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.
0 if each of the tiles is the fractal made by a hexagon with triangles on triangles on triangles which stick out on three sides and stick in on the other 3, like in the picture below. ![tile][1] [1]: https://i.stack.imgur.com/S1sW5.png
If by maximal sidelength you mean the maximum total sidelength rather than the maximum of any one side, then I suspect that a hexagonal tiling is the optimum, which has maximal side length $\sqrt{\frac{2A}{3\sqrt{3}}}$.