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On Parallelohedra of Nil-Space

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On Parallelohedra of Nil-Space ON PARALLELOHEDRA OF NIL-SPACE 1Benedek SCHULTZ, 2Jenő SZIRMAI 1 Budapest University of Technology and Economics, Institute of Mathematics, Department of Geometry, e-mail: [email protected] 2 Budapest University of Technology and Economics, Institute of Mathematics, Department of Geometry, e-mail: [email protected] Received 20 May 2011; accepted 20 May 2011 Abstract: The parallelohedron is one of basic concepts in the Euclidean geometry and in the 3-dimensional crystallography, has been introduced by the crystallographer E.S. Fedorov (1889). The 3-dimensional parallelohedron can be defined as a convex 3-dimensional polyhedron whose parallel copies tiles the 3-dimensional Euclidean space in a face to face manner. This paral- lelohedron presents a fundamental domain of a discrete translation group. The 3-parallelepiped is the most trivial and obvious example of a 3-parallelohedron. Fedorov was the first to succeed in classifying the parallelohedra of the 3-dimensional Euclidean space, while in some non-Euclidean geometries it is still an open problem. In this paper we consider the Nil geometry introduced by Heisenberg’s real matrix group. We introduce the notion of the Nil -parallelohedra, outline the concept of parallelohedra classes analogous to the Euclidean geometry. We also study and visualize some special classes of Nil - parallelohedra. Keywords: Tiling, discrete translation group, parallelohedron, Nil geometry. 2 B. SCHULTZ and J. SZIRMAI 1. Euclidean case Fedorov was the first to successfully classify the parallelohedra in Euclidean 3-space E3 in [7]. These are convex bodies, which allow the tiling of space using only transla- tions. Fedorov’s solution for Euclidean case relied basically on two theorems. The first one was originally a gap in Fedorov’s proof, he never proved, that parallelohedra are centrally symmetric, but used it. This gap was filled by Minkowski. Theorem 1.: Every parallelohedron and its faces are centrally symmetric. The second theorem: Theorem 2.: The faces of a parallelohedron which are parallel to a given edge L form a “closed zone”. The parallel projection of the parallelohedron along the edge L is a parallelogon in the projection plane. Furthermore the same projection of a tiling of the space yields a tiling of the projection plane by parallelogons. Fig. 1 . The truncated octahedron. PARALLELOHEDRA OF NIL SPACE 3 It is a well known fact that in the Euclidean plane only parallelogons are the parallelograms and centrally symmetric hexagons. In dimensions 3 and 4 we have a complete picture. In 3 dimensions Fedorov’s five parallelohedra are the cube, hexagonal prism, rhombic dodecahedron, hexarhombic dodecahedron, truncated octahedron. The truncated octahedron is special among them, because it is maximal in the sense, that it has as many facets as possible (namely 14). 2. Nil-geometry The Nil -geometry can be derived from the famous real matrix group L(R) discovered by Werner Heisenberg. The left (row-column) multiplication of Heisenberg matrices 1 x y1 a c 1 a + x c + xb + z 0 1 z 0 1 b = 0 1 b + y (1) 0 0 1 0 0 1 0 0 1 defines “translations” L(R)={ ( x, y, z), x,y,z ∈ R } on the points of the space Nil ={ ( a, b, c), a, b, c ∈ R }. The matrices K( z ) < L( R ) of the form 1 0 z 0 1 0 → ()0 0 z (2) 0 0 1 constitute the one parametric centre, i.e. each of its elemenst commutes with all elements of L. The elements of K are called fibre translations . The Nil geometry can be projectively interpreted by the "right translations", as the following matrix formula shows, according to [2]: 1 x y z 0 1 0 0 ()()()1 a b c → 1 a b c = 1 x + a y + b z + xb + c . 0 0 1 x 0 0 0 1 The detailed description can be found in article [3]. 4 B. SCHULTZ and J. SZIRMAI 3. Discrete translation group and Nil-lattice We consider Nil -translation defined as above and choose two arbitrary translations 1 2 3 1 2 3 1 t1 t1 t1 1 t2 t2 t2 0 1 0 0 0 1 0 0 τ = , and τ = , (3) 1 1 2 1 0 0 1 t1 0 0 1 t2 0 0 0 1 0 0 0 1 with upper indices as coordinate variables. We define the translation k (τ3 ) ,(k ∈ N,k ≥ 1) by the following commutator: 1 2 1 2 1 0 0 − t2t1 + 1tt 2 k −1 −1 0 1 0 0 ()τ3 = τ2 τ1 τ2τ1 = . (4) 0 0 1 0 0 0 0 1 If we take integers as coefficients, then we generate the discrete group ( τ1,τ 2 ,k) denoted by L(τ1 ,τ2 ,k) . Definition: The Nil point lattice Γ(τ1 ,τ2 ,k) is a discrete orbit of point P in the Nil space under the group L(τ1 ,τ2 ,k) with an arbitrary starting point P (k ∈ N,k ≥ 1). In the following we study only the case k=1. 4. Nil-parallelohedra In this section we consider the possible definition of the Nil -paralleohedra, and visualize two special a cases of such parallelohedra. First we need to specify, what we consider as a polyhedron in Nil -space. The Nil -translations have the fortunate property, that a translated picture of a Euclidean-line (and -plane) in our model is another Euclidean-line (-plane). Keeping this in mind it is viable to consider a Nil - parallelohedron by the definition of discrete translation group from the previous section: PARALLELOHEDRA OF NIL SPACE 5 Definition: A Nil -parallelohedron is a set of points of the Nil space which is realized in our model as a simply connected polyhedron in Euclidean sense and admits isohedral, face to face tiling using only the translates of the prototile by a discrete translation group of Nil . We note here that in some cases, the in Euclidean sense non-coplanar “face” of a Nil -polyhedron is combined by finitely many Euclidean polygons (see Fig. 4-5). Similarly to the Euclidean case we can give a theorem, which gives us considerable help in determining the Nil -parallelohedra, especially in some special cases. The next theorem is the direct consequence of the definitions of the translation and discrete translation group (see (3), (4)). Theorem : If the faces of a Nil -parallelohedron which are parallel to z axis form a “closed zone” then the projection of the parallelohedron along the axis z is a paral- lelogon in the [ x,y ] plane. Furthermore the same projection of a tiling of the space yields a tiling of the projection plane by parallelogons in Euclidean sense. T213 T21 T23 T13 T12 T3 T1 T2 T Fig. 2 . The cube-like Nil-parallelohedron. 6 B. SCHULTZ and J. SZIRMAI First we consider the cube-like parallelohedron, which is illustrated in Fig. 2. The vertices of this polyhedron can be generated by a given discrete translation group L(τ1 ,τ2 ,1) : T = T τ1 , T = T τ1τ 2 , T = T τ2 , T = T τ3 , 1 12 2 3 (5) τ 2τ3 τ1τ3 τ 2τ1 τ 2τ1τ3 T23 = T , T13 = T , T21 = T , T213 = T . τ1 : TT3T23T2 → T1T13T213T21 , τ2 : TT1T13T3 → T2T12T21T23 , τ3 : TT1T12T2T21 → T3T13T21T23T213 , The generator translations induce three L(τ1 ,τ2 ,1) equivalence classes of edges, each provides a so-called defining relation for three generators. The vertices of the parallelohedron fall into one equivalence class. −1 −1 −1 −1 −1 −1 L(τ1 ,τ2 ,1) = {τ1 ,τ 2 −1 = τ1τ3τ1 τ3 = τ 2 τ3τ2 τ3 = τ1τ2 τ3τ1 τ2 }. Now, we are going to show an example of Nil -parallelohedron. Our goal is to construct an exact parallelohedron analogous to the Euclidean cube. Consider two translations and the third translation is given by the commutator (see 5): 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 τ = and τ = ⇒ τ = . 1 0 0 1 1 2 0 0 1 0 3 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 We can choose an arbitrary vertex with coordinates: T1 − − − . 2 2 2 The other vertices are the translated images of T (see (6)). The indices show, which translations were used to compute the given point. PARALLELOHEDRA OF NIL SPACE 7 Fig.3. The cube-like Nil-parallelohedron. In Fig. 3 we show the above computed parallelohedron. The faces are colored so, that the faces with the same color are translated to each other by an element of the discrete translation group. The Fig. 4 shows the prototile and its translated copies by τ1,τ 2 ,τ 3 . Also you can see, that the projection of the tiling onto the [ x,y ] plane gives a parallelogon tiling in the Euclidean plane. Fig. 4. The tiling given by the Nil -parallelepiped. 8 B. SCHULTZ and J. SZIRMAI Next we are going to construct similarly to the cube-like Nil -parallelohedron a hexagonal-like one. In this paper we only show an example for this type of the Nil - parallelohedra, the general case is not detailed here. For this let us consider two translations: 1 3 1 1 0 0 1 0 2 2 0 1 0 0 τ = and τ = 0 1 0 0. 1 0 0 1 1 2 0 0 1 0 0 0 0 1 0 0 0 1 −1 −1 From these we can obtain the τ 3 ,τ 4 translations by (5) and by τ1 τ 2 τ3 = 1 (identity map): 1 3 3 1 − 0 1 0 0 2 2 2 τ = 0 1 0 0 and τ = 0 1 0 0 .
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