Some tilings of the plane whose singular points are uncountable and unbounded

Let I be a tiling of the plane such that for every tile T of I there correspond a tile T of I (not necessarily unique) and an integer k(T, T) (depending on T and T), k(T, T)>2, such that T meets T in k(T, T) connected components. Tiles T and T satisfying this condition are called associated tiles in I. Various properties concerning I and its singular points are obtained. First, it is not possible that every tile in I have a unique associated tile. In fact, there exist infinite families of tiles {F} {F _n:n1} such that F is the unique associated tile for every F _n. Next, if x is a singular point of I, then every neighborhood of x contains uncountably many singular points of I. Finally, the set of singular points of I is unbounded.     

A four-color theorem for periodic tilings

There exist exactly 4044 topological types of 4-colorable tile-4-transitive tilings of the plane. These can be obtained by systematic application of two geometric algorithms, edge-contraction and vertex-truncation, to all tile-3-transitive tilings of the plane.     

A Helly-type theorem for simple polygons

Let be a family of simple polygons in the plane. If every three (not necessarily distinct) members of have a simply connected union and every two members of have a nonempty intersection, then {P:P in } . Applying the result to a finite family of orthogonally convex polygons, the set {C:C in } will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.Supported in part by NSF grant DMS-9207019.     

A Helly-type theorem for countable intersections of starshaped sets

Let k and d be fixed integers, 0kd, and let be a collection of sets in If every countable subfamily of has a starshaped intersection, then is (nonempty and) starshaped as well. Moreover, if every countable subfamily of has as its intersection a starshaped set whose kernel is at least k-dimensional, then the kernel of is at least k-dimensional, too. Finally, dual statements hold for unions of sets.Received: 3 April 2004     

A Helly theorem for intersections of orthogonally starshaped sets

Let $\cal{F}$ be a finite family of simply connected orthogonal polygons in the plane. If every three (not necessarily distinct) members of $\cal{F}$ have a nonempty intersection which is starshaped via staircase paths, then the intersection $\cap \{F : F\; \hbox{in}\; \cal{F}\}$ is a (nonempty) simply connected orthogonal polygon which is starshaped via staircase paths. Moreover, the number three is best possible, even with the additional requirement that the intersection in question be nonempty. The result fails without the simple connectedness condition.     

Families of planar sets having starlike union

Let be a finite family of compact sets in the plane, and letk be a fixed natural number. If every three (not necessarily distinct) members of have a union which is simply connected and starshaped viak-paths, then and is starshaped viak-paths. Analogous results hold for paths of length at most , > 0, and for staircase paths, although not for staircasek-paths.Supported in part by NSF grant DMS-9504249     

A vector-sum theorem in two-dimensional space

Given a finite setX of vectors from the unit ball of the max norm in the twodimensional space whose sum is zero, it is always possible to writeX = {x₁, , x_n} in such a way that the first coordinates of each partial sum lie in [–1, 1] and the second coordinates lie in [–C, C] whereC is a universal constant.     

An acyclicity theorem for cell complexes ind dimension

LetC be a cell complex ind-dimensional Euclidean space whose faces are obtained by orthogonal projection of the faces of a convex polytope ind+ 1 dimensions. For example, the Delaunay triangulation of a finite point set is such a cell complex. This paper shows that the in_front/behind relation defined for the faces ofC with respect to any fixed viewpointx is acyclic. This result has applications to hidden line/surface removal and other problems in computational geometry.Research reported in this paper was supported by the National Science Foundation under grant CCR-8714565     

Equidissections of centrally symmetric octagons

Summary In 1970 Monsky proved that a square cannot be cut into an odd number of triangles of equal areas. In 1988 Kasimatis proved that if a regularn-gon,n 5, is cut intom triangles of equal areas, thenm is a multiple ofn. These two results imply that a centrally symmetric regular polygon cannot be cut into an odd number of triangles of equal areas. We conjecture that the conclusion holds even if the restriction regular is deleted from the hypothesis and prove that it does forn = 6 andn = 8.     

Locally dense finite lattice packings of spheres

We consider finite packings of unit-balls in Euclidean 3-spaceE ³ where the centres of the balls are the lattice points of a lattice polyhedronP of a given latticeL ³E³. In particular we show that the facets ofP induced by densest sublattices ofL ³ are not too close to the next parallel layers of centres of balls. We further show that the Dirichlet-Voronoi-cells are comparatively small in this direction. The paper was stimulated by the fact that real crystals in general grow slowly in the directions normal to these dense facets.The results support, to some extent, the hypothesis that real crystals grow preferably such that they need little volume, i.e that they are locally dense.Dedicated to A. Florian on the occasion of this 60th birthday     

Fractal Penrose tilings I. Construction and matching rules

Summary Quasiperiodic tilings of kite-and-dart type, widely used as models for quasicrystals with decagonal symmetry, are constructed by means of somewhat artificial matching rules for the tiles. The proof of aperiodicity uses a self-similarity property, or inflation procedure, which requires drawing auxiliary lines. We introduce a modification of the kite-and-dart tilings which comes very naturally with both properties: the tiles are strictly self-similar, and their fractal boundaries provide perfect matching rules.     

Convex bodies with extremal volumes having prescribed brightness in finitely many directions

We consider the class of convex bodies in ⁿ with prescribed projection (n – 1)-volumes along finitely many fixed directions. We prove that in such a class there exists a unique body (up to translation) with maximumn-volume. The maximizer is a centrally symmetric polytope and the normal vectors to its facets depend only on the assigned directions.Conditions for the existence of bodies with minimumn-volume in the class defined above are given. Each minimizer is a polytope, and an upper bound for the number of its facets is established.Work partially supported by Istituto di Analisi Globale e Applicazioni, CNR, Firenze.     

On coverings of plane convex sets by translates of strips

In the euclidean planeE ² letS ₁,S ₂, ... be a sequence of strips of widthsw ₁,w ₂, .... It is shown thatE ² can be covered by translates of the stripsS _i if w ₁ ^3/2 = . Further results concern conditions in order that a compact convex domain inE ² can be covered by translates ofS ₁,S ₂, ....This research was supported by National Science Foundation Research Grant MCS 76-06111.     

Ausfüllungen der hyperbolischen Ebene durch kongruente Hyperzykelbereiche

Ohne Zusammenfassung

---

Dem Andenken von Herrn Prof. P. Szász gewidmet