What size rectangle, if tessellated in the bricklayer's pattern will have the centers of the rectangles coincident with a regular hexagon tiling?

Answer: Rectangle of width 1 and height sqrt(3)/2. An equilateral triangle fits just inside it. Possibly useful is to split the rectangle in half vertically, then have a simple grid of them colored two (or seven) different colors, with pairs of horizontally adjacent rectangles the same color denoting one "hexagon".

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