2
$\begingroup$

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
    My guess: hexagonal tiling.2011-11-13

1 Answers 1

4

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 $\sqrt{\frac{2A}{3\sqrt{3}}}$.

  • 0
    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.2011-11-16