Islamic star patterns in absolute geometry

We present Najm, a set of tools built on the axioms of absolute geometry for exploring the design space of Islamic star patterns. Our approach makes use of a novel family of tilings, called "inflation tilings," which are particularly well suited as guides for creating star patterns. We describe a method for creating a parameterized set of motifs that can be used to fill the many regular polygons that comprise these tilings, as well as an algorithm to infer geometry for any irregular polygons that remain. Erasing the underlying tiling and joining together the inferred motifs produces the star patterns. By choice, Najm is build upon the subset of geometry that makes no assumption about the behavior of parallel lines. As a consequence, star patterns created by Najm can be designed equally well to fit the Euclidean plane, the hyperbolic plane, or the surface of a sphere.

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