Penrose Tilings

Penrose tilings are a class of beautiful and fascinating nonperiodic tilings. In addition to their intristic beauty they possess an intriguing mathematical structure. This structure is helping scientists studying crystalography understand quasicrystals, a new breed of strange high-tech materials discovered in 1984 by looking at X-ray diffraction patterns. For more information about quasicrystals, consult Marjorie Senechal's book Quasicrystals and Geometry.

is a fun way to learn more about Penrose tilings and other "quasi-periodic" tilings. Using QuasiTiler, you can create many tilings, including the Penrose tiling above.

### What's the Big Hullabuloo?

Penrose tilings, invented by Oxford mathematician Roger Penrose and author of the popular book The Emperor's New Mind, have made a big splash in the math world in recent years, but you may be asking yourself what the excitement is all about? Aside from being beautiful, Penrose tiles are interesting because they always tile the plane nonperiodically, even though they can be constructed from just two tiles, following a few simple rules. This caught everyone by surprise, because you would think that such a tiling would turn out to be very symmetric, like the wallpaper tilings.

While Penrose tilings nonperiodic, at first glance they seem like they ought to be periodic. If you look more closely, you will see they are almost but not quite periodic. Mathematicians call such tilings quasiperiodic. In a symmetric tiling, you can shift a copy of the tiling around so that it exactly matches up with the original again. With a quasiperiodic tiling, you can can still shift a copy so that it partly matches up with the original, but only right around where you are doing the shifting. Further away, the tilings will inevitably fail to match. Sometimes this is called local symmetry.

### Constructing Penrose Tilings

Penrose tiling are constructed from two tiles, with very specific shapes, illustrated on the left. Each tile is a rhombus or "rhomb" for short. The thin rhomb has angles of 36 and 144 degrees. The thick rhomb has angles of 72 and 108 degrees.

Looking at the tiles you may be confused. The whole point of a Penrose tiling is that the tiles should always tile the plane nonperiodicly, but there are several obvious ways to use these rhombs to make periodic tilings. Can you see a way to make a periodic tiling? (Hint: make it periodic in only one direction).

Even though we can use these rhombs to make periodic tilings, there is more to a Penrose tiling than just the shapes. A Penrose tiling must also follow rules which dictate how you can put the rhombs together. The key to the Penrose rules is the bottom illustration to the left. The rules themselves are:

When constucting a Penrose tiling, two adjecent vertices must both be blank or must both be black. If two edges lie next to each other they must both be blank, or both have an arrow. If the two adjacent edges have arrows, both arrows must point in the same direction.

Once again, we should stress that just using the rhombs alone do not necessarily give you a Penrose tiling. For a Penrose tiling, you have to use both both the rhombs and and the rules.

 Though this be madness, yet there is a method in't. Hamlet, Act 2, Scene 2

The Penrose tilings are one of the more enchanting objects to be found in geometry. We are first lured by the tilings' disoreinting nonperiodicity and then captured by the regularity lurking within this chaos.

Keep your eyes open for "Method in the Madness," a Science U feature article on the mathematics behind Penrose tilings. Coming Soon!

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Page last updated Wed Oct 19 23:11:18 CDT 2005