Smallest polyhedral tilings of 3-tori by parallelohedra
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Smallest polyhedral tilings of 3-tori by parallelohedra Ulrich Brehm1 · Wolfgang Kühnel2 Received: 12 April 2020 / Accepted: 2 September 2020 © The Managing Editors 2020
Abstract There are two parallelohedra in dimension 2 and five in dimension 3. We study smallest polyhedral quotients of the tilings defined by them on tori. For 2-tori the situation is well understood, and the three minimal models are well known. Here we investigate the five 3-dimensional cases with the result: Each of the five parallelohedra admits a minimal (and weakly neighborly) model, moreover in three cases this is unique up to natural equivalence. Altogether there are 14 inequivalent minimal models. Keywords Lattice tiling · Quotient of lattices · Tessellation · Dirichlet-Voronoi cell · Cube · Hexagonal prism · Rhombic dodecahedron · Elongated dodecahedron · Truncated octahedron Mathematics Subject Classification Primary: 52C22; Secondary: 20H15 · 52B10 · 52B70 · 57Q91
1 Introduction A parallelohedron is a convex polytope that can tessellate Euclidean space with facetto-facet contacts via translations. Up to affine transformations, there are exactly two possibilities in dimension 2: The square with two classes of parallel edges and the regular hexagon with three classes of parallel edges. Consequently, all facets of 3dimensional parallelohedra are parallelograms or hexagons with parallel opposite edges. An n-torus is the quotient of Euclidean n-space by a discrete group of translations isomorphic with Zn . In the plane each square of the tessellation {4, 4} is surrounded by 8 squares, and each hexagon of the tessellation {6, 3} is surrounded by
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Wolfgang Kühnel [email protected] Ulrich Brehm [email protected]
1
Fakultät Mathematik, Technische Universität Dresden, 01062 Dresden, Germany
2
Fachbereich Mathematik, Universität Stuttgart, 70550 Stuttgart, Germany
123
Beitr Algebra Geom
6 hexagons. Therefore an associated minimal polyhedral 2-torus must have 9 squares or 7 hexagons, respectively. The latter one is the dual of the unique 7-vertex simplicial torus (see Fig. 1), {6, 3}(2,1) in the notation of McMullen and Schulte (2002). The standard torus with 3 × 3 squares is {4, 4}(3,0) , for another instance with 9 squares see Sect. 3. For other (non-minimal) quotients see (Brehm and Kühnel 2008), for tilings on the torus in general see (Senechal 1988; Kisielewicz and Przelawski 2012). The classification of the 3-dimensional parallelohedra is originally due to E. Fedorov in the 19th century (Fedorov 1885). We state it following Moser (1961) (again up to affine transformations):
Cube Hexagonal prism Rhombic dodecahedron Elongated dodecahedron Truncated octahedron
nr. of parallel edges
nr. of quadrilaterals
nr. of hexagons
3 4 4 5 6
6 6 12 8 6
0 2 0 4 8
Coxeter (1973, p.29) called them the five primary parallelohedra, for an alternative classification see (Fejes Tóth 1965, Sect. I.4.3). The five tessellations defined by translations of these five items show that each tile has • • • • •
26 neighbors f
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