) are some linear combinations of the weights 푣𝑖, (푖 = 1, 2, 3) with an overall factor c. It is easy to find out the vector of 퐷6 corresponding to the inflated vertex of any ABCK tetrahedron 푛 푛 by noting that 푐(휏 푣𝑖) = 푐(푚1́ , 푚́2)푣𝑖. Here we use 휏 = 퐹푛−1 + 퐹푛휏 and we define

1 1 푚́ ≡ 푚 퐹 + (푚 + 5푚 )퐹 , 푚́ ≡ 푚 퐹 + (푚 + 푚 )퐹 , (16) 1 1 푛−1 2 1 2 푛 2 2 푛−1 2 1 2 푛 where 퐹푛 represents the Fibonacci sequence satisfying

푛+1 퐹푛+1 = 퐹푛 + 퐹푛−1, 퐹−푛 = (−1) 퐹푛 , 퐹0 = 0, 퐹1 = 1. (17)

It follows from (16) that 푚́ 1 + 푚́ 2 = even if 푚1 + 푚2 = even, otherwise 푚́ 1, 푚́ 2 ∈ ℤ. This proves that the pair of integers (푚1́ , 푚́2) obtained by inflation of the vertices of the Danzer’s 푛 tetrahedra remain in the subset of 퐷6 lattice. We conclude that the inflated vectors by 휏 in (12) can be obtained by replacing 푚1 by 푚́1 and 푚2 by 푚́2. For example radii of the icosahedra projected by 퐷6 vectors 2휔1, 2휔5, 2휔1 + 2휔5, 2휔1 + 4휔5 (18)

2 3 are in proportion to 1, 휏, 휏 and 휏 respectively. It will not be difficult to obtain the 퐷6 image of any general vector in the 3D space in the form of 푐(휏푝, 휏푞, 휏푟) where 푝, 푞, 푟 ∈ ℤ. Now we discuss the inflation of each tile one by one.

Construction of 흉푲 = 푩 + 푲

The vertices of 휏퐾 is shown in Fig. 8 where the origin coincides with one of the vertices of B as shown in Fig. 2 and the other vertices of K and B are depicted. A transformation is needed to translate K to its new position. The face of K opposite to the vertex 퐷́ 2 having a normal vector

1 (−1, −휎, −휏) outward should match with the face of B opposite to the vertex 퐷́ with a normal 2 0 1 vector (휎, 휏, −1) inward. For this reason, we perform a rotation of K given by 2

0 −1 0 푔퐾 = [0 0 −1] (19) 1 0 0 1 matching the normal of these two faces. A translation by the vector (휏, 휏2, 0) will locate K in 2 its proper place in 휏퐾. This is the simplest case where a rotation and a translation would do the work. The corresponding rotation and translation in 퐷6 can be calculated easily and the results are illustrated in Table 2.

Table 2 Rotation and Translation in 퐻3 and 퐷6.

푯ퟑ 푫ퟔ ̅ 푔퐾 ← rotation→ 푔퐾: (13̅6)(245)

1 2 ←translation→ 1 푡퐾 = (휏, 휏 , 0) [(푚 + 5푚 )푙 + (푚 + 푚 )(푙 + 푙 + 푙 + 푙 − 푙 )] 2 2 1 2 1 1 2 2 3 4 5 6

The rotation in 퐷6 is represented as the permutations of components of the vectors in the 푙𝑖 basis.

Figure 8 Vertices of 휏퐾.

Rhombic triacontahedron generated by 휏퐾 now consists of a “B-Polyhedron” centered at the 휏−1 origin and 30 pyramids of 4K with golden rhombic bases of edge lengths a and heights occupy 2 the 30 inward gaps of “B-Polyhedron” .

Construction of 흉푩 = 푪 + ퟒ푲 + 푩ퟏ + 푩ퟐ

The first step is to rotate B by a matrix

−휎 휏 −1 1 푔 = [−휏 1 −휎], (20) 퐵 2 1 −휎 휏 and then inflate by 휏 to obtain the vertices of 휏푔퐵퐵 as shown in Fig. 9.

(a) top view of 휏푔퐵퐵 (b) bottom view of 휏푔퐵퐵

Figure 9 Views from 휏푔퐵퐵. representation of 휏푔퐵퐵 is chosen to coincide the origin with the 퐷́ 0 vertex of tetrahedron C. It is obvious to see that the bottom view illustrates that the 4K is the pyramid based on the rhombus 1 with edge length (height), 푏( ) as shown in Fig. 4 (c). The 4K takes its position in Fig. 9 by a 2 rotation followed by a translation given by

1 −휎 −휏 1 1 rotation: 푔 = [−휎 휏 1 ], translation: 푡 = (휏, 0,1) . (21) 4퐾 2 4퐾 2 −휏 1 휎

To translate 퐵1 and 퐵2 we follow the rotation and translation sequences as

휎 −휏 1 1 1 rotation: 푔 = [−휏 1 −휎], translation: 푡 = (휏3, 0, 휏2) , 퐵1 2 퐵 2 −1 휎 −휏

−휎 −휏 1 1 1 rotation: 푔 = [ 휏 1 −휎], translation: 푡 = (휏3, 0, 휏2). (22) 퐵2 2 퐵 2 1 휎 −휏

Construction of 흉푪 = 푲ퟏ + 푲ퟐ + 푪ퟏ + 푪ퟐ + 푨

We inflate by 휏 the tetrahedron C with vertices shown in Fig. 10. Top view of 휏퐶 where one of the vertices of 퐾1 is at the origin is depicted in Fig.1.The vertices of the constituting tetrahedra are given as follows 1 1 1 퐾 : {(0,0,0), (1,1,1), (휏, 0,1), (휏2, 휏, 1)}; 1 2 2 4

1 1 1 1 퐾 : { (1,1,1), (휏, 0,1), (휏2, 휏, 1), (휏2, 휏, 1)}; 2 2 2 4 2

1 1 1 1 퐶 : { (1,1,1), (휏, 0,1), (휏2, 휏, 1), (휏2, 휏2, 휏2)} ; (23) 1 2 2 2 2

1 1 1 1 퐶 : { (휏, 0,1), (휏2, 휏, 1), (휏2, 0, 휏), (휏2, 휏2, 휏2)}; 2 2 2 2 2

1 1 1 1 퐴:{ (휏2, 휏, 1), (휏2, 0, 휏), (휏2, 휏2, 휏2), (휏3, 휏2, 휏)}. 2 2 2 2

Figure 10 C-tetrahedron obtained from that of Fig. 2 by translation and inversion.

(a) Top view of 휏퐶 (b) Bottom view of 휏퐶 Figure 11 Views from 휏퐶.

The vertices of 퐾1 can be obtained from K by a rotation 퐾1 = 푔퐾1퐾, where

−휎 휏 −1 1 5 푔 = [−휏 1 −휎], (푔 ) = 1. (24) 퐾1 2 퐾1 1 −휎 휏

1 The transformation to obtain 퐾 is a rotation 푔 followed by a translation of 푡 = (휏2, 휏, 1) 2 퐾2 퐾2 2 where −휎 −휏 −1 1 10 푔 = [−휏 −1 −휎], (푔 ) = 1 . (25) 퐾2 2 퐾2 1 휎 휏

1 To obtain the coordinates of 퐶 and 퐶 first translate C by − (휏, 휏, 휏) and rotate by 푔 and 1 2 2 퐶1 1 푔 respectively followed by the translation 푡 = (휏2, 휏2, 휏2) where 퐶2 퐶1 2

휎 휏 1 1 −휎 −휏 1 1 푔 = [휏 1 휎] , 푔 = [−휎 휏 1 ]. (26) 퐶1 2 퐶2 2 1 휎 휏 휏 −1 −휎

Similarly, vertices of tetrahedron A is rotated by

1 −휎 −휏 1 푔 = [−휎 휏 1 ], (27) 퐴 2 −휏 1 휎

1 followed by a translation 푡 = (휏2, 0, 휏 ). 퐴 2

Construction of 흉푨 = ퟑ푩 + ퟐ푪 + ퟔ푲

It can also be written as 휏퐴 = 퐶 + 퐾1 + 퐾2 + 퐵 + 휏퐵 where 휏퐵is already studied. A top view of 휏퐴 is depicted in Fig.12 with the vertices of 휏퐵 given in Fig. 9.

Figure 12 Top view of 휏퐴 (bottom view is the same as bottom view of 휏퐵- dotted region).

1 1 1 The vertices of C {(0,0,0), (휏, 0,1), (휏2, 1,0), (휏2, 휏, 1)} can be obtained from those in Fig. 2 2 2 2 1 by a translation (−휏, −휏, −휏) followed by a rotation 2

−1 휎 −휏 1 푔 = [ 휎 −휏 1 ]. (28) 퐶 2 −휏 1 −휎

Vertices of 퐾1and 퐾2 are given by

1 1 1 1 퐾 : { (휏, 0,1) , (휏2, 1,0), (휏2, 휏, 1), (3휏 + 1, 휏, 1)}, 1 2 2 2 4

1 1 1 1 퐾 : { (휏, 0,1), (휏2, 휏, 1), (3휏 + 1, 휏, 1), (2휏, −휎, 1)}. (29) 2 2 2 4 2

To obtain 퐾1and 퐾2 with these vertices first rotate K by 푔퐾1and 푔퐾2given by the matrices

휎 휏 −1 휎 휏 1 1 1 푔 = [휏 1 −휎], 푔 = [휏 1 휎], (30) 퐾1 2 퐾2 2 1 휎 −휏 1 휎 휏

1 and translate both by the vector 푡 = (휏, 0,1). The vertices of B in Fig.12 can be obtained by 퐾1 2 a rotation then a translation given respectively by

−휏 −1 휎 1 1 푔 = [−1 −휎 휏], 푡 = (휏3, 0, 휏2). (31) 퐵 2 퐵 2 −휎 −휏 1

All these procedures are described in Table 3 which shows the rotations and translations both in 퐸∥ and 퐷6 spaces.

Table 3 Rotation and Translation in 퐻3 and 퐷6 (see the text for the definition of rotations in 퐸∥ space).

푯ퟑ 푫ퟔ

흉푲 = 푩 + 푲

푔퐾 ← rotation→ (13̅6)(245̅)

푐 2 1 푡퐾 = (휏, 휏 , 0) [(푚 + 5푚 )푙 + 2 ← translation→ 2 1 2 1 (푚1 + 푚2)(푙2 + 푙3 + 푙4 + 푙5 − 푙6)]

흉푩 = 푪 + ퟒ푲 +푩ퟏ + 푩ퟐ

푔퐵 ← rotation→ (3)(1264̅5)

푔4퐾 ← rotation→ (1)(23)(45)(6)

푐 푡 = (휏, 0,1) ← translation→ 푚1푙5 + 푚2(푙1 − 푙2 + 푙3 − 푙4 − 푙6) 4퐾 2

̅ 푔퐵1 (16435)(2) ← rotation→ ̅̅̅ ̅̅̅ 푔퐵2 ← rotation→ (152152), (364364)

푐 푡 = (휏3, 0, 휏2) ← translation→ 1 퐵 2 [(3푚 + 5푚 )푙 + 2 1 2 5 (푚1 + 3푚2)(푙1 − 푙2 + 푙3 − 푙4 − 푙6) 흉푪 = 푲ퟏ + 푲ퟐ + 푪ퟏ +푪ퟐ + 푨 16

̅ 푔퐾1 ← rotation→ (54621)(3)

(identity translation)

̅ ̅ ̅̅̅ ̅ 푔퐾2 ← rotation→ (11)(6345263452)

푐 2 ←translation→ (푚1 + 푚2)(푙1 + 푙5) 푡퐾 = (휏 , 휏, 1) 2 + 2푚2)(푙3 − 푙6)

푐 ←translation→ 1 푡퐶 = − (휏, 휏, 휏) − [(푚 + 3푚 )(푙 + 푙 + 푙 ) 2 2 1 2 1 3 5 +( 푚1−푚2)(푙2 − 푙4 − 푙6)]

̅ 푔퐶1 ← rotation→ (1)(4)(26)(35)

̅ 푔퐶2 ← rotation→ (1)(35642)

푐 2 2 2 ←translation→ (푚1 + 2푚2)(푙1 + 푙3 + 푙5) 푡퐶 = (휏 , 휏 , 휏 ) 2 + 푚2(푙2 − 푙4 − 푙6)

푔퐴 ← rotation→ (1)(6)(23)(45)

푐 2 ←translation→ 1 푡퐴 = (휏 , 0, 휏) [(푚 + 5푚 )푙 2 2 1 2 5 +(푚1 + 푚2)(푙1 − 푙2 + 푙3 − 푙4 − 푙6)]

흉푨 = 푪 + 푲ퟏ +푲ퟐ + 푩 + 흉푩

푐 ←translation→ 1 푡퐶 = − (휏, 휏, 휏) − [(푚 + 3푚 )(푙 + 푙 + 푙 ) 2 2 1 2 1 3 5 +(푚1 − 푚2)(푙2 − 푙4 − 푙6)]

푔퐶 ← rotation→ (11̅ (24̅)(36)(55̅)

̅ 푔퐾1 ← rotation→ (1) (25436)

̅ 푔퐾2 ← rotation→ (1)(4) (26)(35)

푐 푡 = (휏, 0,1) ← translation→ 푚1푙5 + 퐾 2 푚2(푙1 − 푙2 + 푙3 − 푙4 − 푙6)

푔퐵 ← rotation→ (651̅6̅5̅1), (4̅2̅3̅423)

푐 3 2 ← translation→ 1 푡퐵 = (휏 , 0, 휏 ) [(3푚 + 5푚 )푙 + 2 2 1 2 5 (푚1 + 3푚2)(푙1 − 푙2 + 푙3 − 푙4 − 푙6)]

4. Discussions

The 6-dimensional reducible representation of the icosahedral subgroup of the point group of 퐷6 is decomposed into the direct sum of its two 3-dimensional representations described also by the direct sum of two graphs of icosahedral group. We have shown that the subset of the 퐷6 lattice characterized by a pair of integers (푚1, 푚2) with 푚1 + 푚2 = even projects onto the Platonic and Archimedean polyhedra possessing icosahedral symmetry which are determined as the orbits of 푛 the fundamental weights 푣𝑖 or their multiples by 휏 . The edge lengths of the Danzer tiles are related to the weights of the icosahedral group 퐻3 and via Table 1 to the weights of 퐷6, a group theoretical property which has not been discussed elsewhere. It turns out that the tetrahedron K constitutes the fundamental region of the icosahedral group 퐻3 as being the cell of the rhombic triacontahedron. Images of the Danzer tiles and their inflations are determined in 퐷6 by employing translations and icosahedral rotations. This picture gives a one-to-one correspondence between the translations-rotations of 3D space and 6D space. Faces of the Danzer tiles are all parallel to the faces of the rhombic triacontahedron; in other words, they are all orthogonal to the 2-fold axes. Since the faces of the Ammann rhombohedral tiles are orthogonal to the 2-fold axes of the icosahedral group it is this common feature that any tilings obtained from the Ammann tiles either with decoration or in the form of Socolar-Steinhardt model can also be obtained by the Danzer tiles. These geometrical properties have not been studied from the group theoretical point of view, a subject which, is beyond the present work but definitely deserves studying. Note that the inflation introduces a cyclic permutation among the rhombic triacontahedron, “B-polyhedron” and “C-polyhedron” as they occur alternatively centered at the origin. Another novel feature of the paper is to show that ABCK tiles and their inflations are directly related to the transformations in the subset of 퐷6 lattice characterized by integers (푚1, 푚2) leading to an alternative projection technique different from the cut and project scheme. Details of the composition of the ABCK tiles into a structure with long-range quasiperiodic order follows from the inflation matrix (see for instance Baake & Grimm, 2013, pages 229-235).

Acknowledgement

We would like to thank Prof. Ramazan Koc for his contributions to Fig.6 and Fig.7.

References

Alhevaz, A., Baghipur, M. & Shang, Y. (2019). Symmetry 11, 1276. Baake, M. & Grimm, U. (2013). Aperiodic Order, Volume 1: A Mathematical Invitation, Cambridge University Press, Cambridge. Coxeter, H. S. M. (1973). Regular Polytopes, Third Edition, Dover Publications. Conway, J. H. & Sloane, N. J. A. (1999). Sphere packings, Lattices and Groups. Third Edition, Springer-Verlag New York Inc. Danzer, L. (1989). Discrete Math. 76, 1-7. Danzer, L., Papadopolos, D. & Talis, A. (1993). J. Mod. Phys. B7, 1379-86. Di Vincenzo, D. & Steinhardt, P. J. (1991). Quasicrystals: the state of the art, World Scientific Publishers, Singapore. Hann, C., T, Socolar, J. E. S. & Steinhardt, P. J. (2016). Phys. Rev. B94, 014113. Humphreys, J. E. (1992). Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge.

Janot, C. (1993). Quasicrystals: a primer, Oxford University Press, Oxford. Katz, A. (1989). Some Local Properties of the Three-Dimensional Tilings, in Introduction of the Mathematics of Quasicrystals, Vol. 2, p. 147-182. (Edit. Marko V. Jaric), Academic Press, Inc. New York. Koca, M., Koc, R. & Al-Barwani, M. (2001). J. Phys. A: Math. Gen. 34, 11201-11213. Koca, M., Koca, N. & Koc, R. (2015). Acta. Cryst. A71, 175-185. Koca, M., Koca, N., Al-Siyabi, A., & Koc, R. (2018). Acta. Cryst. A74, 499-511. Kramer, P. (1993). Physica Scripta T49, 343-348. Kramer, P. & Andrle, M. (2004). J. Phys. A: Math. Gen. 37, 3443-3457. Math.uni-bielefeld.de/icosahedral tilings in ℝ3: the ABCK tilings. Roth, J. (1993). J. Phys. A: Math. and Gen. 26, 1455-1461. Senechal, M. (1995). Quasicrystals and Geometry, Cambridge University Press, Cambridge. Socolar, J.E.S. & Steinhardt, P. J. (1986). Phys. Rev. B34, 617-647.