## A sponge-like (almost) universal tile (2001)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Bailey01asponge-like,

author = {Duane A. Bailey and Feng Zhu},

title = {A sponge-like (almost) universal tile},

year = {2001}

}

### OpenURL

### Abstract

Abstract. We present an technique for developing a single, aperiodic (or universal) tile that does not overlap. We provide two examples, constructed by converting overlapping regions of Gummelt’s decagon cover to interleaved regions in a porous tile. Each is a bounded, dense, sponge-like set of points in 2 that tiles the plane aperiodically. One tile has measure zero, the other has positive measure everywhere. Many characteristics of the decagon cover are inherited by our tilings. We also discuss how to arbitrarily adjust density of portions of the tile to, for example, support models of physical quasicrystals. A similar approach could be used to eliminate overlap in higher dimensions. 1

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Citation Context ... to, for example, support models of physical quasicrystals. A similar approach could be used to eliminate overlap in higher dimensions. 1 Introduction Since the advent of the first aperiodic tile sets=-=[3, 25]-=- researchers have sought a minimal prototile set that forces a nonperiodic tiling of the plane. A number of aperiodic sets have been found, although the two-tile sets of Penrose are arguably the most ... |

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Citation Context ...chers have sought a minimal prototile set that forces a nonperiodic tiling of the plane. A number of aperiodic sets have been found, although the two-tile sets of Penrose are arguably the most studied=-=[4, 5, 6, 9, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24]-=-. The “A tiles” of Robinson are constructed by cutting Penrose’s kites and darts along their medial lines; their manipulation is often more convenient where inflation and deflation is to be studied. S... |

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Citation Context ...town, Massachusetts 01267. 2 feng@hbs.edu, Program in Information Technology and Management, Harvard Business School, Boston, Massachusetts 02163.sA proof of this can be found in Grünbaum and Shephard=-=[1, 10]-=-. It stems from the observation that every tile in the plane is a member of an ace (a dart capping two adjacent kites). The Penrose tiles enjoy a unique and invertible inflation property that allows o... |

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Citation Context ...s that violate one or more rules of prototile construction) perfectly tile the plane seems likely to shed light on the nature of aperiodicity and its analogy with physical systems[12]. In 1996 Gummelt=-=[11]-=- identified a single decagon that covered the plane aperiodically. The covering process involved overlapping tiles in a small number of ways. The construction, though not a proper tiling, is appealing... |

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Citation Context ...ditional tiles (shapes that violate one or more rules of prototile construction) perfectly tile the plane seems likely to shed light on the nature of aperiodicity and its analogy with physical systems=-=[12]-=-. In 1996 Gummelt[11] identified a single decagon that covered the plane aperiodically. The covering process involved overlapping tiles in a small number of ways. The construction, though not a proper... |

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Citation Context ...n approach essentially encodes the behavior of the cartwheel cover in a single overlapping tile, each tiling has properties that are not readily apparent in the other. Later, in 1997 Bandt and Gummelt=-=[2]-=- and Gelbrich[7] demonstrated that set of Penrose tiles modified to include fractal edges was sufficient to remove the local matching condition. The authors suggested, as well, that in the particular ... |

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Citation Context ...tially encodes the behavior of the cartwheel cover in a single overlapping tile, each tiling has properties that are not readily apparent in the other. Later, in 1997 Bandt and Gummelt[2] and Gelbrich=-=[7]-=- demonstrated that set of Penrose tiles modified to include fractal edges was sufficient to remove the local matching condition. The authors suggested, as well, that in the particular case of the Penr... |

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Citation Context ...darts along their medial lines; their manipulation is often more convenient where inflation and deflation is to be studied. Still, no single aperiodic tile (here, a universal tile) has been discovered=-=[8]-=-. Our premise is that the study of how single nontraditional tiles (shapes that violate one or more rules of prototile construction) perfectly tile the plane seems likely to shed light on the nature o... |

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Citation Context ...ositions. We also demonstrate how to adjust the relative densities of tile regions, perhaps to more accurately model the influence of physical quasicrystal systems. Recent work by Jeong and Steinhardt=-=[13]-=- has studied the stoichiometry of the unit cells of quasicrystals with 10-fold long-range symmetry, and is greatly facilitated by the decagon tiling and its offshoots. Their techniques compute precise... |

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1 | The search for a universal tile - Zhu - 2002 |