Interlocking Dodecagons

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)

Hence,

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)

Which is,

r+(sqrt(r-(r*cos15))*sqrt(2))

This equals to the horizontal spacing of the grid (2x in the first picture).

ghx file

interlocking dodecagons.ghx

Hexagon Based Tesselation

Tomb of Izzettin Keykavus, Sivas/Turkey. 
We can barely see the pattern on the spandrels over the two side windows.

hexagons interlocking from caglar on Vimeo.

hexagon based tesselation point attractor from caglar on Vimeo.

The Concept

Take the mid points of a regular hexagon. Draw three axes as shown in the image(note the directions).

Move the mid-point pairs along these axes. You can as well move the three pair of points with different values. Any polygon you'll draw with this method will be seamlessly tileable. If you move all three starting points up to the hexagonal center, you'll get the Mitsubishi logo made up of three congruent rhombi which are made up of two equilateral triangles.

ghx file

hex tile.ghx