Characterization of Polytopes via Tilings with Similar Pieces

Generalizing results by Valette, Zamfirescu and Laczkovich, we will prove that a convex body K is a polytope if there are sufficiently many tilings which contain a tile similar to K. Furthermore, we give an example that this cannot be improved.     

Tilings of triangles

Let T be a non-equilateral triangle. We prove that the number of non-similar triangles Δ such that T can be dissected into triangles similar to Δ is at most 6. On the other hand, for infinitely many triangles T there are six non-similar triangles Δ such that T can be dissected into congruent triangles similar to Δ. For the equilateral triangle there are infinitely many such Δ. We also investigate the number of pieces in the dissections of the equilateral triangle into congruent triangles.     

Enumerating Cube Tilings

Cube tilings formed by $n$ -dimensional $4\mathbb Z ^n$ -periodic hypercubes with side $2$ and integer coordinates are considered here. By representing the problem of finding such cube tilings within the framework of exact cover and using canonical augmentation, pairwise nonisomorphic 5-dimensional cube tilings are exhaustively enumerated in a constructive manner. There are 899,710,227 isomorphism classes of such tilings, and the total number of tilings is 638,560,878,292,512. It is further shown that starting from a 5-dimensional cube tiling and using a sequence of switching operations, it is possible to generate any other cube tiling.     

Periodic Delone Tilings

Given a periodic point set in 3-dimensional Euclidean space, an algorithm is described for computing the corresponding Delone tiling (and its Delaney symbol). Examples of applications in tiling theory and crystallography are discussed.     

Self-Similar Lattice Tilings

We study the general question of the existence of self-similar lattice tilings of Euclidean space. A necessary and sufficient geometric condition on the growth of the boundary of approximate tiles is reduced to a problem in Fourier analysis that is shown to have an elegant simple solution in dimension one. In dimension two we further prove the existence of connected self-similar lattice tilings for parabolic and elliptic dilations. These results apply to produce Haar wavelet bases and certain canonical number systems.     

Tilings of Decomposable Groups

We study a problem formulated by A. M. Vershik and related to several questions in orbit theory of tilings of finitely generated groups. Let G be decomposed into a free product of two nontrivial groups. Then for any finite subset S of the group G there exists a finite subset P of the group G, PS, such that G is covered by disjoint left translations of the set P. Bibliography: 2 titles.     

Combinatorially Regular Polyomino Tilings

Let T be a regular tiling of ℝ² which has the origin 0 as a vertex, and suppose that φ: ℝ² → ℝ² is a homeomorphism such that (i) φ(0)=0, (ii) the image under φ of each tile of T is a union of tiles of T, and (iii) the images under φ of any two tiles of T are equivalent by an orientation-preserving isometry which takes vertices to vertices. It is proved here that there is a subset Λ of the vertices of T such that Λ is a lattice and φ|_Λ is a group homomorphism. The tiling φ(T) is a tiling of ℝ by polyiamonds, polyominos, or polyhexes. These tilings occur often as expansion complexes of finite subdivision rules. The above theorem is instrumental in determining when the tiling φ(T) is conjugate to a self-similar tiling.     

Tilings of parallelograms with similar triangles

We say that a triangle δ tiles the polygonP ifP can be decomposed into finitely many non-overlapping triangles similar to δ. Let P bea parallelogram with anglesδ andπ -δ (0 <δ ≤π/2) and let δ be a triangle with anglesα, Β, γ (α ≤Β ≤γ). We prove that if δ tilesP then eitherδ ε α,Β,γ,π -γ, π - 2γ or dimL _P =dimL _δ. We also prove that for every parallelogramP, and for every integern (wheren≥ 2,n ? 3) there is a triangle δ so thatn similar copies of δ tileP.     

Quasicrystallic Tilings and Projection Method

Basic notions related to quasiperiodic tilings and Delone sets in Eucledean space are discussed. It is shown how the cut and project method of constructing them is used to calculate their spectra. Special attention is paid to self-similar tilings and the way one can obtain one-dimensional substitutional tilings by the projection scheme. Bibliography: 18 titles.     

Partial Cubes and Archimedean Tilings

Some physicists depicted the molecular structure SnCl₂ · 2(H₂O) by a piece of an Archimedean tiling (4.8.8) that is a partial cube. Inspired by this fact, we determine Archimedean tilings whose connected subgraphs are all partial cubes. Actually there are only four Archimedean tilings, (4.4.4.4), (6.6.6), (4.8.8) and (4.6.12), which have this property. Furthermore, we obtain analytical expressions for Wiener numbers of some connected subgraphs of (4.8.8) and (4.6.12) tilings. In addition, we also discuss their asymptotic behaviors.     

Tilings of polygons with similar triangles

We prove that if a polygonP is decomposed into finitely many similar triangles then the tangents of the angles of these triangles are algebraic over the field generated by the coordinates of the vertices ofP. IfP is a rectangle then, apart from four sporadic cases, the triangles of the decomposition must be right triangles. Three of these sporadic triangles tile the square. In any other decomposition of the square into similar triangles, the decomposition consists of right triangles with an acute angle such that tan is a totally positive algebraic number. Most of the proofs are based on the following general theorem: if a convex polygonP is decomposed into finitely many triangles (not necessarily similar) then the coordinate system can be chosen in such a way that the coordinates of the vertices ofP belong to the field generated by the cotangents of the angles of the triangles in the decomposition.This work was completed while the author had a visiting position at the Mathematical Institute of the Hungarian Academy of Sciences.