Friday, June 10, 2011
The math of Penrose tiles, part 1:
Definitions and angle measures.
Sir Roger Penrose, the world class physicist, is also a recreational mathematician. He came up with several combinations of tiles that could be used to fill the plane with non-repeating patterns before developing the kite and dart system, the two shapes of refrigerator magnets I am using in the posts with the label "Penrose tilings". The words kite and dart are actually standard geometric terminology. A kite is any four sided polygon (quadrilateral) that has two sides of one length and two sides of a different length, the same length sides meet at a corner. A dart is a kite that is concave, or we might say has a dent in it. The dent means an interior angle that is more than 180°. The math term is reflex angle.
The first special thing about the Penrose kite and dart is if we call the side lengths long and short, the long on the kite and dart are the same, as is the short. This means they have several ways of fitting together nicely.
Making such a kite and dart pair is easy if we start with any parallelogram where all the sides have the same length. The standard term for this is a rhombus, but it is also sometimes called a lozenge. (Some books use lozenge to mean only a rhombus whose angles are 45° and 135°.) A rhombus is to a parallelogram as a square is to a rectangle. In fact, rectangles are special parallelograms where all the angles are 90° and a square is a rhombus.
In any case, we can take any old rhombus and cut it in a variety of ways to make a kite and dart pair that will have the same length of short side and the same length of long side.
So there are infinitely many ways to make kite and dart pairings that can be combined into rhombi, and any old rhombus can be use as a tile that when repeated infinitely will fill the entire plane, a method called tesselation in math.
Here is the decision that made Penrose tiles more interesting than your run of the mill kite and dart that make some random rhombus. Sir Roger chose the angles carefully and the one angle both the kite and dart share is 72°. Since 72 times 5 is 360, five of these corners can be put together to fit perfectly, making a ten sided polygon, which is called a decagon. The convex decagon in yellow made of darts is called the same thing both by mathematicians and by actual people, a five pointed star.
Penrose could have chosen another angle that divides evenly into 360 so the kites and darts could be combined to make regular polygons or stars with some number of points, but 72° has some nice properties. The angles of the kite are 72°, 72°, 72° and 144°. The angles of the dart are 72°, 216° for the reflex angle and 36° at both the pointy ends. This means that in some situations, we can replace a 72° angle with two 36° angles put together, and similarly two 72° angles can be replaced in some situations with the 144°.
The math of the angles of the Penrose tiles is really more arithmetic, nothing harder than 36+36=72 and 72+72=144. Choosing these particular angles means the side lengths long and short have a relationship known as the Golden Ratio, or phi, and the math for that steps up from grade school level to high school level. Tomorrow, we will look at phi, the Fibonacci numbers and the several ways these interesting math concepts are linked to the Penrose tiles.