January 16, 2022

The Polygonal method: Jay Bonner (Part 1)

The Polygonal method: Jay Bonner (Part 1)

Examining several examples of ancient geometric patterns and their underlying tessellations shows that the polygonal method constitutes a reference design method. This method, also called polygons in contact, makes it possible to create star patterns by tiling the plane with polygons. As presented in the previous post, the polygonal technique was described by Hankin at the beginning of the twentieth century [1],[2]. In 2005, Kaplan[3] developed an algorithm to automate the generation of Islamic patterns by the polygonal technique.
Jay Bonner presented the polygonal method in systematic[4] and non-systematic [5] forms. This post will cover two systems proposed by J.Bonner as part of the systematic method.  

The systematic method:

Bonner's[6] systematic method uses five finite sets of polygons that can be put together in infinite combinations. Each polygon is associated with the patterns of the four known historical families: acute, median, obtuse, and 2-points.
The five sets of polygons correspond to the following systems proposed by Bonner: 3-4-6-12, 4-8A, 4-8B, 5-10, 7-14.

  • 4-8A System:
    The 4-8 A system is an octagonal system consisting of the polygons shown in Figure 1. Their combination makes it possible to obtain patterns of this family as shown in figures 2 and 3.
Fig 1: Motifs of the 4-8A system (reconstructed from J. Bonner).
  • 4-8 B system:                                                                                                               The 4-8 B system is also an octagonal system. Figure 4 illustrates the polygonal elements of this system, and Figures 5 and 6 show some examples of patterns obtained by the combination of several decorated polygons.
Fig 4: polygons and their corresponding motif of the 4-8B system

[1] E. H. Hankin. Examples of methods of drawing geometrical arabesque patterns. The Mathematical Gazette, 12: 371–373, 1925.                                                 [2] E .H. Hankin. The Drawing of Geometric Patterns in Saracenic Art. Memoirs of the Archaeological Survey of India. Archaeological Survey of India, 1925.           [3] Craig S. Kaplan. Islamic star patterns from polygons in contact. In Proceedings of Graphics Interface 2005, GI ’05, pages 177–185, School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada, 2005. Canadian Human-Computer Communications Society.                                                 [4] Jay Bonner. The historical use of polygonal systems to create Islamic geometric patterns. In Fondation de la mosquée Hassan II de Casablanca, editor, les tracés de l’arabesque géométrique, pages 85–94. Casablanca, 2014.                   [5] J. Bonner. The design of particularly complex non-systematic geometric patterns with multiple centers of localized symmetry. In Fondation de la mosquée Hassan II de Casablanca, editor, les tracés de l’arabesque géométrique, pages 85–94. Casablanca, 2014                                                                                                       [6] Jay Bonner. Polygonal Design Methodology, pages 221–548. Springer New York, New York, NY, 2017.