## About the Penrose Tiling

The Penrose tiling is a basically pentagonal pattern which fills the plane. The positions and neighbor arrangements of the two tile types are statistically similar from one place to another but never repeat exactly, and the tiling is also self-similar at different scales, in the sense that repeating groups of tiles can be identified that play the some role as the single tiles, but at a larger scale. This tiling was discovered by Sir Roger Penrose.

The Penrose tiling that forms the background of my home page is 48x64 units in size, where the "unit" is the edge length of the polygons, a total of 800x600 pixels. If your browser's window is bigger, and if you look carefully, you can see the seam where the image is tiled. The rhombi are saturation modulated, that is, they are all the same brightness and hue with different saturations, specifically, CIELSH 87/*/20 where * (saturation) is 84, 42, 0, 63, 21. Most of the rhombi have the first three colors, about equal numbers. The fourth is much rarer and only one or two have the fifth color. An optimal coloring algorithm would not have needed the fifth color.

Preliminary versions were more conventionally modulated in hue and in brightness. Hue modulation was garish. Brightness modulation interacted with the writing, since the letters are similar in size to the rhombi, making it hard to read. So I tried saturation modulation, and it worked out.

• Penrose tiling with edges visible. Used by permission of Pentaplex Ltd, Brighouse, UK.
• Download the program.
• Tutorial on tiling. This is the section on Penrose tiling.
• Web service which will produce integer lattice tilings for you. It explains the principles used and has references to the non-web literature, specifically:
• N.G.deBruijn, Algebraic theory of Penrose's nonperiodic tilings of the plane, I, II, Nederl. Akad. Wetensch. Indag. Math. 43 (1981) 39-52, 53-66. This is deBruijn's original paper, hard to read.
• Penrose vs. Kimberly-Clark (fascinating! litigatous!)