from barn quilts to mosque tilings

While travelling around southern Ontario recently, I enjoyed seeing barn quilts along the highway as we drove. Some barn quilts are truchet patterns; others, although not limited to using the truchet tile are based on a grid and can be created without a compass.

A truchet pattern that would make a
nice barn quilt

The truchet pattern above is an instance of a star pattern that often comes up in grid doodles - this "inner star" and the related "outer star" can be found, for example, in Froebelian gifts and guides (see The Kindergarten Guide by Maria Kraus-Boelté and John Kraus, 1877).

outer star and inner star (after fig 299 in
The Kindergarten Guide)

The "outer star" can be obtained from the "inner star" by inverting each of its indentations.

outer and inner
superimposed

If we extend all edges of the outer star, we get a new "greater star" that fills a 10x10 grid.

the greater star: an extension of
the outer star

Extending each edge of the great star a unit further and connecting the edges as shown below, we get a 12x12 florette pattern

greater star florette and tiling

Using this pattern as the unit of a tessellation, we get an approximation of a tiling found at the Kairouan Mosque (see here also).