The math of Penrose tiles, part 3: Two proofs of impossible similarity.

I'm about to prove a couple of negatives about Penrose tilings. Recall Donald Rumsfeld proudly and stupidly saying you couldn't prove a negative when it became obvious to everyone the weapons of mass destruction ruse was a complete phony. I had to wonder exactly how many classes he slept through when he got his degree at Princeton.

Of course you can prove a negative. The only place where real proof exists is in math and we prove that things are impossible all the time.

Let me give a couple examples.


It is impossible to build a larger shape similar to a dart using kites and darts.

The dart is the Penrose tile with the dent, and angle of 216°. It is also the only Penrose tile that has the sharp 36° angle. Those angles are adjacent to each other, which means if you need a 36° angle when you are building something, you have to use a dart and you have to plan for the fact the 216° will be right next to it at the distance of short.

If we want to build a bigger dart, it will have to have two 36° angles and a 216° angle, but the distance between these will have to be at least the length of long.

We can't do this with these pieces, or if we achieve this, we will not have a long enough straight line to make the outside of the dart.

This proof takes no math skills really. If you had some Penrose tiles to play with, you would see pretty quickly the problems involved trying to make a shape similar to the dart.



It is impossible to build a shape similar to a kite bigger than Papa Kite.

Yesterday, I showed this picture of a regular kite, a slightly larger kite made of a dart and two kites (a shape I call Mama Kite) and a third larger shape made out of five kites and three darts I call Papa Kite.

Notice this. Each of the straight lines that make up a side of all three of these kites has at most one side of the short length. Because of the angles available, one short is all you can have if you are building a straight line that is empty on one side and completely filled in on the other. The problem is that to make a straight 180° angle from a 72° angle, we need 108°, which in Penrose tiles can only be done by combining a 72° and a 36° angle. Just as we saw in the earlier problem, the 36° angle is a little clumsy when trying to continue a straight line because it is so closely tied to the dent, the 216° angle, known formally in geometry as a reflex angle.

Here is my best attempt at making Granddaddy Kite, the next size up of similarity. The Fibonacci sequence tells me how many pieces I need, 13 kites and 8 darts. I used 12 kites and 7 darts and the shape of the empty space that caused the problem has a 36° angle that we can't negotiate with the shapes available.

Notice that the unfillable space is exactly a Big Dart, the shape we can't make with the two standard Penrose tiles. If a third Penrose tile existed that was the shape of the Big Dart, with side lengths long and long+short, the number of things we could do with the new system would increase dramatically, though it wouldn't help with making a dart bigger than Big Dart. That would still be impossible.

Instead of Big Dart, another "third" Penrose tile that could help in this situation would be a triangle with sides short, short and long, which would have angles 36°, 36° and 108°. With this addition, Big Dart would be these two triangles put side by side along one of the short sides, and suddenly bigger darts and bigger kites would be much, much easier.

In math, we call this "prove or disprove or salvage". When you prove something can't be done, you try to find the simplest changes you could make to the problem where you could do what was asked. The most famous early example of this was Archimedes proving that trisecting any given angle was impossible with a compass and straightedge, but it could be done if you were allowed to put one mark on the straightedge.

This is one of the reasons mathematicians put Archimedes head and shoulders over other ancients like Euclid or Pythagoras. Nobody else was "thinking outside the box" like our Sicilian pal.

Not that I'm telling Sir Roger what to do with his tiles. He is a Big Damn Deal in physics and I'm a blogger.

Not that I'm comparing my salvage to Archimedes' method for trisecting angles. That is a work of stunning beauty.

I'm just sayin'.

And, oh yeah, Donald Rumsfeld is still a pinhead who planned two wars he didn't know how to finish and he can bite me.

I'm just a blogger, but I'm a shitload smarter than he ever was.

If you ever read this, Don, quod erat demonstrandum, you ugly, murderous little pencil pusher.