Saturday, June 11, 2011
The math of Penrose tiles, part 2:
The Golden Ratio phi and its relation to the Penrose tiles.
Yesterday, I discussed the two shapes of the Penrose tiles, the kite and the dart. The dart in this picture is the one in light blue. If alliteration helps you remember, the dart is the one with the dent. The angles on the kite are 72° three times and one obtuse angle of 144°. The dart has one angle of 72°, two sharp angles of 36° and a reflex angle of 216°, which is the one that causes the dent.
There is no hard and fast rule as to how big the two tiles should be, but because they follow the geometric rules of kites (four sides, only two lengths, sides of equal length are adjacent), the ratio between the long and short sides is set in stone. It is phi, also known as the Golden Ratio. The exact value is (1+sqrt(5))/2 and the approximate value is 1.61803398875... on your calculator. Using 1.618 as an approximation is not too bad.
Here are the capital and lowercase versions of phi. Blogger software is .html based and doesn't have a lot of symbols from the Greek alphabet, so I will type out phi every time I mention the number. It's pronounced "fee" not "fie" if we want to be close to the Greek, but some people want it to rhyme with pi. Technically, pi should be "pee" when we say it, but then it would be confused with the letter p in our alphabet.
Phi has many interesting properties, and most of the ways it shows up in the real world involve ratios, some big number divided by a smaller number is equal to the Golden Ratio. Another way phi can be generated mathematically is as the solution to this algebraic expression.
phi² = phi + 1
Phi is not the only number that satisfies the condition that the square of a number is the same as adding 1 to the number, but the other solution is negative, so it can't be the description of a length or an area or some other real physical property.
When we have an equation like the one above, we can use it to find the value of higher powers of phi as well.
phi³ = phi² times phi = (phi + 1) times phi = phi² + phi = 2*phi + 1
Using similar methods to change higher powers of phi into combinations of phi and 1 we get the following pattern.
phi to the fourth power = 3*phi + 2
phi to the fifth power = 5*phi + 3
phi to the sixth power = 8*phi + 5
Some people may recognize the numbers 1, 2, 3, 5, 8... as the start of the Fibonacci sequence.
Here's how phi and the Fibonaccis are tied to the Penrose tiles. Not only is the ratio of the long side to the short equal to the Golden Ratio, but likewise the area of the kite divided by the area of the dart is phi. What this means is that if I want to make a bigger kite that is similar to the original, it can be done, but only by multiplying the side lengths by phi and the area by phi².
For these next statements, remember that long/short = phi and (area of kite)/(area of dart) = phi.
Side lengths: long, short (or phi and 1)
Area: 1 kite (phi)
Side lengths: long + short, long (or phi² and phi)
Area: 2 kites and 1 dart (phi³)
Side lengths: 2 * long + short, long + short (or phi³ and phi²)
Area: 5 kites and 3 darts (phi to the fifth power)
Here's the thing. We can't make the next size up of kite, and there is no way of making a bigger dart with Penrose tiles.
Understandable proofs (knock wood) of these statements tomorrow.