Friday 28 June 2019

Penrose tiles - Deflation

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##.

My adventures with Penrose tiles.

Back in 1977, computers were a rarity and the best way to play with Penrose tiles was to cut out bits of paper and arrange them on a table. But the corners curled up, the slightest breath of wind would misalign them and any pattern would soon go wonky. I had some paper tiles in a tin for a long time and imagined  having a computer program where I could really keep them neat and tidy. I never really studied deflation. I have now. PowerPoint came along in 1987 and in 1990 I met Microsoft Windows for the first time. In those days it used to crash about once every ten minutes. I was writing graphics programs for the beast. PowerPoint 7.0 in 1995 had VBA programming. I caught on to that in 2019 and realised a month ago that it would be handy for drawing Penrose tiles. It is very. All the above pictures and the movie are done with VBA in PowerPoint (screen capture by Debut). The biggest tiling so far is that one at the end of the movie: a whopping 13,854 kites, 8,669 darts - 22,523 tiles in total. That's very near the limit. These big tilings are produced by deflation which was also discovered by the great professor Penrose back in ~1974 about 45 years ago.

More to come - watch this space!

No comments:

Post a Comment