Icosahedral Polyhedra from D6 Lattice and Danzer's ABCK Tiling

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Article Icosahedral Polyhedra from D6 Lattice and Danzer’s ABCK Tiling

1 1, 2, Abeer Al-Siyabi , Nazife Ozdes Koca * and Mehmet Koca † 1 Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud, Muscat 123, Oman; [email protected] 2 Department of Physics, Cukurova University, Adana 1380, Turkey; [email protected] * Correspondence: [email protected] Retired. † Received: 10 November 2020; Accepted: 26 November 2020; Published: 30 November 2020

Abstract: 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 the Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the octahedral tilings in 3D space with icosahedral symmetry H3, and those related transformations in 6D space with D6 symmetry are specified by determining the rotations and translations 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 “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 [1–3]. 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 [4]. 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 [5] 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 [6,7] they can be constructed by the Danzer’s ABCK tetrahedral tiles [8,9], and recently, Hann–Socolar–Steinhardt [10] proposed a model of tiling scheme with decorated Ammann tiles. A detailed account of the Danzer ABCK tetrahedral tilings can be found in [11] and in page 231 of the reference [4] where the substitution matrix, its eigenvalues, and the corresponding

Symmetry 2020, 12, 1983; doi:10.3390/sym12121983 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 1983 2 of 15 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 D6, the point symmetry group of which is of order 256!. See for a review the paper “Modelling of quasicrystals” by Kramer [12] and references therein. Similar work has also been carried out in the reference [13]. Kramer and Andrle [14] have investigated Danzer tiles from the wavelet point of view and their relations with the lattice D6. In what follows, we point out that the icosahedral symmetry requires a subset of the root lattice D6 characterized by a pair of integers (m1, m2) with m1 + m2 = even, which are the coefficients of the orthogonal set of vectors li, (i = 1, 2, ... , 6). Our approach is different than the cut and project scheme of lattice D6, as will be seen in the sequel. The paper consists of two major parts; the first part deals with the determination of Platonic and Archimedean icosahedral polyhedra by projection of the fundamental weights of the root lattice D6 into 3D space, and the second part employs the technique to determine the images of the Danzer tiles in D6. Inflation of the Danzer tiles are related to a redefinition of the pair of integers (m1, m2) 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 D6 to its subset, has not been discussed elsewhere. The paper is organized as follows. In Section2, we introduce the root lattice D6, its icosahedral subgroup, and decomposition of its weights in terms of the weights of the icosahedral group H3 leading to the Archimedean polyhedra projected from the D6 lattice. It turns out that the lattice vectors to be projected are determined by a pair of integers (m1 , m2). In Section3, we introduce the ABCK tetrahedral Danzer tiles in terms of the fundamental weights of the icosahedral group H3. Tiling by n 1+ √5 inflation with τ where τ = and n Z is studied in H3 space by prescribing the appropriate 2 ∈ rotation and translation operators. The corresponding group elements of D6 are determined noting that the pair of integers (m1 , m2) can be expressed as the linear combinations of similar integers with coefficients from Fibonacci sequence. Section4 is devoted for conclusive remarks.

2. Projection of D6 Lattice under the Icosahedral Group H3 and the Archimedean Polyhedra

We will use the Coxeter diagrams of D6 and H3 to introduce the basic concepts of the root systems, weights, and the projection technique. A vector of the D6 lattice can be written as a linear combination of the simple roots with integer coeﬃcients:

X6 X6 X6 λ = niαi = mili, mi = even, ni, mi Z. (1) i=1 i=1 i=1 ∈

Here, αi are the simple roots of D6 defined in terms of the orthonormal set of vectors as α = l l + , i = 1, ... , 5 and α = l + l , and the reflection generators of D act as i i − i 1 6 5 6 6 r : l l + and r : l l . The generators of H can be defined [13] as R = r r , R = r r and i i ←→ i 1 6 5 ←→ − 6 3 1 1 5 2 2 4 R3 = r3r6, where the Coxeter element, for example, can be taken as R = R1R2R3. The weights of D6 are given by ω = l , ω = l + l , ω = l + l + l , ω = l + l + l + l , 1 1 2 1 2 3 1 2 3 4 1 2 3 4 (2) = 1 (l + l + l + l + l l ), ω = 1 (l + l + l + l + l + l ). 2 1 2 3 4 5 − 6 6 2 1 2 3 4 5 6 The Voronoi cell of the root lattice D6 is the dual polytope of the root polytope of ω2 [15] and determined as the union of the orbits of the weights ω1, ω5 and ω6, which correspond to the holes of the root lattice [16]. If the roots and weights of H3 are defined in two complementary 3D spaces E and k E as βi(βí) and vi(ví), (i = 1, 2, 3), respectively, then they can be expressed in terms of the roots and ⊥ weights of D6 as Symmetry 2020, 12, x FOR PEER REVIEW 3 of 16

𝛽 = (𝛼 +𝜏𝛼 ), 𝑣 = (𝜔 +𝜏𝜔 ), Symmetry 2020, 12, 1983 √ √ 3 of 15

𝛽 =β = 1(𝛼(α+𝜏𝛼+ τα)),, v 𝑣== 1 (ω(𝜔+τω+𝜏𝜔), ), (3) 1√√2+τ 1 5 1 √2√+τ 1 5 1 1 β2 = (α2 + τα4), v2 = (ω2 + τω4), (3) √2+τ √2+τ 𝛽 =β = 1(𝛼(α+𝜏𝛼+ τα)), v𝑣== 1 (ω(𝜔+τω+𝜏𝜔). ). 3 √√2+τ 6 3 3 √√2+τ 6 3 1+ √5 ´ √ For the complementary 3D space, replace βi by βi, vi by v´i in (3), and τ = 2 by its algebraic For the complementary 3D space, replace 𝛽 by 𝛽 , 𝑣 by 𝑣́ in (3), and 𝜏= by its 1 √5 1 conjugate σ = − = τ−√. Consequently, Cartan matrix of D6 (Gram matrix in lattice terminology) algebraic conjugate2 𝜎=− =−𝜏. Consequently, Cartan matrix of 𝐷 (Gram matrix in lattice and its inverse block diagonalize as terminology) and its inverse block diagonalize as    2 1 0 0 0 0 2 1 0 0 0 0      −   −   1 2 2 −1 1 0 0 0 0 0 0 0  21 −1 2 0τ 00 0 0 0 0   −⎡ − ⎤ ⎡ − − ⎤   0−1 1 2 2 −11 0 0 0 0 0 −10 2τ −𝜏2 0 0 0 0 0 0  C =  ⎢ ⎥ ⎢ ⎥   ⎢ 0− −1 2− −1 0 0⎥ ⎢ 0 −𝜏− 2 0 0 0⎥  𝐶= 0 0 1 2 1 1  →→  0 0 0 2 1 0   ⎢ 0 0− −1 2− −1− −1⎥ ⎢ 0 0 0 2 −1− 0⎥   0 0 0 1 2 0   0 0 0 1 2 σ   ⎢ 0 0 0− −1 2 0⎥ ⎢ 0 0 0 −1− 2 −𝜎−⎥   ⎣0 0 0 0 0 0 −11 0 0 2 2⎦ ⎣ 00 0 0 0 0 0 0 −𝜎σ 2 2⎦   −   −  (4) 2 2 2 2 1 1 2 + τ 2τ2 τ3 0 0 0 (4)      222211 2 + 𝜏2 2𝜏 2 𝜏 3 0 0 0   ⎡2 4 4 4 2 2⎤  ⎡ 2τ 4τ 2τ 0 0 0 ⎤   244422 ⎢ 2𝜏 4𝜏 2𝜏 0 0 0 ⎥  1 ⎢2 4 6 6 3 3⎥  1  τ3 2τ3 3τ2 0 0 0  C𝐶1==1  ⎢246633⎥ ⟶ 1⎢ 𝜏 2𝜏 3𝜏 0 0 0 ⎥  − 2  246844 2  2 3  2 ⎢2 4 6 8 4 4⎥  →2 ⎢ 00 0 0 0 0 2 + 2 + 𝜎σ 2 σ2𝜎 σ𝜎 ⎥   ⎢123432⎥     1 2 3 4 3 2  ⎢ 00 0 0 0 0 2𝜎 2σ2 4𝜎4σ2 2𝜎2σ3⎥   ⎣123423⎦  ⎣ 0 0 0 𝜎 2𝜎 3𝜎 ⎦   1 2 3 4 2 3   0 0 0 σ3 2σ3 3σ2  FigureFigure 11 shows shows how how one can illustrateillustrate thethe decompositiondecomposition of of the the 6D 6D space space as as the the direct direct sum sum of of twotwo complementary complementary 3D 3D spaces spaces where thethe correspondingcorresponding nodes nodesαi ,𝛼(i,(𝑖=1,2,...,6)= 1, 2, ... , 6), β,i , (𝛽i=, (𝑖1, 2, = 31,) 2, 3) representrepresent the the corresponding corresponding simple simple roots, andand β´𝛽i stands stands for forβi in𝛽 the in complementarythe complementary space. space.

Figure 1. Coxeter–Dynkin diagrams of D6 and H3 illustrating the symbolic projection. Figure 1. Coxeter–Dynkin diagrams of 𝐷 and 𝐻 illustrating the symbolic projection. For a choice of the simple roots of H as β = √2, 0, 0 , β = 1 (1, σ, τ), β = 0, 0, √2 and 3 1 2 √ 3 For a choice of the simple roots of 𝐻 as 𝛽 =√2,0,0,𝛽 =−− 2 (1, 𝜎, 𝜏),𝛽 =0,0,√2 and the similar expressions for the roots in the complementary space, we can√ express the components of thethe similar set of vectorsexpressionsli (i = for1, 2,the... roots, 6) as in the complementary space, we can express the components of the set of vectors 𝑙 𝑖 = 1, 2, … ,6) as    l1   1 τ 0 τ 1 0     −   l𝑙2  1 𝜏1 τ 00 τ𝜏1 −1 0 0   ⎡ ⎤    𝑙  ⎡  − − − ⎤   ⎢l3⎥ 1 −1 0 𝜏 1 τ0 0−𝜏τ −1 0 1   𝑙  = p 1 ⎢  ⎥ −  (5)  ⎢l ⎥ ⎢ 0 0 1 1 𝜏τ 0 0 τ 𝜏 −1⎥ 1   4 = 2(2 + τ)   (5)  ⎢𝑙⎥ 2(2 + 𝜏) ⎢ 0 1 −𝜏− 0 𝜏 1⎥   l5   τ 0 1 1 0 τ   ⎢𝑙⎥ ⎢ 𝜏 0 1 −−1 0 𝜏⎥  l6 ⎣ τ 0 1 1 0 ⎦ τ ⎣𝑙⎦ −𝜏− 0 1 1 0 𝜏 The ﬁrst three and last three components project the vectors li into E and𝑙 E spaces,𝐸 respectively.𝐸 The first three and last three components project the vectorsk 1 into⊥ ∥ and spaces, On the other hand, the weights are represented by the vectors v1 = (1, τ, 0), v2 = √2(0, τ, 0), respectively. On the other hand, the weights are represented by the√ vectors2 𝑣 = (1, 𝜏, 0),𝑣 = v = 1 0, 2, 1 . The generators of H in the space E read √ 3 √ τ 3 √2(0, 𝜏,2 0),𝑣 = (0, 𝜏 ,1). The generators of 𝐻 ink the space 𝐸∥ read √        1 0 0   1 σ τ   1 0 0  −100−  1 1 −𝜎− − −𝜏  1 0 0  =   =   =   R𝑅1 = 00 1 1 0 0,  , R 𝑅2 = −𝜎σ τ 𝜏 1 1,  , 𝑅 R=3 0 01 1 0, 0 , (6)(6)   2 −     00 0 0 1 1   −𝜏τ 1 1 σ 𝜎  0 00 0 −1 1  − − satisfying the generating relations satisfying the generating relations 𝑅 =𝑅 =𝑅 =(𝑅𝑅 ) =(𝑅 𝑅 ) =(𝑅 𝑅 ) =1 (7) 2 2 2 2 3 5 R1 = R2 = R3 = (R1R3) = (R1R2) = (R2R3) = 1 (7)

Symmetry 2020, 12, 1983 4 of 15

Their representations in the E space follows from (6) by algebraic conjugation. The orbits of the ⊥ weights vi under the icosahedral group H3 can be obtained by applying the group elements on the weight vectors as ( ) v1 h 3 = 1 ( 1, τ, 0), ( τ, 0, 1), (0, 1, τ) , √2 2 1 ± ± ± ± ± ± (τ− v2)h n o 3 = 1 ( 1, 0, 0), (0, 0, 1), (0, 1, 0), 1 ( 1, σ, τ), 1 ( σ, τ, 1), 1 ( τ, 1, σ) , (8) 2 √2 2 ± ± ± 2 ± ± ± 2 ± ± ± 2 ± ± ± ( 1 ) τ− v3 h 3 = 1 ( 1, 1, 1), (0, τ, σ), ( τ, σ, 0), ( σ, 0, τ) √2 2 ± ± ± ± ± ± ± ± ± where the notation (v ) is introduced for the set of vectors generated by the action of the icosahedral i h3 group on the weight vi. The sets of vectors in (8) represent the vertices of an icosahedron, an icosidodecahedron, and a dodecahedron, respectively, in the E space. The weights v1, v2, and v3 also denote the five-fold, k (v1) two-fold, and three-fold symmetry axes of the icosahedral group. The union of the orbits of h3 and √2 1 (τ− v3) h3 constitute the vertices of a rhombic triacontahedron. √2 It is obvious that the Coxeter group D6 is symmetric under the algebraic conjugation, and this is more apparent in the characteristic equation of the Coxeter element of D6 given by λ3 + σλ2 + σλ + 1 λ3 + τλ2 + τλ + 1 = 0, (9) whose eigenvalues lead to the Coxeter exponents of D6. The first bracket is the characteristic polynomial of the Coxeter element of the matrices in (6) describing it in the E space, and the second bracket k describes it in the E space [17]. ⊥ Therefore, projection of D6 into either space is the violation of the algebraic conjugation. It would be interesting to discuss the projections of the fundamental polytopes of D6 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 D6 with 60 vertices, which project into 3D space as two icosidodecahedra with 30 vertices each, the ratio of radii of the circumspheres is τ. 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 D6, as mentioned earlier and projects into an icosahedron and two rhombic triacontahedra. Actually, they consist of three icosahedra with the ratio of radii 1, τ, τ2 and two dodecahedra with the radii in proportion to τ. The orbit generated by the weight vector ω3 is a polytope with 160 vertices and constitutes the Voronoi cell of the weight lattice D6∗. 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 vi(ví) as

m1l1 + m2l2 + m3l3 + m4l4 + m5l5 + m6l6 1 = [(m1 m2 + τm5 τm6)v1 + (m2 m3 + τm4 τm5)v2 (10) √2+τ − − − − + (m + m + τm τm )v ] + (v v´ , τ σ). 5 6 3 − 4 3 i → i →

Projection of an arbitrary vector of D6 into the space E , or more thoroughly, onto a particular k weight vector, for example, onto the weight v1 is given by v1 [m1l1 + m2(l2 + l3 + l4 + l5 l6)] [(m1 m2) ω1 + 2m2 ω5] = c( ) (11) − k ≡ − k √2 Symmetry 2020, 12, 1983 5 of 15

Symmetry 2020, 12, x FOR PEER REVIEW 5 of 16 SymmetrySymmetrySymmetry 20 2020202020,,, , 1 11222,,, , xx xx FOR FORFORFOR PEER PEERPEERPEER REVIEW REVIEWREVIEWREVIEW q 555 of ofof 16 1616 Symmetry 2020, 12, x FOR PEER REVIEW 2 5 of 16 And represents an icosahedron where c c(m1, m2) = 2+τ (m1 m2 + 2m2τ) is an overall scale will see that not only icosahedron but also≡ dodecahedron and − the icosahedral Archimedean willwillwillfactor. see see see that thatthat The not subscriptnot not only only only icosahedron icosahedron icosahedronmeans the projection but but but also also also dodecahedron dodecahedron intododecahedron the space E and and and. The the the the expression icosahedral icosahedral icosahedral in Archimedean Archimedean Archimedean(11) shows that willpolyhedra see that can notbe obtained only k icosahedron by relations butsimilar also to dodecahedron(11). The Platonic andk and the Archimedean icosahedral polyhedra Archimedean with polyhedrapolyhedrapolyhedram1 + m2 can cancan= even be bebe obtained obtainedobtained, implying by byby that relations relationsrelations the pair similar similarsimilar of integers to toto (11). (11).(11).( The ThemThe1, mPlatonic PlatonicPlatonic2) are either and andand Archimedean ArchimedeanArchimedean both even or polyhedra bothpolyhedrapolyhedra odd. Wewith withwith will polyhedrapolyhedra cancan bebe obtainedobtained byby relationsrelations similarsimilar toto (11).(11). TheThe PlatonicPlatonic andand ArchimedeanArchimedean polyhedrapolyhedra withwith icosahedralsee that not symmetry only icosahedron are listed but in Table also dodecahedron 1 as projections and of the퐷6 icosahedral lattice vectors Archimedean determined polyhedra by the pair can icosahedralicosahedralicosahedralicosahedral symmetry symmetrysymmetrysymmetry are areareare listed listedlistedlisted in ininin Table TableTableTable 1 111 as asasas projections projectionsprojectionsprojections of ofofof 퐷퐷퐷666 latticelatticelatticelattice vectors vectorsvectorsvectors determined determineddetermineddetermined by bybyby the thethethe pair pairpairpair icosahedral symmetry(푚 , 푚 ). are listed in Table 1 as projections of 퐷6 lattice vectors퐷 determined by the pair ofof integersbe integers obtained ( ((푚푚푚 by,1,,푚푚푚 relations)2)).. . Table Table similar1 1 shows shows to that that (11). only only The a Platonica certain certain subset andsubset Archimedean of of vectors vectors of of polyhedra 퐷 퐷퐷 6 project project with onto onto icosahedral reg regularular ofof integersintegers ((푚푚111,,푚푚222)). . TableTable 11 showsshows thatthat onlyonly aa certaincertain subsetsubset ofof vectorsvectors ofof 퐷퐷666 projectproject ontoonto regregularular oficosahedral integers polyhedra.1 2 Table We 1will shows see thatin the only next a certain section subset that the of vectorsDanzer oftiling 6 alsoproject restricts onto regtheular 퐷6 icosahedralicosahedralicosahedralsymmetry polyhedra.polyhedra.polyhedra. are listed in WeWeWe Table willwillwill1 seeseeassee projections ininin thethethe nextnextnext of sectionsectionsectionD6 lattice thatthatthat vectors thethethe DanzerDanzerDanzer determined tilingtilingtiling alsoalsoalso by the restrictsrestrictsrestricts pair of thethethe integers 퐷퐷퐷6 icosahedralicosahedral polyhedra.polyhedra. WeWe willwill seesee inin thethe nextnext sectionsection thatthat thethe DanzerDanzer tilingtiling alsoalso restrictsrestricts thethe 퐷66 lattice(m , m in) . aTable domain1 shows, where that the only vectors a certain are subset determined of vectors by the of D pairproject of integers onto regular (푚1, 푚 icosahedral2). These6 latticelatticelatticelattice1 in in in in 2 a a a a domain domain domain domain,,, , wherewherewhere the the the vectors vectors vectors are are are determined determined determined by by by the the the pair pair pair6 of of of integers integers integers ( ((푚푚푚111,,,,푚푚푚222)))... . TheseTheseThese latticequantities in astudied domain here, where are potentially the vectors appl areicable determined to many bygraph the indices pair of [18]. integers (푚1, 푚2). These quantitiesquantitiesquantitiespolyhedra. studied studiedstudied We will here herehere see are areare in potentially potentiallypotentially the next section appl applapplicableicable thaticable the to toto Danzermany manymany graph graphgraph tiling indices indicesalsoindices restricts [18]. [18].[18]. the D6 lattice in a domain, quantitiesquantities studiedstudied herehere areare potentiallypotentially applapplicableicable toto manymany graphgraph indicesindices [18].[18]. where the vectors are determined by the pair of integers (m1, m2). These quantities studied here are Table 1. Platonic and Archimedean icosahedral polyhedra projected from퐷 퐷6 (vertices are orbits potentiallyTableTableTable 1. 1. 1. PlatonicPlatonicPlatonic applicable and and and to Archimedean Archimedean Archimedean many graph icosahedral icosahedral icosahedral indices [18 polyhedra polyhedra polyhedra]. projected projected projected from from from 퐷퐷666 (vertices(vertices(vertices(vertices are are are are orbits orbits orbits orbits Tableunder 1.the Platonic icosahedral and group). Archimedean icosahedral polyhedra projected from 퐷6 (vertices are orbits underunderunder the thethe icosahedral icosahedralicosahedral group). group).group). under the icosahedral group). Table 1. Platonic and Archimedean icosahedral polyhedra projected from D (vertices are orbits under Vector of the Polyhedron in 푯ퟑ 6 VectorVectorVector of ofof the thethe Polyhedron PolyhedronPolyhedron in inin 푯푯푯ퟑ VectorVectorthe ofof icosahedral thethe PolyhedronPolyhedron group). inin 푯ퟑퟑ 푫 ퟑ CorrespoCorrespondingnding Vector Vector in in 푫 ퟔ ퟐ CorrespoCorrespondingnding VectorVector inin 푫푫ퟔퟔퟔ 풄 ≡ √ ퟐퟐퟐ (풎 − 풎 + ퟐ풎 흉) Corresponding풎ퟏ, 풎ퟐ Vector∈ ℤ in 푫ퟔ 풄 ≡ √ ퟐ (풎 ퟏ− 풎 ퟐ+ ퟐ풎 흉ퟐ) 풎풎풎ퟏ,,,풎풎풎ퟐ ∈∈∈ℤℤℤ 풄풄≡Vector≡√√ ퟐ of+ 흉 the((풎풎ퟏ Polyhedronퟏퟏ−−풎풎ퟐퟐퟐ++ퟐퟐ풎 in풎ퟐퟐퟐH흉흉3)) Corresponding풎ퟏퟏ, 풎ퟐퟐ ∈ Vectorℤ in D 풄 ≡ √ퟐퟐퟐ++q+흉흉흉 (풎ퟏ − 풎ퟐ + ퟐ풎ퟐ흉) 풎ퟏ, 풎ퟐ ∈ ℤ 6 ퟐ + 흉2 cퟐ + 흉 (m m +2m τ) m1, m2 Z ≡ 2+τ 1− 2 2 ∈ Icosahedron 푚1푙1 + 푚2(푙2 + 푙3 + 푙4 + 푙5 − 푙6) IcosahedronIcosahedronIcosahedronIcosahedron 푚푚푚111푙푙푙푙111+++푚푚푚222(((푙푙푙푙222+++푙푙푙푙333+++푙푙푙푙444+++푙푙푙푙555−−−푙푙푙푙666))) Icosahedron푣1 푐 푚1 푙1 + 푚2 (푙2 + 푙3=+(푚푙4 +−푙5푚−)푙6휔) + 2푚 휔 푣푣푣1Icosahedron푐푐푐 1 1 2 2 =3 (푚41− 푚5 )2휔6 1+ 2푚 휔2 5 푐 푣11 = 푐 (1, 휏, 0) m1l1 + m2(l2 + l3 + l4==+((l푚5푚111l−6−)푚 =푚222())m휔휔1111++m22푚)푚ω2221휔휔+5552 m2ω5 푐푐푐 v11=== 2c(((111,,,,휏휏휏,,,,000))) 푚 , 푚 =∈(2ℤ푚 or− 푚2ℤ)+휔1 + 2푚 휔 푐c√2==2 ((11,, τ휏,, 0)) 푚 ,m1,푚 2m∈ 2ℤ− 1or 2ℤ2+ +11− 2 5 √√√22√22 222 푚푚111,,,푚푚1,222 ∈∈2 2ℤ2ℤ2 Z ororor 22ℤℤ 2Z++11 1 √2 2 푚1, 푚2 ∈ ∈2ℤ or 2ℤ + 1

1 111 [( )( ) 1[[(푚푚1++33푚푚)2(푙푙푙1++푙푙푙2++푙푙푙)3 Dodecahedron 1 2[[((푚푚111++33푚푚222))((푙푙111++푙푙222 ++푙푙333)) DodecahedronDodecahedronDodecahedron 22[(2m[(푚+13+m3)(푚l2)+(푙l1 + l푙2)++푙(3m) m )(l + l + l )] = 푣Dodecahedron3 푐 2 2 1 2 1 2+ (푚3 − 푚1 )(푙 2 + 4푙 +5푙 )]6 푣Dodecahedron푣푣3 푐푐푐 2 2 + (푚 1− 푚 )2−(푙 4+ 푙 5+ 푙 )6]] 푐푣33v3 =푐 c (0, 휏222 , 1) (m ++(m(푚푚11)1ω−−+푚푚222)m)((푙푙푙4ω44++푙푙푙555++푙푙푙666))]] 푐푐푐 c푣3 ====푐(((0000,,,,,휏휏휏τ2,,,,,11 1))) 1 +=((푚2푚1 −6−푚푚2 ))(2휔푙4 3++푙52푚+ 푙휔6 )] √√22 22 2 =−(푚 11− 푚 2)2휔 46+ 2푚5 휔2 6 3 푐√√√222 =222 (0, 휏 , 1) m1, =m=2((푚푚1211Z−−or푚푚222 ))Z휔휔6+66++1 22푚푚222휔휔333 √2 2 푚1, 푚2=∈(∈푚2ℤ1 −or푚 22ℤ)휔+61+ 2푚2휔3 푚푚푚111,,,,푚푚푚222∈∈∈2ℤ2ℤ2ℤ or oror 2 22ℤℤℤ+++111 푚1, 푚2 ∈ 2ℤ or 2ℤ + 1 (푚 + 푚 )(푙 + 푙 ) + 2푚 (푙 + 푙 ) Icosidodecahedron (((푚 1+ 푚 )2))(푙푙 1+ 푙푙 )2 + 2푚 (2푙푙 3+ 푙푙 )4 IcosidodecahedronIcosidodecahedronIcosidodecahedronIcosidodecahedron푣 (푚푚111++푚(푚m2221)(+(푙푙111m++2)(푙푙222l1))+++l22)푚푚+222(2(푙푙m3332++(l3푙푙444+))l4) = IcosidodecahedronIcosidodecahedron푣 2 (푚1 + 푚2)(푙1 + 푙2=) +(푚21푚−2(푚푙32+)휔푙24)+ 2푚2휔4 푐푣푣22v22 = 푐(0, 휏, 0) (m ===m(((푚푚푚)1ω−−−+푚푚푚22m)))휔휔휔ω2 +++222푚푚푚2휔휔휔4 푐푐푐 c푣2 ====푐푐푐(c((0(000,,,,휏휏휏τ,,,000 ))) 1 = (2푚11 2− 푚22 )2휔224 + 2푚22 휔44 푐 √√22= 푐(0, 휏, 0) 푚1, 푚2−∈ 2ℤ1 or 2ℤ2 + 21 2 4 푐√2 = 푐(0, 휏, 0) 푚푚푚111,,,m,푚푚푚122,2∈m∈∈22ℤ2ℤ2ℤ 2 orZ ororor 2 22ℤℤ 2ℤZ++++111 1 √√22 푚1 , 푚2 ∈ 2ℤ∈ or 2ℤ + 1 √2 1 2 ((2푚1 + 푚)2)푙1 +((푚1 + 2푚)2)푙2 ((222푚푚푚111+++푚푚푚222))푙푙푙푙111+++((푚푚푚111+++222푚푚푚222))푙푙푙푙222 Truncated Icosahedron (2푚1 + 푚(22m)푙11++m(2+푚)l1푚1 +2((23m푚푙312+)푙23m푙42)+l2푙+5 − 푙6) TruncatedTruncatedTruncated Icosahedron IcosahedronIcosahedron +++푚푚푚2(((333푙푙푙푙3 +++333푙푙푙푙4 +++푙푙푙푙5 −−−푙푙푙푙6))) Truncated(Truncated푣 + 푣 ) Icosahedron Icosahedron푐 m ( l+=+푚(2푚2l (3+−푙33l 푚+ 3l)푙(4)4휔=++푙55 휔− 푙)66 ) (푣 (1++푣 )2) 푐 2 3 3 ( 3241 35 )2 6 4 1 5 2 6 푐((푣푣111v++1 푣v푣222)) =c푐푐 (1, 3휏, 0) ===((푚푚푚111−−−푚푚푚2−22))(((휔휔휔111+++휔휔휔222))) 푐 (푣c1 + 푣2)== 푐((1,1,, 33τ휏,,, 00)) (m m )(ω=++(2푚푚ω )−( +휔푚2m+)(휔(휔ω) + ω휔 )) 푐푐 √√22 ==2 2((11,,,33휏휏,,,00)) 1 2 +12푚 12(2휔 4+2휔2 )51 4 52 푐 √2 =222 (1, 3휏, 0) − ++22푚푚222((휔휔444++휔휔555)) √√22 푚m, 1푚, m+∈2 22ℤ푚2Z or(휔or 2 2ℤ+Z+휔+11) √2 2 푚푚푚 ,1,,푚푚푚 2∈∈∈2ℤ2ℤ2ℤ∈ or 2oror 2 224ℤℤℤ+++1115 푚푚111,,푚푚222 ∈∈2ℤ2ℤ oror 22ℤℤ++11 1 1 2 111 [( ) ( )( ) 1[[(33푚푚1++33푚푚)2푙푙푙1++(푚푚1++55푚푚)2(푙푙푙2++푙푙푙)3 SmallSmall Rhombicosidodecahedron Rhombicosidodecahedron 2[[((331푚푚[(111 ++33+푚푚222))푙푙111)+++((푚푚(111 +++55푚푚222)))(((푙푙222++푙푙333))) SmallSmallSmall RhombicosidodecahedronRhombicosidodecahedron Rhombicosidodecahedron 222 [(32푚13m+13푚32m)2푙1l+1 (푚m11+ 55푚m2)(l푙22 +l3푙3) Small(푣 Rhombicosidodecahedron1 + 푣3) 푐 + (푚 + 푚 )(푙 + 푙 ) (푣 (v+1+푣v3)) 푐 2 +( + )( + 1) + ( 2 4 )5 ] 푐((푣푣111++푣푣333))==c푐푐( (1, 2+휏 + 1,)1) m1 m2 +++l4(((푚푚푚11l15+++푚푚푚22m2)))1(((푙푙푙푙4443+++m푙푙2푙푙555))l)6 푐 (푣c + 푣 )= 2푐(1,1, 22τ휏 +1,1, 11) + ((푚1 + 푚2 )(−)푙4 +] 푙5 ) 푐푐 1 √√22 3 == 2((11,,,22휏휏++11,,,11)) = (m m )(+ω+(+푚푚ω11−)− +33푚2푚2m)2푙푙(푙4ω]6] +5ω ) 푐 √√√222 =222 (1, 2휏 + 1, 1) 1 2 ++1((푚푚1116−−33푚푚222)2)푙푙666]]3 5 √2 2 = (푚1 − 푚−2)(휔1++(휔푚61)−+32푚푚22)(푙휔6]3 + 휔5) √2 ===(((푚푚푚1 −−−푚푚푚2)))(((휔휔휔1 +++휔휔휔6)))+++222푚푚푚2(((휔휔휔3 +++휔휔휔5))) == (푚푚11 −−푚푚22 )((휔휔11 ++휔휔66))++22푚푚22((휔휔33 ++휔휔55)) 1 1 2 1 6 2 3 5 111 [(3푚 + 5푚 )(푙 + 푙 ) + (푚 + 7푚 )푙 Truncated Dodecahedron 1[[[(((3푚 1+ 5푚 )2))(((푙푙 1+ 푙푙 )2))+ (((푚 1+ 7푚 )2))푙푙 3 Truncated Dodecahedron 1 2[(33푚푚111 ++55푚푚222)(푙푙111 ++푙푙222) ++(푚푚111 ++77푚푚222)푙푙333 TruncatedTruncated DodecahedronDodecahedron 2 [(2232m[1(3+푚51m+2)(5l푚1 +2)l(2푙)1++(푙m21)++7(m푚21)l+3 +7푚(m21)푙+3 3m2)l4 Truncated(Truncated푣2 + 푣3) Dodecahedron Dodecahedron푐 2 + (푚1 + 3푚2)푙4 (푣 (v+2+푣v3)) 푐 + (푚 + 3푚 )푙푙 푐((푣푣222 ++푣푣333))==c푐푐( (0, 3+휏 + 1,)1) +(m++1 ((푚푚m1121)(++l533+푚푚2l226))])푙푙444 푐푐푐 (푣c 2 + 푣3)===2푐(0,((000,, 3,33τ3휏휏휏+++1,111,, 1,111))) ++−((푚푚1 +−3푚푚2))(푙4+푙 )] 푐 √√22 = 2(0, 3휏 + 1, 1) = (m m )(+ω++(((+푚푚푚ω1−−−) +푚푚푚2)2m))(((푙푙푙(ω+5++푙푙푙+)6))]]ω] ) 푐 √√√222 =222 (0, 3휏 + 1, 1) 1 2 +2((푚1116− 푚222))((2푙555+3푙666))]] 4 √2 2 =((푚1 − 푚−)2)(휔2++ 휔푚61)−+푚2푚2 2푙(5휔+3푙+6 휔 4) ===((푚푚푚111−−−푚푚푚222))(((휔휔휔222+++휔휔휔666)))+++222푚푚푚222(((휔휔휔333+++휔휔휔444))) 1 = (푚1 − 푚2)(휔2 + 휔6) + 2푚2(휔3 + 휔4) 111 1[[5((푚1 + 푚)2)푙1 +((3푚1 + 7푚)2)푙2 2[[[555((푚푚푚111+++푚푚푚222))푙푙푙푙111+++((333푚푚푚111+++777푚푚푚222))푙푙푙푙222 222 [5(푚1 + 푚2)푙1 + (3푚1 + 7푚2)푙2 Great Rhombicosidodecahedron 2 1 [5(m + m )l ++((푚3m1 ++97푚m2))푙l3 + (m + 9m )l GreatGreatGreatGreat Rhombicosidodecahedron RhombicosidodecahedronRhombicosidodecahedron Rhombicosidodecahedron 2 1 2 ++1+(((푚푚푚1 +++1 999푚푚푚2))2)푙푙푙푙32 1 2 3 ( ) ( 11 22 ) 33 Great(푣1 v+ Rhombicosidodecahedron1+푣v2++v3 푣3) c 푐 +(m + 5m2)+l++(푚푚(m11 +++95m푚푚222))l푙5푙34++(m(푚1 3+m푚2)2l6)]푙5 (((푣푣푣1c+++푣푣푣2 +++푣푣푣3))=) (푐푐푐1, 4τ + 1, 1) 1 +++4(((푚푚푚1 +++1 555푚푚푚2)))푙푙푙푙4 +++(((푚푚푚11 +++푚푚푚2)))푙푙푙푙5 푐 11 2√22 33 =2 (1, 4휏 + 1, 1) 11 22 44 11− 22 55 푐푐푐 (푣1 + 푣2 + 푣3)=== 푐2(((111,,,,444휏휏휏+++111,,,,111))) = (m1 m2)(ω1+++((ω푚푚211++−ω536푚)푚 +22))2푙푙4m6]+2 (ω(푚3 +1 +ω4푚+2)ω푙5) √2 +++(((푚푚푚1 −−−333푚푚푚2)))푙푙푙6]]]] 푐 √√√222 =222 (1, 4휏 + 1, 1) − ++ (푚푚11 −−33푚푚22 )푙푙66 ] √2 2 == ( 푚(푚1−−푚푚)2()휔(휔1++휔휔21++휔휔)6)+2+262푚푚(2휔(휔3++휔휔4++ == ((푚푚111−−푚푚222))((휔휔111++휔휔222++휔휔666))++22푚푚222((휔휔333++휔휔444++ = (푚1 − 푚2)(휔1 + 휔2 휔+5휔) 6) + 2푚2(휔3 + 휔4 + 휔휔휔555))) 휔5)

3. Danzer’s ABCK Tiles and 푫ퟔ Lattice 3.3.3. Danzer’s Danzer’sDanzer’s ABCK ABCKABCK Tiles TilesTiles and andand 푫푫푫ퟔퟔퟔ LatticeLatticeLattice 3. Danzer’s ABCK Tiles and 푫ퟔ Lattice We introduce the ABCK tiles with their coordinates in Figure 2 as well as their images in lattice WeWeWe introduce introduceintroduce the thethe ABCK ABCKABCK tiles tilestilestiles with withwithwith their theirtheirtheir coordinates coordinatescoordinatescoordinates in ininin Figure FigureFigure 2 22 as asas well wellwell as asas their theirtheir images imagesimages in inin lattice latticelattice 퐷6. ForWe introduce a fixed pair the ABCK of integers tiles with (푚 1their, 푚2) coordinates≠ (0, 0) , let in usFigure define 2 as thewell imageas their of images K by in vertices lattice 퐷퐷퐷666... . ForForFor a a a fixed fixed fixed pair pair pair of of of integers integers integers (((푚푚푚111,,,,푚푚푚222)))≠≠≠(((000,,,,000))),,, , lllletetet us us us define define define the the the image image image of of of KKK bybyby vertices vertices vertices 퐷6. For a fixed pair of integers (푚1, 푚2) ≠ (0, 0) , let us define the image of K by vertices

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𝐷(𝑚,𝑚), 𝐷(𝑚,𝑚), 𝐷(𝑚,𝑚) , and 𝐷 (0,0) in lattice 𝐷 and its projection in 3D space, where 3. Danzer’s ABCK Tiles and D6 Lattice 𝐷 (0,0) represents the origin. They are given as We introduce the ABCK tiles with their coordinates in Figure2 as well as their images in lattice D . For a fixed pair of integers 𝑐𝑣 (m , m ) (0, 0), let us define the image of K by vertices D (m , m ), 6 𝐷(𝑚,𝑚)= =[𝑚1𝑙 +𝑚2 ,(𝑙 +𝑙 +𝑙 +𝑙 −𝑙)]∥ =[(𝑚 −𝑚)𝜔 +2𝑚𝜔]∥ 1 1 2 √2 D2(m1, m2), D3(m1, m2), and D0 (0, 0) in lattice D6 and its projection in 3D space, where D0 (0, 0) represents the origin. They are given as 𝑐𝑣 1 1 𝐷(𝑚,𝑚)= = h [(𝑚 +𝑚)(𝑙 +𝑙)+2𝑚(𝑙 +𝑙)]∥ = [(𝑚 −𝑚)𝜔 +2𝑚𝜔]∥ 2cv√12 2 2 D1(m1, m2) = = m1l1 + m2(l2 + l3 + l4 + l5 l6)] =[ (m1 m2)ω1 + 2m2ω5] (12) √2 h − k − k cv2 1 1 D2(m1, m2) = = 2 (m1 + m2)(l1 + l2) + 2m2(l3 + l4)] = 2 [ (m1 m2)ω2 + 2m2ω4] 2 √2 𝑐𝜏 𝑣 1 k − k 𝐷 (𝑚 ,𝑚 )= 1 = [(𝑚 +𝑚 )(𝑙 +𝑙 +𝑙 )+(−𝑚 +3𝑚 )(𝑙 +𝑙 +𝑙 )] (12) D (m , m) = cτ− v3 = 1 [(2m + m )(l + l + l ) + ( m + 3m)(l + l + l )∥] 3 1 2 √ √2 2 1 2 1 2 3 1 2 4 5 6 2 − k = [2m2ω3 + ( m1 + 3m2)ω6] =[2𝑚𝜔 + (−𝑚− +3𝑚)𝜔]∥ k ( ) which follows from Table 1 1 with with redefinitionsredefinitions ofof thethe pairpair ofof integersintegers (𝑚m1,,𝑚m2 ).. With removal of the D (m , m ) ´ i(1, 2) notation , they represent the vectors in 6D space. A set of scaled vertices defined by D i notation k∥, they represent the vectors in 6D space. A set of scaled vertices defined by 𝐷 ≡ c are used for the vertices of the Danzer’s tetrahedra in Figure2. are used for the vertices of the Danzer’s tetrahedra in Figure 2.

Figure 2. Danzer’s tetrahedra. Figure 2. Danzer’s tetrahedra. It is instructive to give a brief introduction to Danzer’s tetrahedra before we go into details. It is instructive to give a brief introduction to Danzer’s tetrahedra before we go into details. The The edge lengths of ABC tetrahedra are related to the weights of the icosahedral group H3 whose edge 1 1 edge lengths of ABC tetrahedra√2+τ arev1 related √to3 the τweights− v3 of theτ− icosahedralv2 group 𝐻 whose edge1 lengths are given by a = = k k , b = = k k , 1 = k k and their multiples by τ and τ , √2 ‖ √‖ √2 √ √ − lengths are given by 𝑎= = 2,𝑏= = 2 ,1= 2 and their multiples by 𝜏 and 𝜏, where a, b, 1 are the original edge lengths√ introduced √ by Danzer.√ However, the tetrahedron K has edge 1 wherelengths 𝑎,𝑏,1 also involving are the original1 , τ , τ− edge. As lengths we noted introduced in Section by2, the Danzer. vertices However, of a rhombic the tetrahedron triacontahedron K has 2 2 2 1 edge lengths also involving ,(v,1)h . As( τwe− v3 )hnoted in Section 2, the vertices of a rhombic consist of the union of the orbits 3 and 3 . One of its cells is a pyramid based on a golden √2 √2 ( ) ( ) triacontahedron consist of the union of the orbits and . One of its cells is a pyramid rhombus with vertices √ √ based on a golden rhombus with1 vertices 1 v1 1 τ− v3 1 v1 1 τ− v3 1 = (1, τ, 0), = (0, τ, σ), R1 = ( 1, τ, 0), R3 = (0, τ, σ), (13) √2 2= (1, 𝜏, 0),√2 = 2 (0, 𝜏,− −𝜎),𝑅 √ 2= (−1,𝜏,02 − ), 𝑅 √=2 (0,2 𝜏, 𝜎), (13) √ √ √ √

SymmetrySymmetry 2020 2020, 12, 12, x, FOR x FOR PEER PEER REVIEW REVIEW 7 of7 16of 16

AndAnd the the apex apex is isat at the the origin. origin. The The coordinate coordinate of ofth eth intersectione intersection of ofthe the diagonals diagonals of ofthe the rhombus rhombus is is theSymmetrythe vector vector2020 , 12,= 1983=(0,(0, 𝜏, 𝜏,) 0 , )0 and, and its its magnitude magnitude is isthe the in-radius in-radius of ofthe the rhombic rhombic triacontahedron. triacontahedron.7 of 15 √√ Therefore,Therefore, the the weights weights , , , , and and the the origin origin can can be be taken taken as asthe the vertices vertices of ofthe the tetrahedron tetrahedron K. K. √√ √√ √√ AsAndAs such, such, the apexit itis is is atthe the the fundamental origin.fundamental The coordinate region region of of of thethe the intersectionicosahedral icosahedral of thegroup group diagonals from from ofwhich thewhich rhombus the the rhombic isrhombic the vector v2 = 1 (0, τ, 0), and its magnitude is the in-radius of the rhombic triacontahedron. Therefore, triacontahedrontriacontahedron2 √2 2 is isgenerated generated [19]. [19]. The The octahedra octahedra generated generated by by these these tetrahedra tetrahedra denoted denoted by by<𝐵>,< <𝐵>,< 1 𝐶>andthe𝐶>and weights <𝐴><𝐴>v1 , comprise τ comprise− v3 , v2 four andfour copies the copies origin of ofeach can each beobtained obtained taken as by theby a group verticesa group of of oforder the order tetrahedron four four generated generatedK. As by such, bytwo two √2 √2 2𝑅√ 𝑅2 𝑅 𝑅 <𝐾><𝐾> commutingitcommuting is the fundamental generators generators region and and of the or icosahedral ortheir their conjugate conjugate group groups. from groups. which The The octahedron the octahedron rhombic triacontahedron consists consists of is8ofK 8 K generatedgeneratedgenerated by [by19 a]. groupa Thegroup octahedra of of order order eight generated eight cons consisting byisting these of ofthree tetrahedra three commuting commuting denoted generators generators by B , C𝑅 and,𝑅 𝑅, 𝑅Aas asmentionedcomprise mentioned earlier,earlier, and and the the third third generator generator 𝑅 𝑅 is isan an affine affine reflection reflection [20] [20] with with respect respect toh itothe hthe goldeni goldenh rhombici rhombic face face four copies of each obtained by a group of order four generated by two commuting generators R1 and inducedinduced by by the the affine affine Coxeter Coxeter group group 𝐷 𝐷. . R3 or their conjugate groups. The octahedron K consists of 8K generated by a group of order eight One can dissect the octahedron < < 𝐾 𝐾 > into > h i three non-equivalent pyramids with rhombic bases consistingOne can of three dissect commuting the octahedron generators R1, R 3 intoas mentionedthree non-equivalent earlier, and thepyramids third generator with rhombicR0 is an bases 𝑣 −𝑣 ,𝑣 −𝑣 ,𝑣 −𝑣 byabyffi cutting necutting reflection along along the [20 the] lines with lines orthogonal respect orthogonal to the to togoldenthree three planes rhombic planes 𝑣 face −𝑣 induced,𝑣 −𝑣 by,𝑣 the −𝑣 affi. Onene. One Coxeter of ofthe the pyramids group pyramidsDe6 .is is generatedgeneratedOne by can by the dissectthe tetrahedron tetrahedron the octahedron K Kupon upon theK the intoactions actions three of of non-equivalentthe the group group generated generated pyramids by by the with the reflections reflections rhombic 𝑅 bases 𝑅and and h i 𝑅by𝑅. If. cutting Ifwe we call call along 𝑅 𝑅𝐾=𝐾 the𝐾=𝐾 lines as as the orthogonal the mirror mirror image to image three of ofplanesK Kthe the othersv 1othersv2 ,canv 2can bev 3be taken, v 3takenv as1. Oneasthe the tetrahedra of tetrahedra the pyramids obtained obtained is − − − fromgeneratedfrom K Kand and by𝑅 𝑅 the𝐾 𝐾 by tetrahedron by a rotationa rotation Kof uponof 180 180 the around actions around the of the axis the axis group𝑣 .𝑣 A. generatedAmirror mirror image image bythe of of reflections4𝐾 4𝐾 = (2𝐾 = (2𝐾R1 and + 2𝐾) +withR 2𝐾) 3with. respectIfrespect we call to to Rthe 1Kthe =rhombic Krhombic´ as the plane mirror plane (corresponding image (corresponding of K the others to toan canan affine affine be taken reflection) reflection) as the tetrahedracomplements complements obtained it itup fromup to Ktoan an ◦ ´ octahedronandoctahedronR1K by of aof 8 rotationK 8,K as, as we we of mentioned180 mentionedaround above. theabove. axis In Inadditionv2 .addition A mirror to tothe image the vertices vertices of 4K in= in(13) ( 2(13)K the+ the2 octahedronK) octahedronwith respect 8K 8 alsoK to also includestheincludes rhombic the the planevertices vertices (corresponding (0, (0, 0,0) 0 and) and to(0, ( an0, 𝜏, 𝜏,affi 0). ne 0)Octahedral. reflection)Octahedral tiles complements tiles are are depicted depicted it up toin aninFigure octahedronFigure 3. 3.Dissected Dissected of 8K, pyramidsaspyramids we mentioned constituting constituting above. the the In octahedron additionoctahedron to <𝐾> the <𝐾> vertices, as, asdepicted in depicted (13) the in octahedron inFigure Figure 4, have4, 8 haveK also bases, bases, includes two two of the whichof verticeswhich with with (0, 0, 0) and (0, τ, 0). Octahedral tiles are depicted in Figure 3. Dissected pyramids constituting the basesbases of of golden golden rhombuses rhombuses with with edge edge lengths lengths (heights) (heights) 𝜏 𝜏𝑎𝑎,𝑎,𝑎, and, and the the third third is theis the one one with with octahedron K , as depicted in Figure4, have bases, two of which with bases of golden rhombuses with 𝑏(𝑏(). )The. The rhombic rhombich i triacontahedron triacontahedron1 τ τ 1consists consists of of60𝐾 60𝐾 + 60𝐾 + 60𝐾=30(2𝐾+2𝐾 =30(2𝐾+2𝐾 1 ), where), where (2𝐾 (2𝐾 + +2𝐾) form2𝐾) form a a edge lengths (heights) τ a , a − , and the third is the one with b . The rhombic triacontahedron − 2 2 2 cellcell of of pyramid pyramid based based on on a agolden golden rhombus rhombus of ofedge edge 𝜏 𝜏𝑎 𝑎and and height height . Faces. Faces of ofthe the rhombic rhombic consists of 60K + 60K´ = 30 2K + 2K´ , where 2K + 2K´ form a cell of pyramid based on a golden triacontahedrontriacontahedron are are1 the the rhombuses rhombusesτ orthogonal orthogonal to toon one ofe of30 30 vertices vertices of ofthe the icosidodecahedron icosidodecahedron given given rhombus of edge τ− a and height . Faces of the rhombic triacontahedron are the rhombuses orthogonal in (8). 2 toin one(8). of 30 vertices of the icosidodecahedron given in (8).

A-octahedronA-octahedron B-octahedronB-octahedron C-octahedronC-octahedron K-octahedronK-octahedron Figure 3. The octahedra generated by the ABCK tetrahedra. FigureFigure 3. 3.The The octahedra octahedra generated generated by by the theABCK ABCKtetrahedra. tetrahedra.

(a) (b) (c) (a) (b) (c) Figure 4. The three pyramids (a, b, c) composed of 4K tetrahedra. Figure 4.4. TheThe threethree pyramidspyramids ( a(a,–c )b, composed c) composed of 4K of tetrahedra. 4K tetrahedra.

WhenWhen the the 4 4KK ofof Figure Figure 44aa isis rotatedrotated byby the the icosahedral icosahedral group, group, it it generates generates the the rhombic rhombic When the 4K of Figure 4a is rotated by the icosahedral group, it generates the rhombic triacontahedrontriacontahedron as as shown shown in in Figure Figure 55.. triacontahedron as shown in Figure 5.

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Figure 5. A view of rhombic triacontahedron. FigureFigure 5.5. A view of rhombic triacontahedron. The octahedron <𝐵> is a non-convex polyhedron whose vertices can be taken as 𝐷 , 𝜏𝑅 𝐷 𝜏𝐷 ,𝜏𝑅 𝐷 ,2𝜏𝐷 𝐷 ´ ´ The, octahedron B<𝐵>,is and a non-convex the sixth vertex polyhedron is the origin whose vertices. It consists can of be four taken triangular as D1, τ facesR1𝐷D,1 , The octahedron h i is a non-convex polyhedron whose vertices can be taken as τwithD´ , τRedgesD´ , 2𝑎,τ 1 D´𝜏𝑎,, and 𝜏𝑏 the and sixth four vertex triangular is the faces origin withD´ edges 𝜏 .𝑎, It consists 𝑎, 𝑏. Full of action four triangularof the icosahedral faces with 𝜏𝑅3𝐷 , 𝜏𝐷3 3,𝜏𝑅−𝐷 ,2𝜏2 𝐷 , and the sixth vertex is the origin0 𝐷 . It consists of four triangular faces group on the tetrahedron 𝐵 would generate the “B1-polyhedron” consisting of 60𝐵 + 60𝐵 where edgeswith aedges, τa, τb and𝑎, four𝜏𝑎, triangular 𝜏𝑏 and four faces triangular with edges facesτ− witha, a, bedges 𝜏. Full𝑎, 𝑎, action 𝑏. Full of theaction icosahedral of the groupicosahedral on the ( ) ( ) ´ ´ tetrahedrongroup𝐵 =𝑅 𝐵on theisB would tetrahedronthe mirror generate 𝐵image thewould “B -polyhedron” ofgenerate 𝐵. It the has consisting “B -polyhedron”62 vertices of 60B + consisting60(12B likewhere ofB =60𝐵, 30R1 likeB +is the60𝐵 where mirror, 1 √ √ ( ) (τv1)h (τ− v2)h (v3)h ( ) ( ) 3 3 3 image𝐵20 =𝑅 like of𝐵 B .isIt), 180 hasthe edges, 62mirror vertices and image 120(12 faceslike of consisting 𝐵. , It 30 like hasof faces 62 withvertices, 20 triangleslike (12 of like) ,edges180 edges, with, 30 𝜏 likeand𝑎, 120 𝑎, 𝑏. facesThe, √ √2 √2 √2 √ √ ( ) 1 consisting20face like transitive of), faces 180 “ B withedges,-polyhedron” triangles and 120 of facesis edges depicted consistingwith τin− aofFigure, a faces, b. The 6with faceshowing triangles transitive three-fold of “ edgesB-polyhedron” andwith 𝜏five-fold𝑎, is depicted𝑎, 𝑏. axesThe √ insimultaneously. Figure6 showing three-fold and ﬁve-fold axes simultaneously. face transitive “B-polyhedron” is depicted in Figure 6 showing three-fold and five-fold axes simultaneously.

Figure 6. The “B-polyhedron”. Figure 6. The “B-polyhedron”.

The octahedron C is a convex polyhedron,Figure 6. The “ whichB-polyhedron”. can be represented by the vertices The octahedronh is a convex polyhedron, which can be represented by the vertices 1 1 1 1 2 1 2 ( τ<, 0, 𝐶 1) > , (τ, τ, τ), (1, τ, 0), τ , 1, 0 , τ , τ, 1 , (0, 0, 0). (14) The octahedron2 is2 a convex polyhedron,2 2 which can2 be represented by the vertices (𝜏, 0,1), (𝜏, 𝜏, 𝜏), (1, 𝜏,) 0 , (𝜏,1,0), (𝜏,𝜏,1), (0,0,0). (14) This is obtained from that of Figure2 by a translation and inversion. It consists of four triangular (𝜏, 0,1), (𝜏, 𝜏, 𝜏), (1, 𝜏,) 0 , (𝜏,1,0), (𝜏1,𝜏,1), (0,0,0). faces with edges a, b, τband four triangular faces with edgesτ− a, a and b. (14) This is obtained from that of Figure 2 by a translation and inversion. It consists( vof) four triangular(v ) 1 h3 2 h3 The “C-polyhedron” is a non-convex polyhedron with 62 vertices (12 like , 30 like , faces with edges 𝑎, 𝑏, 𝜏𝑏 and four triangular faces with edges 𝜏 𝑎, 𝑎 and 𝑏. √ √ This is obtained from that of Figure 2 by a translation and inversion. It consists of2 four triangular2 (v3)h () () 20faces likeThe with “ 3Cedges)-polyhedron”, 180 𝑎, edges, 𝑏, 𝜏𝑏 and and is a 120 fournon-convex faces triangular consisting polyhedron faces of with only with edges one 62 type vertices𝜏𝑎, of triangular 𝑎 (12 and like 𝑏. face, with 30 like edge lengths, 20 √2 √ √ ( ) ( ) Symmetry1 ( )2020 , 12, x FOR PEER REVIEW 9 of 16 τlike− a, aTheand “)Cb, 180-polyhedron”as canedges, be seenand is 120 ina non-convex Figurefaces consisting7 (see polyhedron also of the only reference with one type62 vertices [ 4of]). triangular It (12 is also like face with transitive, 30 likeedge lengths same, 20 as √ √ √ “B-polyhedron”. () like𝜏 𝑎, 𝑎 )and, 180 𝑏 as edges, can be and seen 120 in facesFigure consisting 7 (see also of theonly reference one type [4]). of triangular It is also faceface transitivewith edge samelengths as √ “B-polyhedron”. 𝜏𝑎, 𝑎 and 𝑏 as can be seen in Figure 7 (see also the reference [4]). It is also face transitive same as “B-polyhedron”.

Figure 7. The “C-polyhedron”.

The octahedron < 𝐴 > is a non-convex polyhedron, which can be represented by the vertices 1 1 1 1 1 (−𝜏, 0,1), (1, −𝜏, 0), (1,1,1), (𝜎, 0, −𝜏), (𝜎, −𝜏,) 1 , (0,0,0), (15) 2 2 2 2 2 as shown in Figure 3. Now, we discuss how the tiles are generated by inflation with an inflation factor 𝜏. First of all, let us recall that the projection of subset of the 𝐷 vectors specified by the pair of integers (𝑚,𝑚) are some linear combinations of the weights 𝑣, (𝑖 = 1, 2, 3) with an overall factor c. It is easy to find out the vector of 𝐷 corresponding to the inflated vertex of any ABCK tetrahedron by noting that 𝑐(𝜏 𝑣)=𝑐(𝑚́ , 𝑚́)𝑣. Here, we use 𝜏 =𝐹 +𝐹𝜏, and we define 𝑚́ ≡𝑚 𝐹 + (𝑚 +5𝑚 )𝐹 , 𝑚́ ≡𝑚 𝐹 + (𝑚 +𝑚 )𝐹 , (16) where 𝐹 represents the Fibonacci sequence satisfying

𝐹 =𝐹 +𝐹, 𝐹 =(−1) 𝐹, 𝐹 =0,𝐹 =1. (17)

It follows from (16) that 𝑚́ +𝑚́ = even if 𝑚 +𝑚 =even, otherwise 𝑚́ ,𝑚́ ∈ℤ. This proves that the pair of integers (𝑚́ , 𝑚́) obtained by inflation of the vertices of the Danzer’s tetrahedra remain in the subset of 𝐷 lattice. We conclude that the inflated vectors by 𝜏 in (12) can be obtained by replacing 𝑚 by 𝑚́ and 𝑚 by 𝑚́. For example, radii of the icosahedra projected by 𝐷 vectors

2𝜔,2𝜔,2𝜔 +2𝜔,2𝜔 +4𝜔 (18)

are in proportion to 1, 𝜏, 𝜏 and 𝜏 , respectively. It will not be difficult to obtain the 𝐷 image of any general vector in the 3D space in the form of 𝑐(𝜏,𝜏,𝜏) where 𝑝, 𝑞, 𝑟 ∈ℤ. Now we discuss the inflation of each tile one by one.

3.1. Construction of 𝜏𝐾 = 𝐵 +𝐾 The vertices of 𝜏𝐾 is shown in Figure 8 where the origin coincides with one of the vertices of B as shown in Figure 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 𝐷 having a normal vector (−1, −𝜎, −𝜏) outward should match with the face of B opposite to the vertex 𝐷 with a normal vector (𝜎, 𝜏,) −1 inward. For this reason, we perform a rotation of K given by 0−10 𝑔 =00−1 (19) 10 0 matching the normal of these two faces. A translation by the vector (𝜏, 𝜏,0) will locate K in its proper place in 𝜏𝐾. This is the simplest case where a rotation and a translation would do the work.

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The octahedron A is a non-convex polyhedron, which can be represented by the vertices h i 1 1 1 1 1 ( τ, 0, 1), (1, τ, 0), (1, 1, 1), (σ, 0, τ), (σ, τ, 1), (0, 0, 0), (15) 2 − 2 − 2 2 − 2 − as shown in Figure3. Now, we discuss how the tiles are generated by inflation with an inflation factor τ. First of all, let us recall that the projection of subset of the D6 vectors specified by the pair of integers (m1, m2) are some linear combinations of the weights vi, (i = 1, 2, 3) with an overall factor c. It is easy to find out the vector of D6 corresponding to the inflated vertex of any ABCK tetrahedron by noting that n n c(τ vi) = c(m´1,m´2)vi. Here, we use τ = Fn 1 + Fnτ, and we define − 1 1 m´1 m1Fn 1 + (m1 + 5m2)Fn, m´2 m2Fn 1 + (m1 + m2)Fn, (16) ≡ − 2 ≡ − 2 where Fn represents the Fibonacci sequence satisfying n+1 Fn+1 = Fn + Fn 1, F n = ( 1) Fn, F0 = 0, F1 = 1. (17) − − − It follows from (16) that m´ 1 + m´ 2 = even if m1 + m2 = even, otherwise m´ 1, m´ 2 Z. This proves ∈ that the pair of integers (m´1,m´2) obtained by inflation of the vertices of the Danzer’s tetrahedra remain n in the subset of D6 lattice. We conclude that the inflated vectors by τ in (12) can be obtained by replacing m1 by m´1 and m2 by m´2. For example, radii of the icosahedra projected by D6 vectors

2ω1, 2ω5, 2ω1 + 2ω5, 2ω1 + 4ω5 (18) 2 3 are in proportion to 1, τ, τ and τ , respectively. It will not be diﬃcult to obtain the D6 image of any general vector in the 3D space in the form of c(τp, τq, τr) where p, q, r Z. Now we discuss the inﬂation ∈ of each tile one by one.

3.1. Construction of τK = B + K The vertices of τK is shown in Figure8 where the origin coincides with one of the vertices of BSymmetryas shown 2020,in 12,Figure x FOR PEER2 and REVIEW the other vertices of K and B are depicted. A transformation is needed10 of 16 to translate K to its new position. The face of K opposite to the vertex D´ having a normal vector The corresponding rotation and translation in 𝐷 can be calculated easily,2 and the results are 1 ( 1, σ, τ) outward should match with the face of B opposite to the vertex D´ with a normal vector illustrated2 − − − in Table 2. 0 1 (σ, τ, 1) inward. For this reason, we perform a rotation of K given by 2 −   Table 2. Rotation and translation in 𝐻 and 𝐷 .  0 1 0   −  gK =  0 0 1  (19) 𝑯𝟑  −  𝑫𝟔  1 0 0  𝑔 ←rotation→ 𝑔:(136)(245) 1 1 2 matching𝑡 the= ( normal𝜏, 𝜏,0) of ← these two faces.→ A translation by the vector( τ, τ , 0 will locate) K in its translation [(𝑚 +5𝑚)𝑙 +(𝑚 +𝑚) 𝑙2 +𝑙 +𝑙 +𝑙 −𝑙 ] proper place in τK. This is the simplest case2 where a rotation and a translation would do the work. The corresponding rotation and translation in D6 can be calculated easily, and the results are illustrated The rotation in 𝐷 is represented as the permutations of components of the vectors in the 𝑙 in Table2. basis.

Figure 8. VerticesVertices of τ𝜏𝐾.K.

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

3.2. Construction of 𝜏𝐵 = 𝐶 + 4𝐾 +𝐵 +𝐵 The first step is to rotate B by a matrix

1 −𝜎 𝜏 −1 𝑔 = −𝜏 1 −𝜎 (20) 2 1−𝜎𝜏

and then inflate by 𝜏 to obtain the vertices of 𝜏𝑔𝐵 as shown in Figure 9.

(a) (b)

Figure 9. Views from 𝜏𝑔𝐵. (a) Top view of 𝜏𝑔𝐵; (b) bottom view of 𝜏𝑔𝐵.

Representation of 𝜏𝑔𝐵 is chosen to coincide the origin with the 𝐷 vertex of tetrahedron C. It is obvious to see that the bottom view illustrates that the 4K is the pyramid based on the rhombus

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The corresponding rotation and translation in 𝐷 can be calculated easily, and the results are illustrated in Table 2.

Table 2. Rotation and translation in 𝐻 and 𝐷.

𝑯𝟑 𝑫𝟔 𝑔 ←rotation→ 𝑔:(136)(245) 1 𝑡 = (𝜏, 𝜏 ,0) ←translation→ [(𝑚 +5𝑚 )𝑙 +(𝑚 +𝑚 )(𝑙 +𝑙 +𝑙 +𝑙 −𝑙 )] 2

The rotation in 𝐷 is represented as the permutations of components of the vectors in the 𝑙 basis.

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Table 2. Rotation and translation in H3 and D6.

H3 D6 g rotation g : 136 245 K ← → K 1 2 1 tK = τ, τ , 0 translation [(m + 5m )l + (m + m )(l + l + l + l l )] 2 ← → 2 1 2 1 1 2 2 3 4 5 − 6 Figure 8. Vertices of 𝜏𝐾. The rotation in D6 is represented as the permutations of components of the vectors in the li basis. Rhombic triacontahedron generated by 𝜏𝐾K now consists of a “B-Polyhedron”B centered at the Rhombic triacontahedron generated by τ now consists of a “ -Polyhedron” centered at the 1 origin and 30 pyramids of 4K with golden rhombic bases of edge lengths a and heights τ− occupy origin and 30 pyramids of 4K with golden rhombic bases of edge lengths a and heights 2 occupy the 30 inwardthe 30 gapsinward of gaps “B-Polyhedron”. of “B-Polyhedron”.

3.2. Construction of 𝜏𝐵 = 𝐶 + 4𝐾 +𝐵 +𝐵 3.2. Construction of τB = C + 4K + B1 + B2 The ﬁrstThe first step step is to is rotateto rotateB byB by a matrixa matrix −𝜎 𝜏 −1 1  σ τ 1  𝑔 = 1−𝜏− − 1 −𝜎 (20) gB =  τ 1 σ  (20) 22 − −   1−𝜎𝜏1 σ τ  − and thenand then inﬂate inflate by τbyto 𝜏 obtain to obtain the the vertices vertices of ofτ g𝜏𝑔BB𝐵as as shown shown in in FigureFigure9 9..

(a) (b)

Figure 9. Views from 𝜏𝑔 𝐵. (a) Top view of 𝜏𝑔 𝐵; (b) bottom view of 𝜏𝑔 𝐵. Figure 9. Views from τgBB.(a) Top view of τgBB;(b) bottom view of τgBB.

𝜏𝑔 𝐵 𝐷 RepresentationRepresentation of τofg BB is chosen is chosen to to coincide coincide the the origin origin withwith the the D´0 vertexvertex of of tetrahedron tetrahedron C. ItC . It is obviousis obvious to see thatto see the that bottom the bottom view view illustrates illustrates that that the the 4K 4isK theis the pyramid pyramid based based on on the the rhombus rhombus with 1 edge length (height), b 2 as shown in Figure4c. The 4 K takes its position in Figure9 by a rotation followed by a translation given by    1 σ τ  1 − −  1 rotation : g4K =  σ τ 1 , translation : t4K = (τ, 0, 1). (21) 2 −  2  τ 1 σ  − To translate B1 and B2, we follow the rotation and translation sequences as   σ τ 1   1  −  1 3 2 rotation : gB =  τ 1 σ , translation : tB = τ , 0, τ , 1 2  − −  2  1 σ τ   − −  (22) σ τ 1   1  − −  1 3 2 rotation : gB2 =  τ 1 σ , translation : tB = τ , 0, τ . 2  −  2  1 σ τ  − 3.3. Construction of τC = K1 + K2 + C1 + C2 + A We inﬂate by τ the tetrahedron C with vertices shown in Figure 10. Top view of τC where one of the vertices of K1 is at the origin is depicted in Figure 11. The vertices of the constituting tetrahedra are given as follows: Symmetry 2020, 12, x FOR PEER REVIEW 11 of 16

with edge length (height), 𝑏( ) as shown in Figure 4c. The 4K takes its position in Figure 9 by a rotation followed by a translation given by

1−𝜎−𝜏 rotation: 𝑔 = −𝜎 𝜏 1 , translation: 𝑡 = (𝜏, 0,1). (21) −𝜏 1 𝜎

To translate 𝐵 and 𝐵, we follow the rotation and translation sequences as

1 𝜎−𝜏1 𝑔 = , 𝑡 = (𝜏,0,𝜏) rotation: 𝐵 2 −𝜏 1 −𝜎 translation: , 1 −1 𝜎 −𝜏 (22) −𝜎 −𝜏 1 rotation: 𝑔 = 𝜏1−𝜎, translation: 𝑡 = (𝜏,0,𝜏). 1𝜎−𝜏

3.3. Construction of 𝜏𝐶 =𝐾 +𝐾 +𝐶 +𝐶 +𝐴 We inflate by 𝜏 the tetrahedron C with vertices shown in Figure 10. Top view of 𝜏𝐶 where one

of the vertices of 𝐾 is at the origin is depicted in Figure 11. The vertices of the constituting tetrahedra are given as follows:

:(0,0,0), (1,1,1), (𝜏, 0,1), (𝜏,𝜏,1)}; 𝐾 Symmetry 2020, 12, 1983 11 of 15 𝐾 : (1,1,1), (𝜏, 0,1), (𝜏,𝜏,1), (𝜏,𝜏,1)}; n 1 1 1 2 o K1 : (0, 0, 0), 2 (1, 1, 1), 2 (τ, 0, 1), 4 τ , τ, 1 ; n 𝐶 :1 ((1,1,1)), 1(𝜏,( 0,1),) (𝜏1,𝜏,12 ), (𝜏1,𝜏2,𝜏)} ; (23) K2 : 2 1, 1, 1 , 2 τ, 0, 1 , 4 τ , τ, 1 , 2 τ , τ, 1 }; n 1 1 1 2 1 2 2 2o C : (1, 1, 1), (τ, 0, 1), τ , τ, 1 , τ , τ , τ ; (23) 𝐶1 :2 (𝜏, 0,1), 2(𝜏,𝜏,1), 2(𝜏,0,𝜏), 2(𝜏,𝜏,𝜏)}; n 1 1 2 1 2 1 2 2 2 C2 : 2 (τ, 0, 1), 2 τ , τ, 1 , 2 τ , 0, τ , 2 τ , τ , τ }; 1 2 1 2 1 2 2 2 1 3 2 o A :{ τ , τ, 1 , τ , 0, τ , τ , τ , τ , τ , τ , τ . 𝐴:{2 (𝜏,𝜏,1), 2 (𝜏,0,𝜏), 2(𝜏,𝜏,𝜏), (2𝜏,𝜏,𝜏)}.

Symmetry 2020, 12Figure, x FOR 10.PEER C-C-tetrahedron tetrahedronREVIEW obtained obtained from from that that of of Fi Figuregure 22 byby translationtranslation andand inversion.inversion. 12 of 16

(a) (b)

FigureFigure 11. 11.ViewsViews from from 𝜏𝐶.τ (Ca.() Topa) Top view view of of𝜏𝐶τ;C (b;()b bottom) bottom view view of of 𝜏𝐶τC. .

The vertices of K can be obtained from K by a rotation K = g K, where The vertices of 𝐾 1 can be obtained from K by a rotation 𝐾1=𝑔K1𝐾, where   −𝜎σ 𝜏 τ −1 1   5 𝑔 = −𝜏1 − 1 −𝜎, − 𝑔 =1. (24) g K =  τ 1 σ , gK = 1. (24) 1 1−𝜎𝜏2 − −  1  1 σ τ  − ( ) The transformation to obtain 𝐾 is a rotation 𝑔 followed by a translation of 𝑡= 𝜏 ,𝜏,1 , The transformation to obtain K is a rotation g followed by a translation of t = 1 τ2,τ, 1 , where where 2 K2 K2 2   −𝜎 σ −𝜏τ −1 1  1 − − −  10 𝑔 = −𝜏 −1 −𝜎, 𝑔 =1 (25) gK2 =  τ 1 σ , gK2 = 1 (25) 2 − − −  1𝜎𝜏 1 σ τ  ( ) To obtain the coordinates of 𝐶and 𝐶, first translate C by − 𝜏,1 𝜏, 𝜏 and rotate by 𝑔 and 𝑔, To obtain the coordinates of C1and C2, ﬁrst translate C by (τ, τ, τ) and rotate by gC and gC , 2 1 2 respectively, followed by the translation 𝑡 = (1𝜏 ,𝜏2 ,𝜏2 )2, where− = respectively, followed by the translation tC1 2 τ , τ , τ , where 1 𝜎𝜏1 1 1−𝜎−𝜏 𝑔 = 𝜏1𝜎 , 𝑔 = −𝜎 𝜏 1 (26) 2 2 1𝜎𝜏 𝜏−1−𝜎 Similarly, vertices of tetrahedron A are rotated by

1 1−𝜎−𝜏 𝑔 = −𝜎 𝜏 1 (27) 2 −𝜏 1 𝜎 followed by a translation 𝑡 = (𝜏,0,𝜏) .

3.4. Construction of 𝜏𝐴 = 3𝐵 + 2𝐶 +6𝐾

It can also be written as 𝜏𝐴 = 𝐶 +𝐾 +𝐾 +𝐵+ 𝜏𝐵, where 𝜏𝐵 is already studied. A top view of 𝜏𝐴 is depicted in Figure 12 with the vertices of 𝜏𝐵 given in Figure 9.

Symmetry 2020, 12, 1983 12 of 15

     σ τ 1   1 σ τ  1  1 − −  gC =  τ 1 σ , gC =  σ τ 1  (26) 1 2  2 2 −   1 σ τ   τ 1 σ  − − Similarly, vertices of tetrahedron A are rotated by    1 σ τ  1 − −  gA =  σ τ 1  (27) 2 −   τ 1 σ  − 1 2 followed by a translation tA = 2 τ , 0, τ .

3.4. Construction of τA = 3B + 2C + 6K

It can also be written as τA = C + K1 + K2 + B + τB, where τB is already studied. A top view of SymmetryτA is depicted 2020, 12, inx FOR Figure PEER 12 REVIEW with the vertices of τB given in Figure9. 13 of 16

FigureFigure 12. 12. TopTop view view of of τ𝜏𝐴A (bottom (bottom viewview isis thethe samesame asas bottombottom viewview ofof τ𝜏𝐵B-- dotteddotted region).region). o 1 ( ) 1 2 1 2 The vertices of C {(0,0,0), 2 (τ𝜏,, 0, 0,1 1),, 2(𝜏τ,1,0, 1, 0),, 2(𝜏τ,𝜏,1, τ, 1)} cancan be be obtained from from those in Figure 22 1 by a translation ( τ, τ, τ) followed by a rotation by a translation 2 (−−𝜏,− −𝜏,− −𝜏) followed by a rotation    1 σ τ  1 − −  gc = −1σ 𝜎τ 1 −𝜏 (28) 12 −  𝑔 =  𝜎−𝜏1τ 1 σ  (28) 2 − − −𝜏 1 −𝜎 Vertices of K1and K2 are given by n 1 1 2 1 2 1 o Vertices of 𝐾and 𝐾K are1 : given(τ, 0, by 1), τ , 1, 0 , τ , τ, 1 , (3τ + 1, τ, 1) , n 2 2 2 4 o (29) K : 1 (τ, 0, 1), 1 τ2, τ, 1 , 1 (3τ + 1, τ, 1), 1 (2τ, σ, 1) . 𝐾 :2 (𝜏,2 0,1), (2𝜏,1,0), 4 (𝜏,𝜏,1), (23𝜏+1,𝜏,1− )}, To obtain K1 and K2 with these vertices, ﬁrst rotate K by gK1 and gK2 given by the matrices     (29)  σ τ 1   σ τ 1  𝐾: (𝜏, 0,1), 1 (𝜏 ,𝜏,1−), (3𝜏+1,𝜏,11 ), (2𝜏, −𝜎, 1)}. gK =  τ 1 σ, gK2 =  τ 1 σ , (30) 1 2 −  2   1 σ τ   1 σ τ  − To obtain 𝐾and 𝐾 with these vertices, first rotate K by 𝑔 and 𝑔 given by the matrices = 1 ( ) and translate both by the vector tK1 2 τ, 0, 1 . The vertices of B in Figure 12 can be obtained by a rotation then a translation given, respectively,𝜎𝜏−1 by 𝜎𝜏1   𝑔 = 𝜏1−𝜎τ 1 σ, 𝑔 = 𝜏1𝜎, (30)   1 − −  1 3 2 gB = 1𝜎−𝜏 1 σ τ , tB = 1𝜎𝜏τ , 0, τ . (31) 2 − −  2  σ τ 1  ( ) and translate both by the vector 𝑡 =− 𝜏,− 0,1 . The vertices of B in Figure 12 can be obtained by a All these procedures are described in Table3, which shows the rotations and translations both in rotation then a translation given, respectively, by E and D6 spaces. k −𝜏 −1 𝜎 𝑔 = −1 −𝜎 𝜏, 𝑡 = (𝜏,0,𝜏). (31) −𝜎 −𝜏 1 All these procedures are described in Table 3, which shows the rotations and translations both in 𝐸∥ and 𝐷 spaces.

Symmetry 2020, 12, 1983 13 of 15

Table 3. Rotation and translation in H3 and D6 (see the text for the deﬁnition of rotations in E space). k

H3 D6 τK=B+K gK rotation 136 245 ← → c 2 1 tK = τ, τ , 0 translation [(m + 5m )l + (m + m )(l + l + l + l l )] 2 ← → 2 1 2 1 1 2 2 3 4 5 − 6 τB=C+4K+B1+B2

gB rotation (3)(12645) ← → g K rotation (1)(23)(45)(6) 4 ← → c t4K = (τ, 0, 1) translation m l + m (l l + l l l ) 2 ← → 1 5 2 1 − 2 3 − 4 − 6 gB rotation (16435)(2) 1 ← → gB rotation (152152), (364364) 2 ← → c 3 2 1 i tB = τ , 0, τ translation [(3m + 5m )l + (m + 3m )(l l + l l l ) 2 ← → 2 1 2 5 1 2 1 − 2 3 − 4 − 6 τC=K1+K2+C1+C2+A

gK rotation (54621)(3) 1 ← → (identity translation)

gK rotation (11)(6345263452) 2 ← → t = c τ2, τ, 1 translation (m + m )(l + l ) + 2m )(l l ) K 2 ← → 1 2 1 5 2 3 − 6 1 c 2 [(m1 + 3m2)(l1 + l3 + l5) + tC = 2 (τ, τ, τ) translation − − ← → (m1 m2)(l2 l4 l6)] − − − gC rotation (1)(4)(26)(35) 1 ← → gC rotation (1)(35642) 2 ← → t = c τ2, τ2, τ2 translation (m + 2m )(l + l + l ) + m (l l l ) C 2 ← → 1 2 1 3 5 2 2 − 4 − 6 gA rotation (1)(6)(23)(45) ← → c 2 1 tA = τ , 0, τ translation [(m + 5m )l + (m + m )(l l + l l l )] 2 ← → 2 1 2 5 1 2 1 − 2 3 − 4 − 6 τA=C+K1+K2+B+τB

1 c 2 [(m1 + 3m2)(l1 + l3 + l5) + tC = 2 (τ, τ, τ) translation − − ← → (m1 m2)(l2 l4 l6)] − − − gC rotation (11) (24)(36)(55) ← → gK rotation (1) (25436) 1 ← → gK rotation (1)(4) (26)(35) 2 ← → c tK = (τ, 0, 1) translation m l + m (l l + l l l ) 2 ← → 1 5 2 1 − 2 3 − 4 − 6 gB rotation 651651 ,(423423) ← → c 3 2 1 tB = τ , 0, τ translation [(3m + 5m )l + (m + 3m )(l l + l l l )] 2 ← → 2 1 2 5 1 2 1 − 2 3 − 4 − 6

4. Discussions The six-dimensional reducible representation of the icosahedral subgroup of the point group of D6 is decomposed into the direct sum of its two three-dimensional representations described also by the direct sum of two graphs of icosahedral group. We have shown that the subset of the D6 lattice characterized by a pair of integers (m1, m2) with m1 + m2 = even projects onto the Platonic and Archimedean polyhedra possessing icosahedral symmetry, which are determined as the orbits of the n fundamental weights vi or their multiples by τ . The edge lengths of the Danzer tiles are related to the weights of the icosahedral group H3 and via Table1 to the weights of D6, 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 H3 as being the cell of the rhombic triacontahedron. Images of the Danzer tiles and their inﬂations are determined in D6 by employing translations and icosahedral Symmetry 2020, 12, 1983 14 of 15

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 two-fold axes. Since the faces of the Ammann rhombohedral tiles are orthogonal to the two-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 the 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 to be studied. 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 D6 lattice characterized by integers (m1, m2), 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 [4]).

Author Contributions: Conceptualization, M.K., N.O.K., A.A.-S.; methodology, M.K., A.A.-S., N.O.K.; software, A.A.-S., N.O.K.; validation, M.K., A.A.-S., N.O.K.; formal analysis, A.A.-S., N.O.K.; investigation, A.A.-S., M.K., N.O.K.; writing-original draft preparation, M.K., A.A.-S., N.O.K.; writing-review and editing, M.K., N.O.K., A.A.-S.; visualization, A.A.-S.; N.O.K.; project administration, M.K.; supervision, M.K., N.O.K., A.A.-S. All authors have read and agreed to the published version of the manuscript. Funding: This research was partly funded by Sultan Qaboos University, Om an. Acknowledgments: We would like to thank Ramazan Koc for his contributions to Figures6 and7. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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