@MISC{Radin94thepinwheel, author = {Charles Radin}, title = {The Pinwheel Tilings of the Plane}, year = {1994} }

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Abstract

this paper, and many long hours of discussions with Dani Berend. 2 CHARLES RADIN means "disordered" in the probabilistic sense used to study patterns in nonlinear dynamics. Within the field of logic, there was a brief line of development in which "very complicated" means "nonrecursive". All published examples, of finite sets of prototiles which can only tile the plane nonperiodically, have the feature that in every tiling each prototile only appears in finitely many orientations. Therefore, for these examples one could add the requirement that copies of a prototile be not just congruent to the prototile, but congruent by a translation; to recover the tilings of these previous examples one might then need to increase the sets of prototiles to larger but still finite sets, effectively declaring certain rotated or reflected prototiles as new distinct prototiles. In other words, the use of congruence in the production of copies of the prototiles was effectively replaceable by translation in all these examples, at the expense of increasing the number of prototiles to a larger finite number. This is not true of the example of this paper. In this example there is a finite set of prototiles, and in every associated tiling of the plane tiles appear in infinitely many orientations; all the connected part of the Euclidean group is needed to analyze the tilings by this set of prototiles, not just its translation subgroup. This constitutes a major advance in the field, which began with models only requiring discrete translations (using "Wang dominoes"), and expanded to continuous translations with models such as those of Penrose; this introduction of rotations adds a distinctly new element to the subject, of particular interest to certain applications such as the physics of quasic...