Ribbon tilings from spherical ones

The problem of classifying all tile-k-transitive tilings of the infinite 2-dimensional ribbon (and pinched-ribbon) is shown to be solvable by classifying certain tile-k-transitive tilings of the sphere, for all k. Complete results are listed for k3.Supported by the Deutsche Forschungsgemeinschaft.     

Crystallographic reptiles

We consider tilings ofR ⁿ by copies of a compact setA under the action of a crystallographic group, such that the union ofk suitably chosen tiles is affinely isomorphic toA. For dimensionn=2 we show that for eachk2 there is a finite number of isomorphy classes of such setsA which are homeomorphic to a disk. We give an algorithm which determines all disk-like tiles for a given group and numberk. The algorithm will be applied to the groupsp2 andp3 withk=3.     

On low-dimensional faces that high-dimensional polytopes must have

We prove that every five-dimensional polytope has a two-dimensional face which is a triangle or a quadrilateral. We state and discuss the following conjecture: For every integerk1 there is an integer f(k) such that everyd-polytope,df(k), has ak-dimensional face which is either a simplex or combinatorially isomorphic to thek-dimensional cube.We give some related results concerning facet-forming polytopes and tilings. For example, sharpening a result of Schulte [25] we prove that there is no face to face tiling of ⁵ with crosspolytopes.Supported in part by a BSF Grant and by I.H.E.S, Bures-Sur-Yvette.     

The classification of 2-isohedral tilings of the plane

Two different methods for enumerating k-isohedral tilings are discussed. One is geometric: by splitting and gluing tiles. The other is combinatorial: by enumerating the appropriate Delaney—Dress symbols. Both methods yield 1270 types of proper 2-isohedral tilings of the plane.     

Domino Tiling Congruence Modulo 4

The number of domino tilings of a region with reflective symmetry across a line is combinatorially shown to depend on the number of domino tilings of particular subregions, modulo 4. This expands upon previous congruency results for domino tilings, modulo 2, and leads to a variety of corollaries, including that the number of domino tilings of a k × 2k rectangle is congruent to 1 mod 4.     

Fractal penrose tiles II: Tiles with fractal boundary as duals of penrose triangles

Summary Suppose given a quasi-periodic tiling of some Euclidean space E _u which is self-similar under the linear expansiong: E_μ→E_μ. It is known that there is an embedding of E_μ into some higher-dimensional space ℝ ^N and a linear automorphism with integer coefficients such that E _u ⊂ ℝ ^N is invariant under andg is the restriction of to E _u . Let E _s be the -invariant complement of E _u , and . If certain conditions are fulfilled (e.g. must be a lattice automorphism,g ^* is an expansion), we construct a self-similar tiling of E _s whose expansion isg ^*, using the information contained in the original tiling of E_μ. The term “Galois duality” of tilings is motivated by the fact that the eigenvalues ofg ^* are Galois conjugates of those ofg. Our method can be applied to find the Galois duals which are given by Thurston, obtained in a somewhat other way for the case that dim E_μ=1, andg is the multiplication by a cubic Pisot unit. Bandt and Gummelt have found fractally shaped tilings which can be considered as strictly self-similar modifications of the kites-and-darts, and the rhombi tilings of Penrose. As one of the examples, we show that these fractal versions can be constructed by dualizing tilings by Penrose triangles.     

Self similar symmetric planar tilings

A new variant of the projection method yields aperiodic tilings of the plane with some rotational symmetry. In particular we display three tilings E _s with full D ₇-symmetry. Each of them is self similar. Further, there is an uncountable number of tilings E without any symmetry, but being almost equivalent to each of the symmetric tiling E _s , i.e. for each R > 0 there is a translation T(E) of E which is equal to E _s in all vertices but a set of error points which are distributed all over the plane but have mutual distance greater than R.      

On approximation of approximately generalized quadratic functional equation via Lipschitz criteria

Abstract

Let G be an Abelian group with a metric d and E ba a normed space. For any f : G → E we define the generalized quadratic di?erence of the function f by the formula

Q_k f (x, y) := f (x + ky) + f (x ? ky) ? f (x + y) ? f (x ? y) ? 2(k² ? 1)f (y)

for all x, y ∈ G and for any integer k with k ≠ 1, ?1. In this paper, we achieve the general solution of equation Q_k f (x, y) = 0, after it, we show that if Q_k f is Lipschitz, then there exists a quadratic function K : G → E such that f ? K is Lipschitz with the same constant. Moreover, some results concerning the stability of the generalized quadratic functional equation in the Lipschitz norms are presented. In the particular case, if k = 0 we obtain the main result that is in [7].     

Packing of Incongruent Circles on the Sphere

 In this paper we provide an upper bound to the density of a packing of circles on the sphere, with radii selected from a given finite set. This bound is precise, e.g. for the system of incircles of Archimedean tilings (4, 4, n) with n ? 6. A generalisation to the weighted density of packing is applied to problems of solidity of a packing of circles. The simple concept of solidity was introduced by L. Fejes Toóth [6]. In particular, it is proved that the incircles of the faces of the Archimedean tilings (4,6,6), (4,6,8) and (4, 6, 10) form solid packings. (Received 21 August 2000; in revised form 21 March 2001)     

The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings

In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of coherent mixed subdivisions of a Minkowski sum ?₁+...+? _r of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding ?(?₁,...,? _r ). In this paper we extend this correspondence in a natural way to cover also non-coherent subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence polytopes provides a new independent proof of the Bohne-Dress theorem on zonotopal tilings. This application uses a combinatorial characterization of lifting subdivisions, also originally proved by Santos. Received February 18, 1999 / final version received January 25, 2000?Published online May 22, 2000     

Sets of tiles with a prescribed number of tilings

Summary For every k2 and r1 there exists a set of k prototiles that admits exactly r distinct tilings. All the tilings obtained are periodic.     

A primer of substitution tilings of the Euclidean plane

This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include self-similar tilings such as the well-known Penrose tilings; for this class there is a substantial body of research in the literature. Combinatorial substitutions are just beginning to be examined, and some of what we present here is new. We give numerous examples, mention selected major results, discuss connections between the two classes of substitutions, include current research perspectives and questions, and provide an extensive bibliography. Although the author attempts to represent the field as a whole, the paper is not an exhaustive survey, and she apologizes for any important omissions.     

Light paths with an odd number of vertices in large polyhedral maps

LetP _k be a path onk vertices. In this paper we prove that (1) every polyhedral map on the torus and the Klein bottle contains a pathP _k such that each of its vertices has degree 6k–2 ifk is odd,k3, (2) every large polyhedral map on any compact 2-manifoldM with Euler characteristic (M)<0 contains a pathP _k such that each of its vertices has degree 6k – 2 ifk is odd,k3, (3) moreover, these bounds are attained. Fork=1 ork even,k2, the bound is 6k which has been proved in our previous paper.     

On the transversal Helly numbers of disjoint and overlapping disks

A family of disks is said to have the property T(k) if any k members of the family have a common line transversal. We call a family of unit diameter disks t-disjoint if the distances between the centers are greater than t. We consider for each natural number k≧ 3 the infimum t_k of the distances t for which any finite family of t-disjoint unit diameter disks with the property T(k) has a line transversal. We determine exact values of t₃ and t₄, and give general lower and upper bounds on the sequence t_k, showing that t_k = O(1/k) as k → ∞. In honour of Helge Tverberg’s seventieth birthday Received: 9 June 2005     

The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, I

We compute the number of rhombus tilings of a hexagon with sidesN,M,N, N,M,N, which contain a fixed rhombus on the symmetry axis that cuts through the sides of lengthM.     

Simplices with congruent k-faces

Let S be a non-degenerate simplex in $\mathbb{R}^{2}$. We prove that S is regular if, for some k $\in$ {1,...,n-2}, all its k-dimensional faces are congruent. On the other hand, there are non-regular simplices with the property that all their (n1)-dimensional faces are congruent.     

On the structure of singular sets of convex functions

Whenf is a convex function ofR ^h, andk is an integer with 0<k, then the set ^k (f)=x:dim(f(x)k may be covered by countably many manifolds of dimensionh–k and classC ² except an _h–k negligible subset.The author is supported by INdAM     

Strukturen geometrischer Ornamente

We consider double-periodic tilings of the whole plane from the view of graph theory, not with respect to symmetry groups. We suppose that the graph is planar and connected and that the fundamental domain contains a finite number of vertices and edges. We assign to every tiling a tableau. There exists a fundamental formula connecting the number of all numbers of the tableau with the sum of the reciprocals of all these numbers and with the number p of lines in the tableau; the formula is proved even if multiple edges or loops occur. By this way we get a graph-theoretic classification of the tilings. We introduce families F of tilings and their ranks. The family F={k₁,k₂...,k_s} (with k₁>k₂>...>k_s>O) is the set of all tilings, the tableau of which contains all the numbers k_j and no others. The smallest number p of lines (which occur for the tilings of the family F) is the rank of F and has special geometric interest. Some open questions are mentioned at the end.

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Herrn Helmut Karzel zum 60. Geburtstag gewidmet     

Entropy numbers and lattice arrangements in l∞ (Γ)

The Banach spaces l_∞(Γ) admit tilings by balls of equal size that are arranged along a lattice. We present classes of bounded sets in spaces l_∞(Γ) whose optimal packings and covers in the sense of inner and outer metric entropy numbers are realized by lattice arrangements. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Approximation of Convex Bodies and a Momentum Lemma for Power Diagrams

 The volume of the symmetric difference of a smooth convex body in and its best approximating polytope with n vertices is asymptotically a constant multiple of . We determine this constant and the similarly defined constant for approximation with a given number of facets by solving two isoperimetric problems for planar tilings. Received 15 May 1997; in revised form 14 August 1997