5-fold system
- Author:
- chris cambré
- Topic:
- Symmetry
successful
From the book Islamic Geometric Patterns of Jay Bonner, you can learn that the 5-fold polygonal system, described by Lu and Steinhardt wasn’t at all the only one, but surely the most successful one. On first sight you don’t expect much from a 5-fold system since you cannot create tilings just with regular pentagons. But the picture changes if you make use of a set of different polygons. Possibilities are endless but out of a regular decagon a limited number of tiles can be deducted to form a homogeneous system without. Different tilings can be created without use of any supplementary polygons.
Forms deducted out of the decagon
The pentagon, the barrel, the extended hexagon and the bow tie aren’t accidental forms. They can easily be constructed starting from a regular decagon.
combine
The forms of the 5-fold system are easy to combine and even follow logically out of the use of decagons in a tiling.
- When matching decagons as close as possible against each other bow ties fill in the remaining space.
- When surrounding decagons by a ring of pentagons you need a barrel as closing piece (as an overlapping of two pentagons) of two rings.
overlap
The extended hexagons and the barrel can be defined starting with overlappings.
- Two partly overlapping decagons define the extended hexagon
- Two partly overlapping pentagons define the barrel.
flexibility
The designers of Islamic geometric patterns developed very soon a strong insight in the possibilities to create a great variety and complexity of patterns. The wall pannels of the Gunbad-i Kabud illustrate the use of overlapping decagons and the alternative filling of a decagon by three hexagons and a bow tie to differ from the most simple solutions. Doing so they created local areas of 5-fold symmetry in the purple 5-pointed stars.
Penrose
A mathematician looking at such a pattern after 1974 easily makes a link with Penrose. You immediately see the purple stars, surrounded by what looks like the Penrose carwheels and local patterns in which the symmetry is broken by alternative fillings. And of course theres the deduction of kite and dart within a decagon, you can find in the stars.
