Technique · Geometry
Penrose tiling
The non-repeating Penrose tiling of fat and thin rhombi, built by repeated subdivision, slowly turning.
Penrose tiling is drawn geometry, rebuilt as it changes. Every piece on Synapse Drip is made from a recipe that layers techniques like this one, and you can change any of it in the Studio.
Good to know
- In a Penrose tiling the fat and thin rhombi appear in the ratio of the golden ratio to one, and any patch you pick, however large, appears again somewhere else in the tiling, but the whole never repeats.
- Dan Shechtman's quasicrystals, found in 1982, are real materials whose atoms sit in a similar non-repeating order, which is why they diffract like a Penrose tiling.
- The tiling has a five-fold symmetry that no repeating pattern can have; the centre of this one is the five-fold wheel of triangles it was built from.
- Roger Penrose found these tilings in 1974, and Martin Gardner introduced them to the public in Scientific American in 1977; in 1997 a company set up to develop Penrose's work sued the makers of Kleenex over quilted toilet tissue embossed with his pattern.
- The entrance paving of the Andrew Wiles Building in Oxford, home of the Mathematical Institute, is a Penrose tiling of two kinds of granite rhombus, laid out to a design by Penrose himself.
The mathematics
- Every round replaces each triangle by two or three smaller ones, shrinking the edges by the golden ratio, 1.618; after 6 rounds the tiles are 0.056 times the size of the first.
- Counting kinds of triangle: a red one is replaced by one red and one blue, a blue one by two blue and one red, so the numbers of the two kinds follow the Fibonacci numbers and their ratio tends to the golden ratio.
- A Penrose tiling is a slice through a five-dimensional lattice of cubes, projected down onto the plane, which is why it has five-fold symmetry yet is never periodic.
In art
- Penrose tilings appear in floors, facades and wallpaper around the world, because they look both ordered and endlessly varied.
- Medieval Islamic craftsmen drew girih patterns of stars and pentagons with a straightedge and compass, tiles that can come close to the same non-repeating order.
More ways to use it
Related techniques
L-system · Space colonisation · Diffusion-limited aggregation · Flow ribbons · Ridgelines · Curve families · Streamlines · Paint strokes