Some Generalizations of the Pinwheel Tiling

@article{Sadun1998SomeGO,
  title={Some Generalizations of the Pinwheel Tiling},
  author={Lorenzo A Sadun},
  journal={Discrete \& Computational Geometry},
  year={1998},
  volume={20},
  pages={79-110}
}
  • L. Sadun
  • Published 24 December 1997
  • Mathematics
  • Discrete & Computational Geometry
Abstract. We introduce a new family of nonperiodic tilings, based on a substitution rule that generalizes the pinwheel tiling of Conway and Radin. In each tiling the tiles are similar to a single triangular prototile. In a countable number of cases, the tiles appear in a finite number of sizes and an infinite number of orientations. These tilings generally do not meet full-edge to full-edge, but can be forced through local matching rules. In a countable number of cases, the tiles appear in a… 
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References

SHOWING 1-9 OF 9 REFERENCES
The pinwheel tilings of the plane
"prototiles") in Euclidean n-space, En, for n > 2. The prototiles are usually required to be rather nice topologically, at least homeomorphs of the closed unit ball. One then makes arbitrarily many
MATCHING RULES AND SUBSTITUTION TILINGS
A substitution tiling is a certain globally de ned hierarchical structure in a geometric space; we show that for any substitution tiling in En, n > 1, subject to relatively mild conditions, one can
Space tilings and substitutions
We generalize the study of symbolic dynamical systems of finite type and ℤ2 action, and the associated use of symbolic substitution dynamical systems, to dynamical systems with ℝ2 action. The new
What is ergodic theory
Ergodic theory involves the study of transformations on measure spaces. Interchanging the words “measurable function” and “probability density function” translates many results from real analysis to
Symmetry and Tilings, Notices of the
  • AMS
  • 1995
It is a pleasure to thank Chaim Goodman-Strauss, Yoram Last, Tom Mrowka, Johan Råde, Charles Radin and Felipe Voloch for useful discussions , and Steven Janowsky for assistance with computer graphics
  • It is a pleasure to thank Chaim Goodman-Strauss, Yoram Last, Tom Mrowka, Johan Råde, Charles Radin and Felipe Voloch for useful discussions , and Steven Janowsky for assistance with computer graphics
The tilings with rational z all satisfy the hypotheses of Goodman-Strauss's theorem, implying that they can be forced through local matching rules
  • The tilings with rational z all satisfy the hypotheses of Goodman-Strauss's theorem, implying that they can be forced through local matching rules
distribution of size and orientation suggests that the tiling has a purely absolutely continuous spectrum
  • distribution of size and orientation suggests that the tiling has a purely absolutely continuous spectrum