In this paper, we study the  self-assembly of rectangles in a non-cooperative, 3D version of Winfree's abstract Tile Assembly Model . We prove four main results -- the first two negative and the last two positive. First, we use Catalan numbers and a restricted version of the Window Movie Lemma (Meunier, Patitz, Summers, Theyssier, Winslow and Woods, SODA 2014) to prove two new lower bounds on the minimum number of unique tile types required for the self-assembly of rectangles. We then give a general construction for the efficient self-assembly of thin rectangles. Our construction is non-cooperative and " just-barely " 3D in the sense that it places tiles at most one step into the third dimension. Finally, we give a non-cooperative, just-barely 3D optimal encoding construction that self-assembles the bits of a given binary string along the perimeter of a thin rectangleof constant height .
In this paper, we study the minimum number of unique tile types required for the self-assembly of thin rectangles in Winfree's abstract Tile Assembly Model (aTAM), restricted to temperature-1. Using Catalan numbers, planar self-assembly and a restricted version of the Window Movie Lemma , we derive a new lower bound on the tile complexity of thin rectangles at temperature-1 in 2D. Then, we give the first known upper bound on the tile complexity of `` just-barely '' 3D thin rectangles at temperature-1, where tiles are allowed to be placed at most one step into the third dimension. Our construction, which produces a unique terminal assembly, implements a just-barely 3D , zig-zag counter, whose base depends on the dimensions of the target rectangle, and whose digits are encoded geometrically, vertically-oriented and in binary .
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minimum number of unique tile types required for the
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rectangles in a non-cooperative, 3D version of
thin rectangles in
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. We prove four main results -- the first two negative and the last two positive. First, we use Catalan numbers
(aTAM), restricted to temperature-1. Using Catalan numbers, planar self-assembly
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(Meunier, Patitz, Summers, Theyssier, Winslow and Woods, SODA 2014) to prove two new lower bounds on the minimum number of unique tile types required for the self-assembly of rectangles. We then give a general construction for the efficient self-assembly of thin rectangles. Our construction is non-cooperative and "
, we derive a new lower bound on the tile complexity of thin rectangles at temperature-1 in 2D. Then, we give the first known upper bound on the tile complexity of ``
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"
''
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in the sense that it places tiles
thin rectangles at temperature-1, where tiles are allowed to be placed
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Finally, we give a non-cooperative,
Our construction, which produces a unique terminal assembly, implements a
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optimal encoding construction that self-assembles the bits of a given binary string along the perimeter of a thin rectangleof constant height
, zig-zag counter, whose base depends on the dimensions of the target rectangle, and whose digits are encoded geometrically, vertically-oriented and in binary