More ways to tile with only one shape polygon

@article{Socolar2007MoreWT,
  title={More ways to tile with only one shape polygon},
  author={Joshua E. S. Socolar},
  journal={The Mathematical Intelligencer},
  year={2007},
  volume={29},
  pages={33-38}
}
  • J. Socolar
  • Published 1 March 2007
  • Mathematics
  • The Mathematical Intelligencer
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 desired by increasing the aspect ratio of the monotile. Aside from illustrating some elegant peculiarities of the hexagonal parquet tiling, the constructions demonstrate three points.1.Monotiles with arbitrarily large isohedral number do exist;2.The additional topological possibilities afforded in 3D… 

An aperiodic hexagonal tile

Construction operations to create new aperiodic tilings : local isomorphism classes and simplified matching rules

Title: ‘Construction operations to create new aperiodic tilings: local isomorphism classes and simplified matching rules’ Author: David Fletcher This thesis studies several constructions to produce

Aperiodic Tilings with One Prototile and Low Complexity Atlas Matching Rules

  • D. Fletcher
  • Computer Science, Mathematics
    Discret. Comput. Geom.
  • 2011
A constructive method is given that can decrease the number of prototiles needed to tile a space by exchanging edge-to-edge matching rules for a small atlas of permitted patches.

Forcing Nonperiodicity with a Single Tile

Some features of a prototile the authors recently introduced that is an einstein according to a reasonable definition are reviewed and clarified.

Asymptotical Unboundedness of the Heesch Number in $${\mathbb {E}}^d$$ E d for

It is shown that, if darrow d → ∞ , then there is no uniform upper bound on the set of all possible finite Heesch numbers in the space E d, and given any nonnegative integer n, the authors can find a dimension d in which there exists a hypersolid whose Heesh number is finite and greater than  n.

Mapping the aperiodic landscape, 1982–2007

The discovery of quasicrystals galvanized mathematics research in long-range aperiodic order, accelerating the dissolution of the the ancient periodic/non-periodic dichotomy begun by Penrose, Ammann,

References

SHOWING 1-10 OF 14 REFERENCES

Simple octagonal and dodecagonal quasicrystals.

  • Socolar
  • Physics
    Physical review. B, Condensed matter
  • 1989
Penrose tilings have become the canonical model for quasicrystal structure, primarily because of their simplicity in comparison with other decagonally symmetric quasiperiodic tilings of the plane.

A simpler approach to Penrose tiling with implications for quasicrystal formation

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

Penrose tilings as coverings of congruent decagons

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

SCD patterns have singular diffraction

Among the many families of nonperiodic tilings known so far, SCD tilings are still a bit mysterious. Here, we determine the diffraction spectra of point sets derived from SCD tilings and show that

New Conjectural Lower Bounds on the Optimal Density of Sphere Packings

An optimization procedure is precisely the dual of a primal linear program devised by Cohn and Elkies to obtain upper bounds on the density, and hence has implications for linear programming bounds, and proves that the density estimate can never exceed the Cohn– Elkies upper bound.

Weak matching rules for quasicrystals

Weak matching rules for a quasicrystalline tiling are local rules that ensure that fluctuations in “perp-space” are uniformly bounded. It is shown here that weak matching rules exist forN-fold

A strongly aperiodic set of tiles in the hyperbolic plane

We construct the first known example of a strongly aperiodic set of tiles in the hyperbolic plane. Such a set of tiles does admit a tiling, but admits no tiling with an infinite cyclic symmetry. This

Tilings and Patterns

"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

Algebraic theory of Penrose''s non-periodic tilings

Quasicrystals: a new class of ordered structures

A quasicrystal is the natural extension of the notion of a crystal to structures with quasiperiodic, rather than periodic, translational order. We classify two- and three-dimensional quasicrystals by