Paul Gailiunas

Statement

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 have previously (2020) explored what new Islamic geometric patterns might become available if angle relationships are allowed to be approximate. Recently I have used the same approach with the specific aim of finding patterns that are simple. The approximate arrangement of a pentagon, hexagon and octagon meeting at a vertex has been quite productive.

Artworks

There are at least three patterns with four-fold symmetry based on the arrangement of a pentagon, hexagon and octagon around a vertex. Two are already known in the historical tradition. This one seems to be new. The choice of an angle of 36° between the pattern lines helps to disguise the slight irregularity in the five-pointed stars. In addition a small adjustment allows segments of pattern lines to come into alignment in groups of three. This is probably not immediately obvious, but gives an added feeling of coherence to the design.

There seem to be no historical patterns with the arrangement of a pentagon, hexagon and octagon around a vertex that have six-fold symmetry. There are several ways to deal with the grouping of seven hexagons that appears around centres of six-fold symmetry. In this example they have been modified following a well-known device to create large free hexagons comparable in scale to the octagonal regions.