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In this paper we explore the power of tile self-assembly models that extendthe well-studied abstract Tile Assembly Model (aTAM) by permitting tiles ofshapes beyond unit squares. Our main result shows the surprising fact that anyaTAM system, consisting of many different tile types, can be simulated by asingle tile type of a general shape. As a consequence, we obtain a singleuniversal tile type of a single (constant-size) shape that serves as a"universal tile machine": the single universal tile type can simulate anydesired aTAM system when given a single seed assembly that encodes the desiredaTAM system. We also show how to adapt this result to convert any of a varietyof plane tiling systems (such as Wang tiles) into a "nearly" plane tilingsystem with a single tile (but with small gaps between the tiles). All of theseresults rely on the ability to both rotate and translate tiles; by contrast, weshow that a single nonrotatable tile, of arbitrary shape, can produceassemblies which either grow infinitely or cannot grow at all, implyingdrastically limited computational power.  On the positive side, we show how to simulate arbitrary cellular automata fora limited number of steps using a single nonrotatable tile and a linear-sizeseed assembly.

One Tile to Rule Them All: Simulating Any Turing Machine, Tile Assembly System, or Tiling System with a Single Puzzle Piece