Wednesday, January 6, 2010

Wednesday Math, vol. 104: Penrose tiles

Touch the tips of your forefingers together and also the tips of your thumbs. If you imagine these are perfectly straight lines, you can line them up to make a triangle or move your thumbs to make a four sided polygon. All these four sided objects would be classified as kites, which are quadrilaterals with two adjacent sides of length a and two adjacent sides of length b. If you move your thumbs up so the shape has a dent in it, like the blue shape in this drawing, the concave kite is often called a dart.

Sir Roger Penrose (1931-) is a Big Damn Deal in physics. In 1988, he shared the prestigious Wolf Prize with Stephen Hawking. Penrose also enjoys recreational mathematics, and is credited with the invention of Penrose tiles, shapes you can use to tile the plane. I had an earlier post about tiling of the plane with regular polygons, including equilateral triangles, squares and hexagons. There are also ways to mix and match different regular polygons, all of which share the same side length.

Penrose made sets of non-regular shapes that tile the plane. His first set of tiles had six different shapes, but after fooling around with the system for a while, he came up with a new and cleaner system that had just two shapes, known as the Penrose kite (green) and the Penrose dart (blue). Since they fit precisely into part of a regular pentagon, the two side lengths a and b have a certain ratio, known as the golden ratio, and the angles are a fixed size as well. The kite has three angles measuring 72° and the large angle measures 144°. The dart angles are 36° at the sharp points, 72° at the top and 216° for the angle that makes the dent. You might notice that 2×36 = 72, 2×72 = 144, 3×72 = 216 and 5×72 = 360. This means that if you put enough of these angles together you should be able to add up to a full circle at every meeting point.

He's a clever guy, that Sir Roger Penrose.

Clever Sir Roger decided on just one rule for Penrose tilings, and that is that you can't put a dart and a kite together like they are in the picture above to make a rhombus. Tiling the plane with a rhombus is way too easy, as is tiling the plane with any parallelogram. Instead, the Penrose tiles can be put together in many different non-repeating ways to fill up the entire plane. Non-repeating patterns are known as aperiodic in math. Here is part of one such pattern.

I've been hunting around online for Penrose tiles, but I've only found one supplier, and compared to other geometry toys you can get, the prices are steep. This is a shame, because if you wander around the 'Net, you can find some very pretty patterns people have made with Penrose tiles, and I think they would be fun to play with.


Anonymous said...

Hey Matt!

Happy New Year from Rob Zdybel!

You may remember me from such 2600 hits as "Pigs in Space" ;-)

Just to let you know I follow your blog and think of you (and the "good old" days) often.

Not to take anything away from Dr. Penrose here. He *is* a pretty bright guy, no question. But it seems to me that two Penrose tiles back-to-back simply cover the same ground as two equilateral (hey, you're the math prof, did I spell that right? :) triangles would. Not so surprising that they fill space really - even with the extra restrictions.

Fascinating to find it in nature too!

Keep up the excellent blog!

Anonymous said...

So now I go and actually *read* the damn thing to find that they are isoceles.


Oh well, the heartfelt greeting is still sound. ;-)

Matty Boy said...

Hey, Rob! Of course I remember you. Nice of you to comment.

Any triangle can tile the plane, it doesn't have to be equilateral or isosceles. Take two copies of any triangle and you can put them together to make a parallelogram, and any parallelogram can also tile the plane.

Penrose tiles are fun because patterns don't repeat. I'm looking into getting a passel of them made out of acrylic or maybe flexible magnetic sheeting.