# One butterfly at a time

It’s not often that I will finish a book, that it leaves me amazed at the richness and expanse of its ideas … and that I have to confess I did not understand fair chunks of it. *Butterfly in the Quantum World* is such a book. Please understand: I have no problem making that confession. Because even if there were parts I had to skim over—mere mathematical dabbler that I am—I was also filled with an increasing wonder at the niches explored, the connections made. In that sense, Indubala Satija, its author, made me feel that in merely reading her book, I had become a partner in her own journey of wonder.

The story begins with a certain mathematical construct, if you will—though that word implies a mundanity that really does not apply. In playing with number sequences as a mathematically inclined teenager, the cognitive scientist Doug Hofstadter (author of *Gödel, Escher, Bach*) stumbled on a particular pattern that concealed within itself, of all things, a copy of itself. (See my earlier column here, “Butterfly on the wall”, (bit.ly/2lS0cVI). Now in a very real way, mathematics lives on patterns, and its practitioners—teenagers included—thrive on them. Entire careers have flowered in the search for meaning in such patterns, maybe even more so when they can be defined and described recursively, in terms of themselves, as Hofstadter’s pattern was.

Years later, as a PhD student in solid-state physics, Hofstadter was studying the behaviour of electrons in crystals subjected to magnetic fields. When he stumbled on a deep connection to his teenaged discovery with numbers, you can imagine how astonished and delighted he was. Who would have thought it?

He captured the relationship between the energy levels of these electrons and the magnetic field in a graph he called “Gplot”. Thing is, as far as I can tell, only Hofstadter has ever used that name. Pretty much every other scientist who has explored its charms—and there are many such, both charms and scientists—knows it as the “Hofstadter Butterfly.” If you take a look at it, you’ll know why. Especially when coloured as in Indu Satija’s book, it looks a whole lot like one of those beautiful insects that flutter by, catching the sun and filling us with, yes, wonder.

Satija is a physicist at George Mason University in Virginia. She ran into Hofstadter’s Butterfly early in her career, and found that it fluttered by over and over through the years. The connections thrilled her, to the extent that she began compiling them into a book. In 2014, she sent Hofstadter a draft. It thrilled him too, to the extent that he wrote a prologue, a “guest chapter” and worked closely with Satija to chisel the draft into the finished product. At one point, he writes of his butterfly and of all that’s in this book:

“Little did I suspect that from these humble beginnings would flow so many other results in the coming decades. Although I didn’t participate in those discoveries, I have watched them from the sidelines with great interest, and it gives me a feeling of pride and privilege to have had the good fortune of playing a role in the launching of this fertile, multifaceted area of research in physics.”

“Multifaceted” is right on the money. As Satija tells us, the butterfly makes an appearance in quantum physics. In a certain arrangement of circles named for the ancient Greek mathematician Apollonius (“Apollonian gasket”). In the “anholonomous” behaviour of Foucault’s pendulum, which demonstrates Earth’s rotation by not returning to its initial state. In topology, the study of surfaces. In the behaviour of light. There’s plenty more, including hints at research and results still unknown, still to be discovered.

I mean, there’s almost no limit to the connections—which is itself a reminder of the recursion at the heart of Hofstadter’s original discovery with his number sequences. My father loved the Marx Brothers’ films because he said the brothers managed to wring every possible joke out of a situation, a conversation, sometimes even a word. I thought of that seemingly effortless wringing all through this book. Because there’s endless meaning and analogy to be drawn from Hofstadter’s Butterfly, and actually you don’t even have to wring it out.

**The book is littered with examples of Satija’s wide-eyed wonder at the endless delights of the Hofstadter butterfly. It’s worth dipping into purely for them.**

But yes, it’s true: I know too little physics and mathematics to understand various parts of this book. Even so, I enjoyed it a great deal, and I realize that probably sounds like a contradiction. So I’m going to try to explain why, and why I think you will similarly enjoy the book.

Early in the book, Satija offers us this quote: “A mathematician … who is not at the same time a bit of a poet will never be a full mathematician.” That spirit runs through the book like a sparkling chain of jewels. I always think that if more of us can come to see mathematics that way, to feel that love for it—instead of the fear that seems more common—this world we inhabit would be a better place by far.

Consider just two examples.

In 1885, a Swiss schoolteacher called Johann Jakob Balmer found a simple formula that described mathematical patterns that he loved observing in nature. It might have stayed as obscure as Balmer’s name probably is to most of us.

But a few decades later, the great Danish physicist Niels Bohr theorized an explanation for certain dark lines in the spectrum of hydrogen, involving the energy levels of its electrons. Bohr won the Nobel Prize in Physics for this discovery in 1922.

But amazingly, Balmer’s equation actually gives us those lines, which is why they are now called Balmer lines. As Satija explains, “[A]ll of a sudden, the world understood *why* the Balmer formula was the way it was, and with that, the profound mysteries of the atom were starting to be unlocked.”

Look around you for patterns and who knows—one day somebody may connect them to the “profound mysteries of the atom” and win a Nobel Prize. Wow.

Satija devotes several chapters to exploring the butterfly’s connection to the “quantum Hall effect”, a remarkable property, discovered in 1980, of how certain materials conduct electricity while subjected to a magnetic field.

The effect, thus, turns out to be related to the Apollonian gasket I mentioned above. “How unexpected”, she writes, “that a beautiful and abstract piece of mathematics from well over 2000 years ago” is connected so intimately to a 1980 discovery. In fact, when we observe the quantum Hall effect, “we are seeing … a reincarnation of an Apollonian gasket from way back in 300 BC!”

The book is littered with examples, like these, of Satija’s wide-eyed wonder at the endless delights of the Hofstadter butterfly. It’s worth dipping into purely for them.

And then there are the “contributions” of Hofstadter himself. His own delight is no less evident, as he pursues ever-more intricate patterns and their implications. Like the time when he spends several pages explaining and deriving something called Harper’s equation. On the way, he shows us an interim equation to make your eyes glaze over (mine did, I admit), but comments: “This feels magical, like pulling a rabbit out of a hat, does it not?” A few lines later, he says the equation would likely “leave most cosine-savvy high-school students choking in the dust. … But I think that it’s important to point out and to savour such magically fluid thinking.”

My eyes glazed over, certainly. But I also savoured the thinking. For the number of times “magic”, “poetry” and their variants appear in this book tells a story by itself.

So it’s only appropriate that the last chapter has several poems: four winners in a Butterfly contest Satija organized, and one by Satija and Hofstadter (“It’s one, yet it’s infinity/Eternity, sublimity/Divinity and mystery/It’s raga, yet it’s poetry.”)

And in a Coda, Satija finds a connection to her Indian roots as well: in lines from Bhaskara’s *Lilavati*, “a little gem of poetic verse where love, beauty and mathematics are braided together.” It’s a puzzle about a swarm of bees that Bhaskara poses in those lines, and he ends it thus: “Say, lovely woman, the number of bees.”

Honestly, I didn’t need to understand everything in this book. It’s in its spirit, its scientific and mathematical curiosity, its willingness to be surprised at every turn, that it speaks loudest to me. It’s the reason mathematics is such an elegant, rewarding pursuit. It explains what I feel about Satija and Hofstadter: jointly, one butterfly at a time, making this a better world.

Butterfly in the Quantum World: The Story of the Most Fascinating Quantum Fractal *by Indubala I. Satija with contributions by Douglas Hofstadter. Morgan & Claypool, 2016.*

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