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What is the minimal possible value of the maximal total side length shared by any two tiles in a tiling of the plane if all tiles have the same area $A$?

$\text{Total side length} = \text{Length-integral of the curve formed by the intersection of two tiles}$

i) Using a finite set of tiles
ii) Using any set of tiles.

This is known for $3D$ as http://mathworld.wolfram.com/KelvinsConjecture.html
I have found no info on the planar case, so it may be trivial, in which case I want to see the proof.

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    I fail to see how this has anything to do with elementary set theory...2011-11-10
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    My guess: hexagonal tiling.2011-11-13

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