Optimal packing of a tile without rotation or reflection

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Let's say we have a shape. We will call this shape $A$ and we will say that $A$ is some finite subset of the regular square tiling. Similar to a polyomino, except we do not require that $A$ has a connected interior.



Now I would like to a the optimal way to pack copies of $A$ on the plane without rotation or reflection. For example if $A$ were:



Shape



one optimal packing would be:



Tiling



another would be:



Tiling 2



despite the fact that if we were allowed to rotate or reflect $A$ we would be able to tile the plane perfectly.



I've been trying to come up with an algorithm to find an optimal packing for arbitrary $A$, but I am rather stumped and looking through literature I was not able to find any work on this problem.




geometry tiling packing-problem




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    up vote
    2
    down vote

    favorite












    Let's say we have a shape. We will call this shape $A$ and we will say that $A$ is some finite subset of the regular square tiling. Similar to a polyomino, except we do not require that $A$ has a connected interior.



    Now I would like to a the optimal way to pack copies of $A$ on the plane without rotation or reflection. For example if $A$ were:



    Shape



    one optimal packing would be:



    Tiling



    another would be:



    Tiling 2



    despite the fact that if we were allowed to rotate or reflect $A$ we would be able to tile the plane perfectly.



    I've been trying to come up with an algorithm to find an optimal packing for arbitrary $A$, but I am rather stumped and looking through literature I was not able to find any work on this problem.




    geometry tiling packing-problem




    share|cite|improve this question
























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      Let's say we have a shape. We will call this shape $A$ and we will say that $A$ is some finite subset of the regular square tiling. Similar to a polyomino, except we do not require that $A$ has a connected interior.



      Now I would like to a the optimal way to pack copies of $A$ on the plane without rotation or reflection. For example if $A$ were:



      Shape



      one optimal packing would be:



      Tiling



      another would be:



      Tiling 2



      despite the fact that if we were allowed to rotate or reflect $A$ we would be able to tile the plane perfectly.



      I've been trying to come up with an algorithm to find an optimal packing for arbitrary $A$, but I am rather stumped and looking through literature I was not able to find any work on this problem.




      geometry tiling packing-problem




      share|cite|improve this question














      Let's say we have a shape. We will call this shape $A$ and we will say that $A$ is some finite subset of the regular square tiling. Similar to a polyomino, except we do not require that $A$ has a connected interior.



      Now I would like to a the optimal way to pack copies of $A$ on the plane without rotation or reflection. For example if $A$ were:



      Shape



      one optimal packing would be:



      Tiling



      another would be:



      Tiling 2



      despite the fact that if we were allowed to rotate or reflect $A$ we would be able to tile the plane perfectly.



      I've been trying to come up with an algorithm to find an optimal packing for arbitrary $A$, but I am rather stumped and looking through literature I was not able to find any work on this problem.





      geometry tiling packing-problem





      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Aug 17 at 3:32

























      asked Aug 17 at 1:50









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