# Perfectly Repeating Square Grid on a Hexagonal Tile Base

I am attempting to discover what the sizing ratios of squares and hexagons are. What I want to do with this information is determine (if possible) what size squares in a grid I should place over a hexagonal tiled floor so that the grid repeats correctly per tile with each hex tile having an identical segment of the square grid pattern imprinted on it. Essentially, the hexagonal tiles should be identical (including whatever portion of square grid), but when put together should display a complete square grid that would be imprinted onto the tiles.

• The symmetry of a hexagonal tiling is not commensurate with the symmetry of a square tiling, so this is not possible. You would need more than one kind of pattern on the hexagonal tiles (two should do I think). – Dan Rust Oct 20 '18 at 0:09
• It can be done if you're willing to use irregular hexagons. I found a pattern using hexagons with angles of $90, 135, 135, 90, 135, 135$ and sides of length $1$ and $\sqrt2$. – Steve B Oct 20 '18 at 7:30
• 2 patterned hexagons will create a grid of $1.5$ x $\sqrt3$ rectangles. $\sqrt3$ is about $1.73$ – Steve B Oct 20 '18 at 8:14
• Sorry I should mentioned that I meant regular hexagon when I said hexagon. Obviously you can create a hexagonal tiling from a square tiling with the same translational symmetries by just cutting and pasting the requisite notches. Somehow I think the the OP meant to restrict to regular hexagons though. – Dan Rust Oct 20 '18 at 12:28