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### On the structure of multiple translational tilings by polygonal regions [article]

Mihail N. Kolountzakis
1999 arXiv   pre-print
We study the problem of when such a polygon K tiles the plane multiply when translated at the locations Λ, where Λ is a multiset in the plane.  ...  After establishing a necessary and sufficient condition for K to tile with a given lattice Λ (which was first found by Bolle for the case of convex polygons-notice that all convex polygons that tile, necessarily  ...  Introduction In this paper we study multiple tilings of the plane by translates of a polygonal region of a certain type, the polygons with the pairing property of Definition 2 below.  ...

### On the Structure of Multiple Translational Tilings by Polygonal Regions

M. N. Kolountzakis
2000 Discrete & Computational Geometry
We study the problem of when such a polygon K tiles multiply the plane when translated at the locations , where is a multiset in the plane.  ...  We consider polygons with the following "pairing property": for each edge of the polygon there is precisely one other edge parallel to it.  ...  Rewrite then On the Structure of Multiple Translational Tilings by Polygonal Regions 547 Proof of Theorem 3.  ...

### Periodic structure of translational multi-tilings in the plane [article]

Bochen Liu
2019 arXiv   pre-print
Our second result is a new characterization of convex polygons that multi-tile the plane by translations.  ...  As a corollary, together with a result of Kolountzakis, it implies that any convex polygon that multi-tiles the plane by translations admits a lattice multi-tiling, of a possibly different multiplicity  ...  Besides the structure of P , one can also study the structure of the discrete multi-set Λ such that P + Λ multi-tiles. In 2000, Kolountzakis ([13] ) proved the following result in the plane.  ...

### Forcing Nonperiodic Tilings with One Tile Using a Seed [article]

Bernhard Klaassen
2021 arXiv   pre-print
In this paper a class of spiral tilings is linked to a weaker form of the einstein problem where one or several seed tiles are used.  ...  So far, researchers come only close to this demand by defining decorated prototiles forcing nonperiodicity of any generated tiling using matching rules.  ...  Let p be a point where one of the marked arc begins. Translation by vector τ yields point p .  ...

### An integer representation for periodic tilings of the plane by regular polygons

José Ezequiel Soto Sánchez, Tim Weyrich, Asla Medeiros e Sá, Luiz Henrique de Figueiredo
2021 Computers & graphics
We describe a representation for periodic tilings of the plane by regular polygons.  ...  Our approach is to represent explicitly a small subset of seed vertices from which we systematically generate all elements of the tiling by translations.  ...  This research was done in the Visgraf Computer Graphics laboratory at IMPA, at FGV EMAp, and at the Department of Computer Science, University College London.  ...

### Page 4046 of Mathematical Reviews Vol. , Issue 2001F [page]

2001 Mathematical Reviews
The authors consider the problem of tiling an L-shape (six-sided polygon) with squares. They classify the regions that can be tiled using two types of tiles: squares and anti-squares.  ...  An anti-square is a tile with a negative value instead of a positive one so that when it is placed over an ordinary tile we get cancellation in the region of overlap.  ...

### Can You Pave the Plane with Identical Tiles?

Chuanming Zong
2020 Notices of the American Mathematical Society
Meanwhile, Qi Yang and Chuanming Zong made a series of unexpected discoveries in multiple tilings in the Euclidean plane.  ...  To find the list and to show the completeness of the list is a unique drama in mathematics which has lasted for more than one century, and the completeness of the list has been mistakenly announced more  ...  This work was supported by the National Natural Science Foundation of China (NSFC11921001) and the National Key Research and Development Program of China (2018YFA0704701).  ...

Ablimit Aji, Fusheng Wang, Hoang Vo, Rubao Lee, Qiaoling Liu, Xiaodong Zhang, Joel Saltz
2013 Proceedings of the VLDB Endowment
Hadoop-GIS supports multiple types of spatial queries on MapReduce through spatial partitioning, customizable spatial query engine RESQUE, implicit parallel spatial query execution on MapReduce, and effective  ...  Our experiments have demonstrated the high efficiency of Hadoop-GIS on query response and high scalability to run on commodity clusters.  ...  For example, spatial join on tiles with multiple assignment based partitioning can be corrected by eliminating duplicated object pairs from the query result.  ...

### Hadoop-GIS: A High Performance Spatial Data Warehousing System over MapReduce

Ablimit Aji, Fusheng Wang, Hoang Vo, Rubao Lee, Qiaoling Liu, Xiaodong Zhang, Joel Saltz
2013 Proceedings of the VLDB Endowment
Hadoop-GIS supports multiple types of spatial queries on MapReduce through spatial partitioning, customizable spatial query engine RESQUE, implicit parallel spatial query execution on MapReduce, and effective  ...  Our experiments have demonstrated the high efficiency of Hadoop-GIS on query response and high scalability to run on commodity clusters.  ...  Acknowledgments This work is supported in part by PHS Grant UL1RR025008 from the CTSA program, R24HL085343 from the NHLBI, by Grant Number R01LM009239 from the NLM, by NCI Contract No.  ...

### Geometric segmentation of perspective images based on symmetry groups

Yang, Shankar Rao, Kun Huang, Wei Hong, Yi Ma
2003 Proceedings Ninth IEEE International Conference on Computer Vision
The result of such a segmentation is a hierarchy of geometric primitives, called symmetry cells and complexes, whose 3-D structure and pose are fully recovered.  ...  Based on three fundamental principles that summarize the relations between symmetry and perspective imaging, namely, structure from symmetry, symmetry hypothesis testing, and global symmetry testing, we  ...  these identified regions (e.g., as congruent tiles on the same 3-D plane etc.).  ...

### Structure results for multiple tilings in 3D [article]

Nick Gravin, Mihail Kolountzakis, Sinai Robins, Dmitry Shiryaev
2012 arXiv   pre-print
We study multiple tilings of 3-dimensional Euclidean space by a convex body.  ...  In a multiple tiling, a convex body P is translated with a discrete multiset Λ in such a way that each point of the space gets covered exactly k times, except perhaps the translated copies of the boundary  ...  There is a vast literature on the study of coverings of Euclidean space by a convex body, and an equally vast body of work on classical tilings by translations of one convex body, which must necessarily  ...

### Tilings, tiling spaces and topology

2006 Philosophical Magazine
To understand an aperiodic tiling (or a quasicrystal modeled on an aperiodic tiling), we construct a space of similar tilings, on which the group of translations acts naturally.  ...  The topology of the space of tilings, particularly the Cech cohomology, gives information on how the original tiling can be deformed.  ...  The pullback map σ * n is multiplication by 1 on H 0 and multiplication by 2 on H 1 , so the dyadic solenoid has H 0 = Z, H 1 = Z[1/2].  ...

### Veech groups and polygonal coverings

Pascal Hubert, Thomas A. Schmidt
2000 Journal of Geometry and Physics
Introduction One of the most accessible problems in physics and mathematics would seem to be that of the dynamics of a particle elastically re ected by the walls of a euclidean polygon.  ...  In particular, this allows us to show that even if one polygon tiles another, the respective V eech groups are not necessarily commensurable.  ...  Let Q be a polygon tiled by the polygon P and let MQ and MP be their associated surfaces.  ...

### From 2D hyperbolic forests to 3D Euclidean entangled thickets

S.T. Hyde, C. Oguey
2000 European Physical Journal B : Condensed Matter Physics
structure  ...  The generated graphs are compared with some organo-metallic molecular crystals with multiple frameworks and molecular mesophases found in copolymer melts.  ...  Any part of the forest, lying on one side of a bounding geodesic, can be translated relative the rest (on the other side of the geodesic) by an arbitrary translation along this geodesic.  ...

### Escherization

Craig S. Kaplan, David H. Salesin
2000 Proceedings of the 27th annual conference on Computer graphics and interactive techniques - SIGGRAPH '00
Our solution works by using a simulated annealer to optimize over a parameterization of the "isohedral" tilings, a class of tilings that is flexible enough to encompass nearly all of Escher's own tilings  ...  We demonstrate the use of these tools-along with several additional techniques for adding decorations to tilings-with a variety of original ornamental designs.  ...  Branko Grünbaum motivated the use of isohedral tilings as a powerful and manageable system for ornamental design. Douglas Zongker sat in on many early discussions.  ...
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