Penrose Tiles

[Brian Olewnick]

So, one of the side-effects (I hesitate to call it a benefit) of moving to the near suburbs of Jersey City is that we get a backyard. The term as it applies here may draw snickers of derision from those of you with vast wooded expanses behind your dwelling, but in the environs of NYC, a plot measuring 20 x 30 feet is not something to scoff at. Now myself, I'd be happy with 600 square feet of grass in which to loll about, cavort with Nanook, take naps, etc. However, I'm married to a maniacal gardener harboring delusions of the Tuileries. Thus, plans are underway to convert this verdant mini-paradise into a bunch of flower beds. And a patio.

After considering various ways to go, we opted for paving stones, the ones that are sorta octagons with a square protruding from one side. Attractive enough, but while stacking them in various formations prior to laying them, I got to thinking about Penrose Tiles.

I've little idea how well known these things are; I bet not very, though they should be. They were created by the British mathematician Roger Penrose (an extremely fascinating fellow, someone who's provided some of the stronger--though not unanswerable, imho--arguments against "strong AI" in his fine book, "The Emperor's New Mind"). He was interested in tiling patterns, ie, tesselations that completely cover an arbitrarily expansive plane. Obviously, there are many regular shapes which accomplish this goal from squares to hexagons to more elaborate figures. All of them, however, result in regular, repeatable patterns. Penrose sought to find figures which would produce irregular patterns yet still cover a plane 100%. He found them in a 2-piece set known as Kite and Dart. Here's an example:



For me, the fascinating thing is that one's mind seeks to see some sort of regular pattern and one is almost convinced that it's there but it's not. As a whole, it has something of the slipperiness of peripheral vision, like being able to see a faint star out of the corner of your eye but not when you look directly at it, as if the sought regularity is just outside your field of vision.

I'm tempted to think of Penrose Tiles as a visual concretization of the sort of arguments (aesthetic taste, philosophical, etc.) that beset us. It's hard to explicate exactly how, but there's something about them......

[Uli]

That be a nice patio!

[clinthopson]

I can imagine seeing the Penroses after a couple of stiff Scotches. They will vibrate, baby.

We're redoing our courtyard with flagstone. It's a stone from Montana called Three Rivers with varying shades of grey, brown and green with many fossils. I'll post a pic when it's done.

[Slurpy]

That is fascinating. Penrose, I believe, also worked with Stephen Hawking and utilizing their advanced mathematics, came up with the singularity theory and then postulated phenomena like black holes, which we now are almost certain exist. I hate people that smart Had a buddy in advanced calculus in high school that, if he couldn't use a known theorem to solve a problem, invented his own theorem! Went on to become a ceramic engineer for NASA.....wanker.

[bluenoter]

Originally Posted by Brian Olewnick

He was interested in tiling patterns, ie, tesselations that completely cover an arbitrarily expansive plane. Obviously, there are many regular shapes which accomplish this goal from squares to hexagons to more elaborate figures. All of them, however, result in regular, repeatable patterns. Penrose sought to find figures which would produce irregular patterns yet still cover a plane 100%. He found them in a 2-piece set known as Kite and Dart.

Brian, I once "got" that concept, but to my embarrassment, I find that I no longer do. In your illustration, the figures along the borders of the square aren't complete but can be viewed as either truncated or different figures entirely, so how can the two pieces in the set be said to cover the whole square?

For your patio, you'd have to bust up some of the tiles.

[Brian Olewnick]

Originally Posted by bluenoter

Brian, I once "got" that concept, but to my embarrassment, I find that I no longer do. In your illustration, the figures along the borders of the square aren't complete but can be viewed as either truncated or different figures entirely, so how can the two pieces in the set be said to cover the whole square?

For your patio, you'd have to bust up some of the tiles.

Rita, it won't be a "square", or any figure with straight sides, that gets covered, just any abstract, borderless 2-dimensional space (hexagons won't cover a square either, eg). If I was actually using Penrose Tiles for our patio (and you'd think someone would've marketed these things by now), I'd simply have irregular borders! (like Linda would allow something like that, yeah...)

[SinginSumo]