# Inflation

You can see from the picture below that every kite can be thought of as being made up of two smaller kites plus two small 'half darts'. Similarly, a dart can be cut into a small kite plus two small 'half darts'.

This suggests the possibility that any fragment of a Penrose tiling can be used to generate a larger tiling by replacing every kite with the coloured shape on the left (made of two kites and one complete dart) and every dart with the coloured shape on the right (made of one kite and one complete dart).

In order for this to be possible, we must show that these curious shaped pieces fit together accpording to the Penrose rules.

Now there are just five ways of putting kites and darts together in pairs:

and, amazingly, all five can be reproduced using our inflated kites and darts eg:

This process in effect constitutes a proof that Penrose kites and darts will tile the plane. Simply start with any valid tiling - a single kite will do - replace every kite and dart with the smaller versions as described and then magnify the result so that the new kites and darts are the dame size as the old ones. (The required ratio is, of course 1.618, the Golden ratio). Now you have a valid tiling which is larger than the original one. Repeat the process ad infinitum and you have a tiling which covers an infinite area!

Another interesting consequence arises. Let us assume that every sufficiently large tiling has p kites to every dart and that we have a tiling with pN kites and N darts. The process of inflation turns a kite into 2 kites and 1 dart and a dart into 1 kite and 1 dart so the new tiling will have 2pN + N kites and pN + N darts. The ratio of these, however, must still be p so we can write

2pN + N = p.( pN + N )

from which it follows that p = ( √5 +1) / 2 or the Golden ratio 1.618...

It also follows that, since this number is irrational, no finite region of any tiling can have exactly this ratio of kites to darts and therefore no infinite tiling can consist of a periodic repetition of a 'cell' of finite size. In other words - every Penrose tiling is aperiodic!