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. Because Najm is built using only the
axioms of absolute geometry, which makes no assumption about the
behaviour of parallel lines, star patterns created by Najm can be
designed equally well to fit the Euclidean
plane, the hyperbolic plane, or the surface of a sphere.