Complex Numbers

Laziness is the Milkman

One of the things that we tend to take for granted -- if we think about it at all, that is -- is the ability of mathematics to model the natural world.  

I don't really understand advanced math or physics, but this sort of thing is endlessly fascinating to me. I chose the word "endlessly" deliberately, there; today's link involves infinite, non-periodic patterns.

http://nautil.us/issue/69/patterns/impossible-cookware-and-other-triumphs-of-the...

Impossible Cookware and Other Triumphs of the Penrose Tile
Infinite patterns that never repeat have moved from fantasy to reality.

I first found out about Penrose tiles back in the 80s. I always wanted to cover a floor with them, but I'm entirely too lazy to do the work involved.

In 1974, Roger Penrose, a British mathematician, created a revolutionary set of tiles that could be used to cover an infinite plane in a pattern that never repeats. In 1982, Daniel Shechtman, an Israeli crystallographer, discovered a metallic alloy whose atoms were organized unlike anything ever observed in materials science. Penrose garnered public renown on a scale rarely seen in mathematics. Shechtman won the Nobel Prize. Both scientists defied human intuition and changed our basic understanding of nature’s design, revealing how infinite variation could emerge within a highly ordered environment.

Sometimes, scientists discover something and have to figure out the math. Other times, the pure math comes first, followed by related discoveries. The latter happens quite a lot.

Penrose turned to five-axis symmetry, the pentagon, to create his plane of non-repeating patterns, in part, he has said, because pentagons “are just nice to look at.” What was remarkable about Penrose tiles was that even though he derived his tiles from the lines and angles of pentagons, his shapes left no awkward gaps. They snugged together perfectly, twisting and turning across the plane, always coming close to repetition, but never quite getting there.

The idea that complexity can arise from simplicity is something I've been interested in for a long time. It's right there in the blog intro.

The linked article goes on to explain, in words, what Penrose tiling is. But it helps to see more examples. Here is a more visual explanation.  

In fact, Penrose tiles bridged the golden ratio, the math we invent, and the math in the world around us.

As much as I appreciate Nautilus, the promise of the headline is never realized in the text I linked above. It's only mentioned as an aside:

Their irregular atomic configuration, for instance, gives them a low surface energy, which means that not much sticks to them. As a result, quasicrystal coatings have found their way into nonstick cookware. (When Penrose created his novel tiles, there was no reason to think it would have a bearing on crystallography, let alone on frying an egg.)

The point being that all of this stuff seems really esoteric, until you realize that it has real-world, practical applications -- including one of my favorites, an invention that keeps me from having to do too much work.

They say necessity is the mother of invention. I say laziness is the milkman.