Patternless Patterns

I just read this article from Quanta Magazine about aperiodic tables, and my head hurts—in a good way. I’m going to tell you in a few sentences what I think I’ve learned about aperiodic tables, but I’m warning you: I’m going to get it wrong. So take the time to click on the link. After I tell you what I think I learned, I’m going to improvise a little bit.

---

There’s a whole area of mathematics built around tiling—patterns that develop when you put tiles of different colors or shapes together on a two dimensional plane. Think chessboards and bathroom tiles. Think Escher prints. Think (some) quilts. Most tile patterns repeat themselves. You create a block of tiles laid on a floor, stack another block next to it, and the pattern repeats. Those are called periodic tiles. But in the 1970s, the mathemetician Roger Penrose developed a pattern that cannot repeat. You place his two tiles on the floor. To place the next two tiles on the floor, you have to shift them. Fine, you think. Keep doing that, and eventually you’ll come to a bigger, repeatable block. But you can’t. Penrose discovered a pattern that can’t be repeated, a patternless pattern.

---

Pattern is rhythm. We can’t think without patterns, without arbitrary beginnings and endings that give shape to our experiences of the world. We walk, swim, ride our bikes through dots of data, and our minds draw lines to connect some of them and discard others, graphing our lives. Thoughts are either metaphors or similes. Say, “That’s a beautiful sunset,” and you are saying two things: 1) What I’m seeing is beautiful, and 2) What I’m seeing is both the same and different from other sunsets I’ve seen. Every thought is comparative. Every thought is This is That, or This is not That.

---

Some of us find it easier to think: This is That. Others of us find it easier to think: This is not That.

---

Every time a pattern breaks, new thoughts pour out.

---

Idries Shah opens his book, Tales of the Dervishes with a story about three fish, all of whom try to outwit a fisherman on the bank. I read it as a parable against dogmatism, but is parable the right word? The parables I grew up with were simpler. This parable (if it is a parable) switches back and forth from the fish’s point of view to the fisherman’s point of view. What other parable does that?

---

A fish sees the fisherman with his net, can’t find a place to hide in time, so it leaps up on the bank and plays dead. The fisherman doesn’t want a dead fish, so he throws it back. A second fish is a student to the first fish. It jumps up on the bank, tries to play dead, but forgets to hold its breath. The fisherman puts it in the basket, but the fish manages to escape. A third fish jumps onto the bank and plays dead— but by this time the fisherman doesn’t care if the fish is dead or alive. He puts the fish in the basket, remembering to latch it up. He feeds the fish to his cat.

Three fish jump up on the bank, but the pattern refuses to form, because the context changes at each leap. It’s a patternless pattern.

---

Is the fish story a narrative version of an aperiodic table? The story of the first fish cannot be repeated. The second fish is not like the first fish. And by the time the third fish arrives, the fisherman has changed.

---

Idries Shah traces the story back to the 14th Century, but he dates this particular version from 1813.

---

Sacagawea was already dead by then. She died in 1811 at Manuel Lisa’s “Fort Lisa” on the Missouri River. She was 24 years old. She had been 18 years old when she joined the Lewis and Clark expedition, traveling with her infant son on her back. Why am I thinking of Sacagawea right now?

---

What makes a pattern is a sense of beginning and a sense of ending.

---

Would you rather binge watch a periodic tile or an aperiodic tile?

---

As part of my graduate study, I was introduced to a Welsh poetic form that I really liked. It was syllabic, which meant counting syllables rather than stresses, and it combined end rhymes with internal rhymes. I can’t tell you what was called, because I can’t find the book. It’s supposed to be right there on my shelf next to Yang Ye’s Chinese Poetic Closure but, no. I suppose I could Google it. You can if you want.

What I liked about this Welsh form— the name of which I don’t remember— was that those internal rhymes seemed to soften the end rhymes. I hadn’t become acclimated to end rhymes yet, and they always seemed harsh to me, like a heavy drum beat. I wanted the end of the lines to be more subtle, and having a rhyme in the middle seemed to prepare my ear for what was to come, soften the bang.

I don’t mind hard rhymes as much now as I did then, but I still find them grating sometimes. Before I read Alexander Pope, I give myself a little self-talk: this is good, this is more nuanced than it sounds.

---

Patterns are how we know the world, hear our music, find that book on the shelf. We can’t live without patterns. We can’t live well without breaking them.

---

In the lead-up to Gulf War II, I was still establishing my small-town Alaskan bookstore. It was quieter then, and I had more time to chat with customers. I remember one really kind man. He brought up the subject of Iraq, and I asked him, did he think Saddam Hussein was involved in the 9/11 terrorist attacks? He held out his hand palm up. “David, I don’t think it. I know it. I know it like I know the palm of my hand right here.”

He knew it because the idea that Iraq was responsible fit a pattern for him.

But. Do we really know the palms of our hands?

---

Before the Lewis and Clark expedition started, both Lewis and Clark had to be away from the camp in present-day Wood River. They put Sergeant Ordway in charge. It didn’t go well. Some of the men, including John Colter, wanted to leave camp to go drinking in the village. Ordway said no. Colter drew a gun.

Why does it feel like I know both Ordway and Colter? Their early 19th Century lives are completely inaccessible to me, but it feels like I’ve met them. The junior officer who can’t control his men, because rank is just not enough. The rebellious private who can barely manage to accept the authority of the captains, but who reels under Ordway’s command.

And a part of me wonders if the expedition would have run more smoothly if Colter had been in charge. Colter knew how to travel light. Many months later, a pack horse would slip and tumble down a hill, crashing against a tree. The horse was all right, but William Clark’s heavy writing desk was obliterated. If John Colter had been in charge, they wouldn’t have been carrying that heavy writing desk across the wilderness.

---

But what do I know?