Difference between revisions of "Open Problems"

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<li>In [[Self-Assembly of Discrete Self-Similar Fractals]], Patitz and Summers proved several results about self-assembling discrete self-similar fractals, but it is still an open questions as to whether or not there exists a discrete self-similar fractal that can be self-assembled in the [[Abstract Tile Assembly Model (aTAM) | aTAM]].  Also, are there discrete self-similar fractals which are not pinch-point fractals that are provably impossible to strictly self-assemble (e.g. the Sierpinski carpet)?</li>
 
<li>In [[Self-Assembly of Discrete Self-Similar Fractals]], Patitz and Summers proved several results about self-assembling discrete self-similar fractals, but it is still an open questions as to whether or not there exists a discrete self-similar fractal that can be self-assembled in the [[Abstract Tile Assembly Model (aTAM) | aTAM]].  Also, are there discrete self-similar fractals which are not pinch-point fractals that are provably impossible to strictly self-assemble (e.g. the Sierpinski carpet)?</li>
  
<li>In [[Self-Assembly_with_Geometric_Tiles]], a construction was shown in which 2D tiles with disconnected geometries which are forced to stay within the plane as they combine, are capable of assembling $n \times n$ squares using only $O(\log(\log(n)))$ tile types.  Can a similar construction be shown with connected geometries (and staying in 2D)?</li>
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<li>In [[Self-Assembly with Geometric Tiles]], a construction was shown in which 2D tiles with disconnected geometries which are forced to stay within the plane as they combine, are capable of assembling $n \times n$ squares using only $O(\log(\log(n)))$ tile types.  Can a similar construction be shown with connected geometries (and staying in 2D)?</li>
  
<li>In [[Computability_and_Complexity_in_Self-Assembly]], it was shown that for every computably enumerable language $L \subset \mathbb{N}$, a pattern representing $L$ [[Weak_Self-Assembly | weakly self-assembles]] along the x-axis, but with the points spread out roughly quadratically.  Can those points instead be spread out by only a constant factor? (Or with no space between them as with decidable languages as shown in [[Self-Assembly_of_Decidable_Sets]]?)</li>
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<li>In [[Computability and Complexity in Self-Assembly]], it was shown that for every computably enumerable language $L \subset \mathbb{N}$, a pattern representing $L$ [[Weak_Self-Assembly | weakly self-assembles]] along the x-axis, but with the points spread out roughly quadratically.  Can those points instead be spread out by only a constant factor? (Or with no space between them as with decidable languages as shown in [[Self-Assembly of Decidable Sets]]?)</li>
  
 
==References==
 
==References==

Revision as of 19:20, 27 May 2014

The following are a list of open problems in self-assembly: