More ways to tile with only one shape polygon

  title={More ways to tile with only one shape polygon},
  author={Joshua E. S. Socolar},
  journal={The Mathematical Intelligencer},
  • 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

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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.

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

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