Interlocking Arrows 2

Gök Medrese, Tokat, Turkey

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

sixfold pattern on hex-grid

sixfold pattern on hex-grid from caglar on Vimeo.


A common hexagon based pattern found in beautiful tilings of Yeşil Cami (Green Mosque) in Bursa/Turkey. Equilateral triangles are arrayed around a regular hexagon.

 

ghx file

sixfoldpattern.ghx