# Paul Gailiunas

I am interested in any visually interesting manifestation of
mathematics, and have used a variety of media at various times: paper
or card models, geometrical drawings, computer images, and I have used
mathematically inspired designs in leather bookbindings.

I attended Jay Bonner’s workshop in 2012 on “Creating Non-Systematic
Islamic Geometric Patterns with Complex Combinations of Star Forms”,
and it seemed to me that there are many possibilities beyond the
handful of known examples of these patterns. Combinations of star
forms that seem to be incompatible depend on angle sums that are only
approximate, and I have found a way to explore the full range of
possible approximate triangles.

This unusual design is based on a rectangle with 10-gons at its centre and corners. There is an approximate triangle with angles from 7-, 10-, and 12-gons, but, since the diagonal of a rectangle is not a line of mirror symmetry, the 12-stars must lie on centres of 2-fold symmetry, mid-way between the 10-stars. This fixes the positions of the 7-stars on the rectangles’ edges, just off the perpendicular mirror lines, so they occur in pairs, and only the region of overlap is used. Further small adjustments have been made. There are two designs with the same symmetry (2*22 or cmm) at the entrance of the Sultan Hasan complex in Cairo (1356-63 C.E.), but they have different structures, in both cases based on octagons.