Çifte Minareli Medrese, Sivas/Turkey
interlocking dodecagons from caglar on Vimeo.
Setting up the rectangular grid, showing where the ratio Sx/Sy = 1/sqrt(3) comes from
If you want the dodecagons meet in their sides mid-points as in the original pattern , the proportions are as follows, where r is the radius of dodecagonal circumcircle and 2a is the size of one side. From the equation for the size of the inradius of a polygon;
inradius = r*cos(π/n)
sqrt(r²-a²) = r*cos(π/12)
a = sqrt(r-(r*cos15))
We found the relationship between the side length and the radius of a dodecagon
We now need to define a grid in form of r and a. We know if we want the dodecagons to meet up in their sides midpoints distance between their centers would be r+a*sqrt(2)
This equals to the horizontal spacing of the grid (2x in the first picture).