A Sponge-Like (Almost) Universal Tile
نویسندگان
چکیده
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.
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