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.