From circles to polygons

In his article The Search of Quasi-Periodicity on 5-Fold Islamic Ornament from 2009 Peter R. Cromwell describes the transition from a rhombic line pattern  with tangent circles to the 'Polygons in Contact' (PIC) of Hankin. Many early Islamitic patterns show a square or triangle grid in which 6-, 8- or 12-pointed star patterns are created. Cromwell illustrates how to create 10-pointed stars upon a rhombic grid with 72° and 108° angles.
  • Draw tangent circles in the rhombic grid centred at the gridpoints.
  • Draw a 10-pointed star whose points coincide with the tangent points of the circles. Some points of the star don’t touch another circle.
  • Draw regular decagons within the circles who’s edges are perpendicular to the points of the star.
  • Substitute the circles by the tangent decagons and hide the rhombic grid
  • Reproduce the star pattern in al circles. In between the decagons bowtie like polygons appear.
  • Extend the lines that define the star into the bow ties until they intersect. In the bow ties appear, congruent with those in the stars.
  • Hide the grid.
According to Cromwell the transition from circles to polygons seems small but in fact it isn’t because it allows generalisations with less constraints for the designers. With the ability to combine polygons freely more patterns and star forms can be created. What’s important is the angle between the lines and the edges of the polygons.