My Bathroom Floor

This wasn't the first Penrose tile floor in the world, but it had to be up there. I started getting the tile made in 1997, and laid the floor a year later as my new house was being finished. I thought of putting radiant heating in the floor, knowing it would be cold in the winter and I didn't want to obscure it with a bathmat, but money gets scarce as you finish up a house, and I never did it. The floors at Miami (Ohio) University and Carlton College preceded this one, and there are newer ones at Meredith College and The University of Western Australia.

Someday, of course, I will move out of this house and someone else will have the floor. Despite the unconvential inspiration for the floor, I think this is something that will be appreciated by future owners, although whether they choose to obscure it with a bathmat during the winter months, only time will tell. I say this because the floor has stood up very well to the toughest critics out there, and that is the trades workers who actually built the house. Many of them saw it, and they went out of their way to comment favorably on it. Any of you who have had a house built know this is very unusual - they have seen some very fancy houses, and for the most part, nothing fazes them.

Penrose tilings are named after the English Mathematician, Roger Penrose. He knew when he discovered these tilings 30 years ago that people would want to use them as I have, so he patented them. I waited for the patent to run out before I made this floor - Kimberly Clark did the same thing when they introduced Penrose tile toilet paper about a year later, but Sir Roger (he's a knight of the realm) made such a fuss that they pulled it from the market. It was never available in the US, I would have happily purchased a lifetime supply.

There are many features that Penrose tilings have, but what makes them especially interesting is that with only two tiles you can be guaranteed that the pattern is acyclic, i.e. that it will never repeat (what this means will be explained in some detail below). The two shapes that you see here are the ones that are most commonly used and they are a slight simplification of what is needed to guarantee acyclicality, but you can make up for that by the simple rule that you cannot ever form a rhombus by putting the "inside" angle of the "dart" (the smaller shape, colored blue on my floor) against the "top" angle of the kite (the larger shape, colored white on the floor).

To get an idea of what acyclicity means, imagine you are visiting a wallpaper store. You want some interesting wallpaper that one could look at for a long time without getting bored. Some of the paper has patterns that repeat very frequently - you would only have to move the paper a few centimeters left or right, up or down to get the pattern to repeat, that is, so that the pattern would superimpose perfectly and out to infinity onto where it had been in its initial position. There might be some more high priced wallpaper that depicts a scene that takes several meters to repeat; often such wallpaper features scenes of things like people in a village and serves as a sort of substitute for a mural.

An acyclic tiling will never repeat. If you pick it up and superimpose it over itself it will never line up perfectly. That this could be done with a finite number of shapes had been known for a long time, but Penrose proved you could do it with as few as two. This was then, and remains today, a remarkable result, and this floor is a tribute to that theorem.

There are many other remarkable facts about Penrose tilings. If you look closely, you will see "bowtie" patterns, both small and large. These often appear adjacent to one another, but can never appear more than three in a row. Also, if you continue the tiling to infinity, being careful to follow the rule not to create any two-tile rhombus's, the pattern you get will be the same as the one anyone else would have gotten no matter how they began or proceeded with theirs. Either pattern would superimpose perfectly over the other. Moreover, every possible finite sub-pattern will occur infinitly often.

The golden mean, that ratio that figures so prominently in architecture and nature, is very important here too. The ratio of the areas of the two shapes is the golden mean. The ratio of the number of the two shapes is the golden mean. The ratio of the long side to the short side is the golden mean.

The kite-dart pattern I have used is not the only Penrose pattern. When using ceramic tile, a design that uses the two rhombi might be preferable, since the large angle in the dart led to manufacturing difficulties. If I were to do it over, I would explore techniques other than ceramic, including the terazzo that most others have used. Here are some other links to Penrose tilings that I know of.

Incidentally, there is a flaw in the layout. Can you find it?