Icosahedral Polyhedra from D6 lattice and Danzer's ABCK tiling

It is well known that the point group of the root lattice D6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H3, its roots and weights are determined in terms of those of D6. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D6 determined by a pair of integers (m1, m2) in most cases, either both even or both odd. Vertices of the Danzer's ABCK tetrahedra are determined as the fundamental weights of H3 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 (m1, m2) with coefficients from Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral tiling and theoctahedral tiling in H3 and the corresponding D6 spaces are specified by determining the rotations and translation in 3D and the corresponding group elements in D6 . The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of the K-polyhedron, B-polyhedron and the C-polyhedron generated by the icosahedral group have been discussed.


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, 2009). 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 Amman tiles of acute and obtuse rhombohedra. Later it was proved that (Danzer, Papadopolos & Tallis, 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 Amman tiles. Amman rhombohedral and Danzer ABCK tetrahedral tilings are intimately related with the projection of six dimensional cubic lattice 6 and the root and weight lattices of 6 (Koca, Koca & Koc, 2015). Kramer and Andrle (Kramer & Andrle, 2004) has investigated Danzer tiles from the wavelet point of view and their relations with the 6 lattice. In what follows we point out that the icosahedral symmetry requires a subset of the root lattice 6 characterized by a pair of integers ( 1 , 2 ) as coefficients of the orthogonal set of vectors , ( = 1, 2, … ,6). The paper is organized as follows. In Sec. 2, we introduce the root lattice 6 , its icosahedral subgroup, decomposition of its weights in terms of the weights of the icosahedral group 3 leading to the Archimedean polyhedra projected from the 6 lattice. It turns out that the lattice vectors to be projected are determined by a pair of integers ( 1 , 2 ). In Sec. 3 we introduce the ABCK tetrahedral Danzer tiles in terms of the fundamental weights of the icosahedral group 3 . Tiling by inflation with where = 1+√5 2 and ∈ ℤ is studied in 3 space by prescribing the appropriate rotation and translation operators. The corresponding group elements of 6 are determined noting that the pair of integers ( 1 , 2 ) can be expressed as the linear combinations of similar integers with coefficients from Fibonacci sequence. Sec. 4 is devoted for conclusive remarks.

Figure 1
Coxeter-Dynkin diagrams of 6 and 3 illustrating the symbolic projection.
Their representations in the ⊥ space follows from (6)  where the notation ( ) ℎ 3 is introduced for the set of vectors generated by the action of the icosahedral group on the weight .
The sets of vectors in (8) represent the vertices of an icosahedron, an icosidodecahedron and a dodecahedron respectively in the ∥ space. The weights 1 , 2 and 3 denote also the 5-fold, 2fold and 3-fold symmetry axes of the icosahedral group. The union of the orbits of constitute the vertices of a rhombic triacontahedron.
It is obvious that the Coxeter group 6 is symmetric under the algebraic conjugation and this is more apparent in the characteristic equation of the Coxeter element of 6 given by whose eigenvalues lead to the Coxeter exponents of 6 .The first bracket is the characteristic polynomial of the Coxeter element of the matrices in (6) describing it in the ∥ space and the second bracket describes it in the ⊥ space (Koca, Koc, Al-Barwani, 2001). Therefore projection of 6 into either space is the violation of the algebraic conjugation. It would be interesting to discuss the projections of the fundamental polytopes of 6 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 ω 1 is a polytope with 12 vertices called 6D octahedron and represents an icosahedron when projected into either space. The orbit of weight ω 2 constitutes the "root polytope" of 6 with 60 vertices which projects into 3D space as two icosidodecahedra with 30 vertices each, the ratio of radii of the circumspheres is . The dual of the root polytope is the union of the three polytopes generated by ω 1 , ω 5 and ω 6 which constitute the Voronoi cell of the lattice 6 as mentioned earlier and projects into an icosahedron and two rhombic triacontahedra. Actually they consist of three icosahedra with the ratio of radii 1, , 2 and two dodecahedra with the radii in proportion to . The orbit generated by the weight vector ω 3 is a polytope with 160 vertices and constitutes the Voronoi cell of the weight lattice 6 * . 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 (́) as 1 1 + 2 2 + 3 3 + 4 4 + 5 5 Projection of an arbitrary vector of 6 into the space ∥ , or more thoroughly, onto a particular weight vector, for example, onto the weight 1 is given by and represents an icosahedron where ≡ ( 1 , 2 ) = √ 2 2+ ( 1 − 2 + 2 2 ) is an overall scale vector. The subscript ∥ means the projection into the space ∥ . The expression in (11) shows that 1 + 3 2 = even implying that the pair of integers ( 1 , 2 ) 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 6 lattice vectors determined by the pair of integers ( 1 , 2 ). Table 1 shows that only certain subset of vectors of 6 project onto regular icosahedral polyhedra. We will see in the next section that the Danzer tiling also restricts the 6 lattice in a domain where the vectors are determined by the pair of integers ( 1 , 2 ).

Table1
Platonic and Archimedean icosahedral polyhedra projected from 6 (vertices are orbits under the icosahedral group).

A-tetrahedron B-tetrahedron C-tetrahedron K-tetrahedron
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 3 whose edge lengths are given by = and the apex is at the origin. The coordinate of the intersection of the diagonals of the rhombus is the vector 2 2√2 = 1 2 (0, , 0) and its magnitude is the in-radius of the rhombic triacontahedron. and the origin can be taken as the vertices of the tetrahedron 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 < >, < > and < > comprises 4 copies of each obtained by a group of order 4 generated by two commuting generators 1 and 3 or their conjugate groups. The octahedron < > consists of 8K generated by a group of order 8 consisting of three commuting generators 1 , 3 as mentioned earlier and the third generator 0 is an affine reflection (Humphrey,1992) with respect to the golden rhombic face induced by the affine Coxeter group ̃6 . One can dissect the octahedron < > into three non-equivalent pyramids with rhombic bases by cutting along the lines orthogonal to three planes 1 − 2 , 2 − 3 , 3 − 1 . One of the pyramid is generated by the tetrahedron K upon the actions of the group generated by the reflections 1 and 3 . If we call 1 =́ as the mirror image of K the others can be taken as the tetrahedra obtained from K and 1 by a rotation of 180 0 around the axis 2 . A mirror image of 4 = (2 + 2 ) 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, , 0). Octahedral tiles are depicted in Fig. 3. Dissected pyramids constituting the octahedron < > as depicted in Fig.4 have bases, two of which with bases of golden rhombuses with edge lengths(heights) −1 ( 2 ) , ( 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 < > is a non convex polyhedron whose vertices can be taken as ́1 , 1́1 , ́3 , 3́3 , 2 −1́2 and the sixth vertex is the origin ́0 . It consists of 4 triangular faces with edges , , and 4 triangular faces with edges −1 , , . Full action of the icosahedral group on the tetrahedron would generate the "B-polyhedron" consisting of 60 + 60́ where ́= 1 is the mirror image of . It has 62 vertices (12 like The face transitive "B-polyhedron" is depicted in Fig. 6.
The octahedron < > is a convex polyhedron which can be represented by the vertices This is obtained from that of Fig.2 by a translation and inversion. It consists of 4 triangular faces with edges , , and 4 triangular faces with edges −1 , and .
The "C-polyhedron" is a convex polyhedron with 62 vertices (12 like ), 180 edges and 120 faces consisting of only one type of triangular face with edge lengths −1 , , . It is also face transitive same as "B-polyhedron". The "C-polyhedron" is depicted in Fig. 7.
It follows from (16) that ́1 + 3́2 = even if 1 + 3 2 = even, otherwise ́1,́2 ∈ ℤ. This proves that the pair of integers ( 1 , 2 ) obtained by inflation of the vertices of the Danzer's tetrahedra remain in the subset of 6 lattice. We conclude that the inflated vectors by in (12) can be obtained by replacing 1 by 1 and 2 by 2 . For example the radii of the icosahedra projected by the 6 vectors 2 1 , 2 5 , 2 1 + 2 5 , 2 1 + 4 5 are in proportion to 1, , 2 and 3 respectively. It will not be difficult to obtain the 6 image of any general vector in the 3D space in the form of ( , , ) where , , ∈ ℤ. Now we discuss the inflation of each tile one by one.

Construction of = +
The vertices of is shown in Fig. 8 where the origin coincides with one of the vertex of B as shown in Fig. 2 matching the normal of these two faces. A translation by the vector 1 2 ( , 2 , 0) will locate K in its proper place in . This is the simplest case where a rotation and a translation would do the work. The corresponding rotation and translation in 6 can be calculated easily and the results are illustrated in Table 2. The rotation in 6 is represented as the permutations of components of the vectors in the basis.

Construction of = + + +
The first step is to rotate B by a matrix and then inflate by to obtain the vertices of as shown in Fig. 9.
(a) top view of (b) bottom view of Figure 9 Views from .
This representation of is chosen to coincide the origin with the ́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 with edge length (height), ( ) as shown in Fig. 4 (c). The 4K takes its position in Fig. 9  The vertices of 1 can be obtained from K by a rotation 1 = 1 , where ( 1 ) 5 = 1.

Construction of = + +
It can also be written as = + 1 + 2 + + where is already studied. A top view of is depicted in Fig.12 with the vertices of given in Fig. 9.

Figure 12
Top view of (bottom view is the same as bottom view of − dotted region).
The vertices of C {(0,0,0), To obtain 1 and 2 with these vertices first rotate K by 1 and 2 given by the matrices and translate by the vector 1 = 1 2 ( , 0,1). The vertices of B in Fig.12 can be obtained by a rotation then a translation given respectively by All these procedures are described in Table 3 which shows the rotations and translations both in ∥ and 6 spaces.

Table 3
Rotation and Translation in 3 and 6 (see the text for the definition of rotations in ∥ space).