Tiling with polyominoes and combinatorial group theory

@article{Conway1990TilingWP,
  title={Tiling with polyominoes and combinatorial group theory},
  author={John H. Conway and Jeffrey C. Lagarias},
  journal={J. Comb. Theory, Ser. A},
  year={1990},
  volume={53},
  pages={183-208}
}
Geometric and algebraic properties of polyomino tilings
In this thesis we study tilings of regions on the square grid by polyominoes. A polyomino is any connected shape formed from a union of grid cells, and a tiling of a region is a collection of
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Spaces of domino tilings
TLDR
A criterion to decide if two tilings are in the same connected component, a simple formula for distances, and a method to construct geodesics in this graph, which is a CW-complex whose connected components are homotopically equivalent to points or circles.
Tilings with trichromatic colored-edges triangles
Domino Tilings of the Torus
We consider the problem of counting and classifying domino tilings of a quadriculated torus. The counting problem for rectangles was studied by Kasteleyn and we use many of his ideas. Domino tilings
Hard and Easy Instances of L-Tromino Tilings
TLDR
This work studies tilings of regions in the square lattice with L-shaped trominoes and shows that deciding the existence of a tiling remains NP-complete; yet, if a region does not contain so-called “forbidden polyominoes” as subregions, then there exists a polynomial time algorithm.
Computational complexity and decidability of tileability
For finite polyomino regions, tileability by a pair of rectangles is NP-complete for all but trivial cases yet can be solved in quadratic time for simply connected regions. Through a series of
Conway's tiling groups
John Conway discovered a technique using infinite, finitely presented groups that in a number of interesting cases resolves the question of whether a region in the plane can be tessellated by given
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References

SHOWING 1-10 OF 45 REFERENCES
Undecidability and nonperiodicity for tilings of the plane
This paper is related to the work of Hao Wang and others growing out of a problem which he proposed in [8], w 4.1. Suppose that we are given a finite set of unit squares with colored edges, placed
Conway's tiling groups
John Conway discovered a technique using infinite, finitely presented groups that in a number of interesting cases resolves the question of whether a region in the plane can be tessellated by given
Combinatorial Group Theory
Chapter I. Free Groups and Their Subgroups 1. Introduction 2. Nielsen's Method 3. Subgroups of Free Groups 4. Automorphisms of Free Groups 5. Stabilizers in Aut(F) 6. Equations over Groups 7.
Polynomials and Polyominoes
J. B. Kelly † American Mathematical Monthly 73 (1966) 464–471 1. The associated polynomial. Let S be a finite set of lattice points (i.e. points with integral coordinates) in k-dimensional Euclidean
Computers and Intractability: A Guide to the Theory of NP-Completeness
Horn formulae play a prominent role in artificial intelligence and logic programming. In this paper we investigate the problem of optimal compression of propositional Horn production rule knowledge
Generators and relations for discrete groups
1. Cyclic, Dicyclic and Metacyclic Groups.- 2. Systematic Enumeration of Cosets.- 3. Graphs, Maps and Cayley Diagrams.- 4. Abstract Crystallography.- 5. Hyperbolic Tessellations and Fundamental
Topology . Elements of the topology of plane sets of points
This book provides an elementary introduction to the ideas and methods of topology by the detailed study of certain topics. There are elegant but rigorous proofs of many of the basic theorems, and
A packing theory
...
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4
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