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

total sidelength = 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... –  Asaf Karagila Nov 10 '11 at 18:29
    
My guess: hexagonal tiling. –  Tim Seguine Nov 13 '11 at 15:09

1 Answer 1

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]: http://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 $\sqrt{\frac{2A}{3\sqrt{3}}}$.

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I think you got the maximal side length wrong. If the side of the hexagon is $s$, the area is $A = 3\sqrt{3}s^2/2$, so the maximal sidelength is $\sqrt{2A/3\sqrt{3}}$. –  Craig Nov 15 '11 at 3:57
    
Thanks craig for pointing that out. The boundary between any two tiles must be a straight line, the shortest length connecting the two endpoints of the boundary curve. –  Angela Richardson Nov 16 '11 at 17:26

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