# Forcing Nonperiodicity with a Single Tile

@article{Socolar2010ForcingN, title={Forcing Nonperiodicity with a Single Tile}, author={Joshua E. S. Socolar and Joan M. Taylor}, journal={The Mathematical Intelligencer}, year={2010}, volume={34}, pages={18-28} }

An aperiodic prototile is a shape for which infinitely many copies can be arranged to fill Euclidean space completely with no overlaps, but not in a periodic pattern. Tiling theorists refer to such a prototile as an "einstein" (a German pun on "one stone"). The possible existence of an einstein has been pondered ever since Berger's discovery of large set of prototiles that in combination can tile the plane only in a nonperiodic way. In this article we review and clarify some features of a… Expand

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#### References

SHOWING 1-10 OF 31 REFERENCES

An aperiodic hexagonal tile

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 2011

A two-dimensional, hexagonal prototile with markings that enforce local matching rules is proven to be aperiodic by two independent methods. Expand

Hexagonal parquet tilings: k-isohedral monotiles with arbitrarily large k

- Physics, Mathematics
- 2007

This paper addresses the question of whether a single tile with nearest neighbor matching rules can force a tiling in which the tiles fall into a large number of isohedral classes. A single tile is… Expand

A simpler approach to Penrose tiling with implications for quasicrystal formation

- Physics
- Nature
- 1996

QUASICRYSTALS1have a quasiperiodic atomic structure with symmetries (such as fivefold) that are forbidden to ordinary crystals2,3. Why do atoms form this complex pattern rather than a regularly… Expand

A Small Aperiodic Set of Planar Tiles

- Computer Science, Mathematics
- Eur. J. Comb.
- 1999

A simple set of two tiles that can only tile aperiodically is given that is invariant under any infinite cyclic group of isometries and among the smallest sets ever found. Expand

Undecidability and nonperiodicity for tilings of the plane

- Mathematics
- 1971

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… Expand

The sphinx: a limit-periodic tiling of the plane

- Physics
- 1989

The sphinx is a non-periodic tiling of the plane made of one type of tile. In order to characterise the type of order found in this tiling, the computation of its diffraction spectrum is considered.… Expand

Penrose tilings as coverings of congruent decagons

- Mathematics
- 1996

The open problem of tiling theory whether there is a single aperiodic two-dimensional prototile with corresponding matching rules, is answered for coverings instead of tilings. We introduce… Expand

More ways to tile with only one shape polygon

- Mathematics
- 2007

ConclusionI have exhibited several types of monotiles with matching rules that force the construction of a hexagonal parquet. The isohedral number of the resulting tiling can be made as large as… Expand

The Mathematics of Long-Range Aperiodic Order

- Mathematics
- 1997

Preface. Knotted Tilings C.C. Adams. Solution of the Coincidence Problem in Dimensions d smaller than or equal to 4 M. Baake. Self-Similar Tilings and Patterns Described by Mappings C. Bandt. Delone… Expand

Tilings and Patterns

- Mathematics
- 1990

"Remarkable...It will surely remain the unique reference in this area for many years to come." Roger Penrose , Nature "...an outstanding achievement in mathematical education." Bulletin of The London… Expand