Friday, 28 June 2019

What are Penrose Tiles?

I first came across Penrose tiles in Martin Gardner's "Mathematical Games" column in Scientific American in 1977. They were invented shortly before that by Professor Roger Penrose who was a brilliant British mathematician and cosmologist. Among many other things he "revolutionised the mathematical tools that we use to analyse the properties of spacetime".

There are two types of tiles: a kite and a dart:
 A kite and dart on the left                                       An ace or fool's kite on the right.
They are constructed from a rhombus (equal sided parallelogram) with the angles 72° and 144°. From this it follows that they fit together as an ace, as shown on the right. It also follows that each can tile round a vertex because 5 x 72 = 360.

There are four other ways to tile kites and darts round a vertex without coming unstuck.

You can also see from the cut rhombus that the lengths of the sides as either Short or Long. It turns our that the ratio of the long side to the short side is the golden ration known as $\phi = 1.61803...$ (phi) to the ancient Greeks. They would have written it$$\phi=\frac{\left(1+\sqrt5\right)}{2}$$They probably got it from considering a rectangle that contained another rectangle inside it with the same aspect ratio. The smaller rectangle is drawn with a dashed line and the equation for  $\phi$ follows directly:
From the ratio of the sides of Penrose tiles it is easy to prove the the ratio of the areas of a kite to a dart is also $\phi$.