The mathematician Tadashi Tokieda loves to explore the special mathematical and physical properties of the simple objects that he calls “toys” — and he’s passionate about sharing what they can teach us about the world. In this episode, he takes host Steven Strogatz on a conversational tour of some of his toys’ surprises and talks about his life as an artist and classical philologist before he became a professor of mathematics at Stanford University. This episode was produced by Dana Bialek. Read more at QuantaMagazine.org. Production and original music by Story Mechanics.
Steven Strogatz: Hello, hello, Professor.
Tadashi Tokieda: Hello, hi, hi. How are you doing?
Strogatz: Good, good.
Tokieda: It’s before lunchtime here, which means that probably you will be in a cheerful mood and I will be grumpy.
Strogatz: Oh, well, should we get you something?
Tokieda: [LAUGHS] No, no, that’s okay, that’s okay, but it is —
Strogatz: Do you need some tea or something, or get an apple?
Strogatz [narration]: From Quanta Magazine, this is “The Joy of x.” I’m Steve Strogatz. In this episode, Tadashi Tokieda.
Tokieda: It’s amazing how conditioned we humans are by, by food and, and sleep, and things like this.
Strogatz: Oh, yeah, especially, I, I am very sensitive to the need to eat. Actually — so, in my case, I kind of shut down and get dysfunctional and just stare blankly. And my wife can tell when it’s happening. [LAUGHS]
Tokieda: Oh, that’s interesting. But I am in a cursed position: I never feel hungry.
Strogatz: Mm —
Tokieda: Now, I’m, I’m not claiming that I push myself to the limit —
Tokieda: — but I don’t remember the last time I felt hungry, since early childhood.
Tokieda: But —
Tokieda: And nor do I feel full, actually, so I …
Strogatz: Yeah, that’s a dangerous combination. [LAUGHS]
Tokieda: Eating is a purely intellectual decision for me, and often a social one, yeah. But the — but, indeed, my wife can tell that, you know, I become grumpy and — grumpy and so forth.
Tokieda: So, if I start sort of behaving that way, please forgive me and point it out, and I will …
Strogatz: [LAUGHS] I, all right, I will, I will forgive you, and point it out. [LAUGHTER]
Tokieda: That, that’s right, that’s right. In fact, people — my friends don’t seem to mind pointing it out.
Strogatz: All right, now, we don’t do too much poetry on this show, but I’m going to try something with you. It’s from William Blake, “Auguries of Innocence”:
To see a world in a grain of sand and a heaven in a wildflower,
When I think of Tadashi Tokieda, I think about Blake’s grain of sand, except that for Tadashi, it’s not the world in a grain of sand; it’s the world in a toy. Tadashi sees in toys, — just like children’s toys, little playthings — the whole universe that’s in there, the principles of physics. He is a mathematician, originally from Japan, working in mathematical physics, and his passion is toys: inventing them, collecting them, explaining them, studying them. Because these are toys that can reveal real-world surprises about math and physics, but in his hands, they’re like magic tricks. And I believe that almost as soon as we met, he started doing these magic tricks for my kids and me and my wife.
Strogatz: So, I, I’m thinking back to the day, in 2012 —
Strogatz: — a, a sunny day in Cambridge, England, when I reacquainted myself with you. And I had my two little daughters with me, and my wife.
Tokieda: Yes, yes, of course I remember.
Strogatz: And, yeah, good, so you remember. And my memory is that you had a toy —
Strogatz: — that Americans would call a Slinky —
Tokieda: Yes, that’s right.
Strogatz: — maybe worldwide — is it known as a Slinky everywhere?
Tokieda: Uh, yes, uh, it was a registered trademark, at some point, yeah.
Strogatz: Yeah, okay. So, a Slinky, for anyone who doesn’t know —
Strogatz: A Slinky is a very loose spring, or how would you define it?
Tokieda: Very, very long.
Strogatz: Long and loose.
Tokieda: And very, a very sort of thickly coiled spring.
Tokieda: So that, when you suspend it, it goes way, way down. You know, if you think about the spring, maybe you are thinking of something that’s a foot long or something, but it’s actually very, very long, and very soft, and really wobbly —
Tokieda: — that kind of spring.
Strogatz: That’s right, that’s right, very good. And so, I think what you did, if I can recall, is that you, you had the Slinky, you asked my daughters, who were then about, let’s see, they would’ve been 12 and 10, Leah and Jo — and you said, “I have, I have a Slinky. I wanted to show you something. Now, watch carefully.” And then you held the Slinky, as you mentioned, suspending it.
Strogatz: So you held it — did you stand on, on something?
Tokieda: Yes, I stood on a, on a —
Strogatz: On a chair, maybe?
Tokieda: — on a table —
Strogatz: On a table.
Tokieda: — I think, yeah, on a table because I wanted to have the Slinky to be suspended to its full length, which is very long indeed, yeah.
Strogatz: Ah, okay, so then you presumably held way — stretched your arm up over your head —
Tokieda: Yeah, yeah, that’s right.
Strogatz: — let the Slinky hang down. It’s, I don’t know, six feet long or something when it’s stretched out?
Tokieda: Yeah, yeah.
Strogatz: And, well, you tell me, what would’ve been your instructions, do you remember?
Tokieda: Well, I said, “Well, you know, this is the kind of toy that you might have played with,” because usually people let it go down a staircase, flonk, flonk, flonk. It’s a really wonderful thing to watch. By the way, somebody called [Michael Selwyn] Longuet- Higgins analyzed this motion, in terms of shockwave. And so, it’s actually very, very beautiful .
Strogatz: And it was a standard — that was the TV commercial for the Slinky, when I was a kid.
Tokieda: Is that right? Oh, I see, I see.
Strogatz: Oh, absolutely, when I was a kid and they were trying to sell you a Slinky, they would, there was a —
Tokieda: Ah —
Strogatz: I can remember the theme song [sings tune] —
Tokieda: [LAUGHS] I see —
Strogatz: My sound engineer is nodding and smiling.
Tokieda: Ah —
Strogatz: He remembers it, too. We’re the same age, I think. But, so, yeah —
Tokieda: I see, uh —
Strogatz: “Tada, a, a Slinky, a Slinky, something, wonderful toy …” And anyway, and they would show it doing this motion where it’s shaped like elbow macaroni, and one of the feet of the elbow would be on a stair, and then the next foot would somehow magically —
Strogatz: — jump over the top —
Strogatz: — and climb down to the next stair —
Strogatz: — and so on. It would just walk downstairs.
Tokieda: Oh, I see. Anyway, so what I would have suggested to your delightful daughters is, is to hang the Slinky, and say, “Well, we’ll do something else.” And, you know, if I drop an object, I think I would have probably started with some more usual solid object. For example, I might have taken maybe a case or a book or something, and if I drop it, as soon as I release my, my hand and let go, it will start dropping, right? It will just start accelerating, it will shoot down
Tokieda: Yes, they say, yes, yes, and I probably would have demonstrated this. But this Slink– Slinky behaves in a rather different fashion. So, I’m going to let go of the top, and of course the top will start going down, but watch the bottom of the Slinky, just keep focusing. And I would have probably put the bottom of the Slinky, hang it so that the bottom of the Slinky is at eye level —
Tokieda: — uh, eye level. And just watching them and then, “Please watch, and I’m going to say, ‘3, 2, 1, 0,’ and then let go.” But you’ll see that for a space of time, very brief moment of time but noticeable space of time, the bottom of the Slinky will seem to hover, levitate and not to move, although I release the top. So, it’s just hanging in, in midair while, apparently — this is a rough description — the top of the Slinky will come collapsing down to the bottom. But — and only when the, the spring has collapsed to its shortest length would whole thing start falling down. And so, I would have described this.
Strogatz: So, so, is that really your memory, that you would have told them what to look for, before doing it?
Tokieda: Yes, in, in —yeah. And the — but in fact, I would have done it … So, I’m saying this in the, in this language, in this order, because, well, people who are listening to this will not be watching any object.
Tokieda: And because the, the point is for them to notice a new phenomenon rather than for me to give the punchline.
Strogatz: Exactly, exactly.
Tokieda: And besides, by the way, it’s children and adults alike: If they really try to look out for a new phenomenon, they might find something that the person showing, for example, like me, may not have expected. And that’s so much the better, of course.
Strogatz: Mm —
Tokieda: Yeah, so, anyway, so, yeah, I would have described — and I would have invited them to watch the bottom, and then see what, what happens. And then, and let go, and then, and they would have noticed something.
Strogatz: Absolutely. And so this is what happened, and it was so astonishing, it’s —
Tokieda: Yes —
Strogatz: — the bottom is hovering —
Strogatz: — and then it only seems to notice that its top is no longer, you know, holding it up, and then it starts to fall.
Tokieda: Yeah, yeah, exactly.
Strogatz: It seems unbelievable.
Strogatz: To — first of all, it reminded me of this childhood … [CHUCKLES] Funny —
Strogatz: — all these things bring back childhood memories.
Tokieda: Mm-hmm, mm-hmm.
Strogatz: So, this, in this case, it’s another bit of American pop culture, the cartoon of Roadrunner and Coyote. Um —
Tokieda: Aha, I see, oh, yes, yes, I know, I know, and —
Strogatz: You can guess what I’m thinking of, here?
Tokieda: Yeah, yeah, it’s really wonderful. I like this. And then, the question is, with whom do you sympathize? With this, I always sympathize with the coyote.
Strogatz: [LAUGHS] Well, there’s — right. So, I’m thinking of the specific thing that used to happen frequently, that Coyote would be chasing Roadrunner, and then, somehow, Roadrunner would evade capture. And the coyote, who’s running so furiously, would end up running off a cliff, and then be suspended in midair, not realizing that he was now off the cliff. And he’d look around for a while, and everything was fine until he would look down, and then [LAUGHS], then he would start to drop. So it was almost the same kind of experience, watching the Slinky. It’s — the bottom is hovering, and then it only seems to notice that its top is no longer, you know, holding it up, and then it starts to fall.
Tokieda: Yeah, but the crucial thing in this, in this entire experience — and in fact, your daughters did this — is then I invited your daughters to take turns to do the hanging themselves.
Tokieda: And then the rest of us watched. Because obviously, the first time it happens, I am cheating, right? That’s the first, shall I say, as, as Bayesians would, would say, “the reasonable prior.” That would be the default assumption, that I am cheating, I’m kind of magician —
Tokieda: — and I’m showing something, and so on. So they have to do it themselves, and I have to sit back and watch, and, you know, convince them, and in fact, they have to sort of discover for themselves. And in fact, it works for them. It, it’s, in fact, something that is in nature. And that is, I think, key. You see, the — it’s, all of these surprises are of course wonderful and — in fact, that’s key, as you say, and we shall discuss this, if you like. But the … There is a difference between what the magician — and this is kind of a magic trick, if you like —
Tokieda: — broadly construed. But the magic tricks, as performed by professional magicians, have the property that the magician has to do something really clever and skillful in order for something interesting to happen, some surprise to be precipitated.
Tokieda: In other words, the intelligence or the information content of the magic trick is in the hands of the magician.
Strogatz: Ah —
Strogatz: Good, yes?
Tokieda: Whereas the kind of thing that I would like to pursue, and I really enjoy sort of sharing with people is also a magic trick, but I’m in fact not doing anything.
Tokieda: All I’m doing is to, so, to point out a phenomenon of nature, which may have escaped notice until now, or maybe have, have been noticed but by not many people, and which is under our noses all the time. Provided we are careful, and provided we are, we watch, watch with imagination.
Strogatz: Yeah, good, nice phrase.
Tokieda: But you see the, you see the — and it works every time, and I keep all my hands open, and so forth, and yet some surprise is precipitated, because the information content and the intelligence of the magic trick is in the hands of nature.
Strogatz: Mm —
Tokieda: So I’m doing absolutely nothing. All I’m doing is to introduce nature to the spectator. “Spectator, Nature. Nature, Spectator — please meet.” And that’s, that’s all there is, there is.
Tokieda: So in some sense, that is a very, very different kind of, a brand of magic. There is, this is therefore a special brand of magic, and it is very curious —
Tokieda: — and it’s unusual brand of magic for professional magicians, as well. But it’s also has some other properties. First of all, I’m not doing anything, and secondly, um, it’s, it’s kind of modular and it can be built up. Because, you know, after we did this, um, you know, we went on to see some other things, and then we tried to do this and that. And then, the, people might say, “Well, what, if you try this, okay, let’s try this,” and so forth, and then you can build up. And you can learn from what you have learned before, and then you look for new surprises, and so on.
Tokieda: So you can actually piece together these little bits of — or sometimes big bits of magic, magic tricks, curated by nature, and then build up something for ourselves, and look for even bigger surprises.
Strogatz: Mm —
Tokieda: And so this is a very special brand of magic, and in fact, this brand of magic has a traditional name.
Strogatz: Oh, really?
Tokieda: It’s called “Science.”
Strogatz: Ah, you got me. [LAUGHS] Walked right into that one.
Strogatz: When we see something dropped, we expect it to fall right away. And the idea that the bottom of the Slinky could just hover in space without moving for quite a perceptible length of time, it comes as a shock. It seems like it’s defying the laws of physics, though, of course, it can’t — you know, nothing really defies the laws of physics, that’s why they’re laws. But still, this combination of surprise and pleasure is something that we normally associate with magic. But through Tadashi’s work, we see that it’s right there in science, too, that nature itself can produce those same feelings of surprise and pleasure.
Science is not Tadashi’s only love: He also loves languages, and in an earlier life, he taught himself 10 different languages, at least. He was a practicing philologist, that was his profession at one time. A philologist is someone who loves languages, or more specifically, loves words. It’s right there in the name: philo for love and logos for word.
Before I had my conversation with Tadashi, we almost had a little negotiation. There was a back-and-forth exchange of e-mails: what would we talk about, you know, what was fair game? Normally, with him, it’s all about visuals — he likes to do experiments right before your eyes and surprise you, but you can’t really do that in a podcast. So in place of that, he thought of a list of surprising or interesting things that we could discuss, and it looked fantastic. I, you know, was very happy to do that. But in the course of that e-mail, he introduced an exotic word, at least exotic to me, a French word, causerie, which means, apparently — I had to look it up — an informal chat, a, a very lighthearted chat about this or that.
Tokieda: Yeah, so this is the kind of thing that — I mean, I just generated this without any thought, but the — this is the kind of thing that of course goes in, goes into what I describe as causerie. And hopefully — well, in this case, no, discussing various … chit, chitchatting about this and that about the universe, over a glass of wine or something like this.
Strogatz: Is that so … yeah, I was curious about the word. I looked it up on the internet —
Strogatz: — but tell us what you — what, what, how, how do you understand it?
Tokieda: Well, causerie is obviously a —
Tokieda: Yeah, causerie is obviously a French word, and you know, the … one of the generators of, shall I say, culture is, is conversations in salons of the, of the old aristocratic homes and so forth. And in the — in France, in particular, there was a tradition, as you know, of … Let’s say, the lady of the, of the family to host a bunch of guests. It is really a party, but the party not the American style of standing around and, and nibbling things and, and drinking cocktails.
Tokieda: But everyone is sort of seated, or maybe roaming about, and so forth in a very comfortable salon, and then, you know, discussing this, that. The rule, however, is, as Talleyrand said — you know, the, the, the great French minister, and a rather suspicious character — anyway — said that, the, the sign of intelligence is the ability to speak lightly of heavy things and heavily of light things. And that, that —
Strogatz: Oh, good.
Tokieda: — and so, you should not be completely serious, but there should be some, some piquancy to, to what you say. It, it shouldn’t be trite, and so forth. Anyway, it’s, it’s extremely difficult to achieve, and of course I don’t — but the — that’s the kind of thing that is very nice. And the, the word causerie has been used in other contexts. For example, when Einstein — after general theory of relativity — was touring Europe and the world, indeed, he came to Paris — and by the way, there is a story about how Einstein is addressed in various countries. In France, of course, he was Monsieur Einstein, Monsieur Professeur. In Germany, he was Herr Professor Doktor Einstein. And the question is, “How was he addressed in America?”
Strogatz: Mm, Mister.
Strogatz: Oh, Albert, okay, of course, better. [LAUGHS]
Tokieda: That’s right. But anyway, so, in France, there was a — you would be interested in this, actually — a causerie that was organized by the Academy of Sciences between Einstein and, and Élie Cartan and the, a few other people about this and that. So this was a fairly —
Strogatz: Since it’s possible someone listening to this will not be as familiar with Élie Cartan as the typical mathematician —
Tokieda: Ah. So Élie Cartan is one of the fathers of modern studies of the area of differential geometry — that is, the study of geometry that uses, makes wholesale use of differential integral calculus.
Tokieda: And he really developed a very original approach, which was not so well understood at the time, but which really took over the entire field, and in particular the theory of differential forms and, and so forth. And —
Strogatz: Okay, so he’s hosting a salon.
Tokieda: Well, it’s kind of a, it’s a —
Strogatz: Or the, the —
Tokieda: — a more artificial setup, because now, aristocratic homes are now gone, long gone, turned into museums, and so forth. But I think it was Académie des Sciences in Paris which hosted this causerie and instead of having Einstein give a lecture, which he did elsewhere, but there was an occasion when, where Cartan and Einstein, and some other interesting people — were sitting around and just discussing and talking about various things, while they were surrounded by an audience listening.
Strogatz: Ah —
Tokieda: It’s a bit like a panel discussion, but it’s much, much more informal, and much faster and, and sort of wittier than a, a panel discussion.
Tokieda: You know, a panel discussion has this, kind of, usually, row of people behind, behind the table.
Tokieda: It looks like a jury —
Tokieda: — some kind of an examination board, and so forth, and it is coordinated by some, somebody who’s running the show, and so on. But the causerie really depends on the spontaneity and the taste of each, each person. And apparently, it was really interesting.
Strogatz: What Tadashi is really known for is revealing subtle yet deep principles of, of math and physics, through incredibly down-to-earth homespun materials, things that — he calls them toys. He uses these ordinary materials to reveal extraordinary things about our universe. I also have found, in talking to him, that he’s argumentative in a pleasant way. His instinct seems to be to say, “No,” and then — not, not so much in a corrective — well, yeah, maybe in a corrective way.
But, but that whatever direction you try to take something, or if you think you anticipate where he’s going, he likes to stop you cold and say, “No,” and then, he’ll straighten you out and take it where he wants to go. So I have found that a little unnerving in the past, to be constantly getting jammed, you know. Like in improv, they, they say you’re supposed to say yes, “Yes, and …”I find him as a “No — no, and …” person. [LAUGHS]
Tokieda: One maybe key feature of surprise is that — the expectation. It goes hand-in-hand, but it is the obverse side of surprise. So you are expecting something, or you are expecting the universe to work in a certain way, or people around you to behave in a certain way, and the universe or the people don’t.
Strogatz: Yeah, yeah.
Tokieda: Suddenly — and you have to change your perception, your, your take on the universe, ever so slightly —
Tokieda: — but substantially, that’s —
Tokieda: — that’s kind of what surprise is. “Jeez, this does not fit what I have been thinking, what I have been assuming …
Tokieda: “So, I have to do something about it.” And that’s a surprise, and it’s, it’s pleasurable, but it’s also slightly disturbing.
Tokieda: And the mathematicians like to — and in fact, have codified this, this kind of thing, at least partly, and this is codified in the, in the concept and the word of “counterexample.”
Strogatz: Ah, okay.
Tokieda: So you sort of assume that under this condition and this condition, this and this conclusion must hold. Logically, it’s inescapable. But then, somebody comes along and says, “Well, look, here is a situation that satisfies all your conditions, but your conclusion doesn’t hold.”
Tokieda: And you say, “Ah, that’s a counterexample.” You have to actually revisit your logic, and you have to think a bit more carefully.
Strogatz: I see. Nice, so, so it’s not just an example; it’s counter to your expectation.
Tokieda: That’s right, it’s the counterexample.
Tokieda: So it is in fact really superbly psychologically important. So as, as such, I think counterexample —
Tokieda: — is really central to, if you like, the human civilization, if I may make a grandiloquent speech, and … You know, because that’s how we progress.
Strogatz: Oh, that’s nice.
Tokieda: We have to have counterexamples, and otherwise, we just keep assuming, assuming, assuming and just — we, we die the death of, you know, assuming everything. So counterexample is really a funny — and in some sense a subtle, and really an annoying concept, I think, for daily logic.
Strogatz: I agree, yeah, it’s, it’s extremely unforgiving, very inflexible.
Tokieda: Exactly. And, but —
Strogatz: I mean, it’s the highest possible standard of evidence, to demand no counterexamples.
Tokieda: Exactly, and that’s the world in which we mathematicians must work, and — this is, it, I think the con– the concept and the word “counterexample” has so much advertising, you know, to show people that there is such a world. And in fact, you can sometimes, if needs be, operate with that standard of rigor. You know, another example of this is — everyone knows, intuitively, and of course it’s used all the time in, in hard sciences — that “if a, then b” is logically equivalent to “not b implies not a.”
Tokieda: “If not b, then not a” — it’s called contrapositive. So the statement and its contrapositive are absolutely equivalent, and there is no difference between them.
Tokieda: And this is used intuitively and instinctively by everyone, and so on. However, let’s consider the statement and accept the statement: “If Steve doesn’t scold Tadashi, then Tadashi doesn’t work.”
Strogatz: If I don’t scold you —
Tokieda: Okay, so we —
Strogatz: — then you don’t work?
Tokieda: “If, if Steve doesn’t scold Tadashi, then Tadashi doesn’t work.”
Tokieda: Okay, so this is a statement we can accept, and we also understand what the intuitive background is, probably because Tadashi is a lazy bum and, you know, he needs to be scolded by, driven by Steve, and so on.
Strogatz: [LAUGHS] Okay, yes?
Tokieda: But it should be logically equivalent to say: If Tadashi works, then Steve scolds him.
Strogatz: [LAUGHS] Okay?
Tokieda: Which doesn’t sound equivalent at all, right?
Strogatz: [LAUGHS] No, it doesn’t.
Tokieda: “If, if Steve doesn’t scold Tadashi, then Tadashi doesn’t work,” it should be equivalent to say, “If Tadashi works, then Steve scolds him.”
Strogatz: Well, yes, although there feels like there’s a tense issue: “I may — I must have scolded you previously.”
Tokieda: Exactly. So, the, the point is that in, in daily language, an implication “if, then” statement always comes with a nuance of chronology —
Strogatz: Ah —
Tokieda: — as you say, tense or causation.
Strogatz: Ah, oh, okay, okay, uh-huh?
Tokieda: But in mathematics, and in fact in implications, logical implications … In a harder sense, there is no such nuance. All you are saying is that, “If you are facing this situation, then you can conclude this and that,” unambiguously and always, from that situation. That’s all there is. And so indeed, in order to make the second statement equivalent to the, sound equivalent to the first intuitively, we should have said: “If Tadashi works, then Steve must have scolded him —”
Strogatz: Right, right, right.
Tokieda: “— before.” So, but it can happen earlier or later in space-time, it doesn’t matter as, as far as mathematics is concerned. But, you know, people do of course grow up as human beings, and the natural instinct for causation and chronology, so they find this implication very, very confusing.
And I have, I’ve attended especially seminars among expert mathematicians where confusion arose about logic, because of this issue.
Strogatz: Hm —
Tokieda: So it turns out that even among professional mathematicians, but let alone, you know, among non-mathematicians, it often helps to explain things if you imbed a logical chain of reasoning into some kind of story.
Strogatz: After the break, how calculus explains the emotional nature of the seasons. Also, Tadashi thinks he’s figured out a puzzle from the fifth century B.C. We’ll be right back.
Strogatz: I wanted to know how Tadashi found his way into math.
Tokieda: I came into mathematics quite late in my — well into my 20s.
Tokieda: Before then, I had no exposure to mathematics.
Strogatz: It’s amazing.
Tokieda: In particular, I had never heard of calculus until well past my 20s.
Tokieda: And I started studying calculus, at some point, yeah, to teach myself calculus.
Strogatz: Sometime in your, in your 20s, maybe creeping up on 30.
Tokieda: Yeah. Yeah, and I was already, you know, I, I was lecturing in classical philology and so forth. I was a grownup, but I decided to study something else. So I taught myself calculus, and in fact before then, I had to teach myself, you know, high school algebra and analytical geometry, and all that.
Tokieda: But anyway … So, when I learned calculus, I learned the definition of the derivative, and that is the rate of change, if you like.
Tokieda: And before I could do anything with this, this definition — you know, you learn that, for example, derivative of x cubed is 3 times x squared, and things like that, later on. But before I could do anything with the derivative, because I didn’t have practice yet, I learned the definition of derivative, and I suddenly thought I understood something for the first time. We all feel — and generations of literature from all countries say — that autumn and spring have some special status. During those seasons, you feel that time is passing, fleeting very fast, and then you feel nostalgic about things, and you have romantic feelings, and so on, right? You know, flowers come, and leaves fall, and so forth, and so forth, yes.
Strogatz: Yeah, and Tennyson has that line: “In the spring, a man’s heart lightly turns to thoughts of love.”
Tokieda: Exactly. So in, in, in contrast, in summer, in midsummer and midwinter, you think that things are standing still, you have this feeling.
Tokieda: And I understood why we all have that feeling. Because if you plot anything that’s relevant like temperature or humidity and so on and so forth, as a function of time through, through the year, well, summer and winter are maximum and minimum points of that graph.
Tokieda: That’s where the rate of change, that is, the derivative, is nearly zero. And so where is the derivative maximum — positively maximum negatively maximum? Of course, in between: spring and autumn. That’s where things are changing very fast, and of course that’s when we have this fleeting and sort of ephemeral feeling, and therefore nostalgia and of course a very emotional reaction.
Strogatz: Oh, that’s an interesting point, I see. So that’s, so that’s a nice — well, really? So, is the claim that spring and fall are the — well, what, the most nostalgic months, the — I mean, because certainly winter is a very emotional time, as is summer.
Tokieda: That’s it, but the winter, because you, you kind of think about … But I think it’s true that people have this fleeting feeling and ephemeral feeling —
Strogatz: The fleeting feeling, yes, the ephemeral, okay.
Tokieda: — in, in spring and — and so, it can be sometimes positive, sometimes negative, or whatever, it, it, it can’t be classified along one axis, positive and negative. So I think spring and autumn have this, this, this feeling. And also I understood that — if now you take that graph, for example, if you sort of, kind of graph throughout the year, and then you stretch it very, very high and low, vertically —
Tokieda: — that means that you are living in a country where it’s very hot in the summer and very cold in winter, for example.
Strogatz: Oh, yeah.
Tokieda: Then that means that in spring and autumn, the, the slope or the derivative rate of change becomes bigger, right?
Tokieda: So, that means — and also, for a given range of temperature — let’s talk about temperature — that the period during which you fall into that temperature will be shorter, because the slope is higher.
Tokieda: So that means that things happen much faster and much more quickly. And so in countries like Russia, you have a much more intense feeling of fleetingness, and so … and I think it influences the Russian literature, as well.
Strogatz: What a nice idea. [LAUGHS]
Tokieda: But, but anyway, you understand all of this, without being able to calculate anything with derivative, just merely by learning that there is this way of looking at the universe: derivative.
Strogatz: Tadashi is always offering new ways to look at the universe. Sometimes, he asks us to look far back in history, to make sense of how we understand nature.
Tokieda: There is something interesting about the — clockwise about the hem— hemispheres. You know, why is clockwise clockwise?
Strogatz: Yes, I have no idea.
Tokieda: Why not the other way around?
Strogatz: I don’t know.
Tokieda: Why not the other way around? I think the only explanation is that, a long time ago, clocks were, were sundials.
Tokieda: So there is this stick standing, it’s called a gnomon, in the traditional term. And in the northern hemisphere, the sun rises east and kind of goes south and sets west. And if you think about how the shadow of the gnomon on the ground moves, it starts by being cast toward west, because the sun is east, and then goes north and then finally ends up being east. And that is clockwise.
Strogatz: Aha, so you think it’s a product of clocks somehow being a northern hemisphere —
Tokieda: I think — because on the southern hemisphere, the shadow moves counterclockwise.
Strogatz: Uh-huh? Hm —
Tokieda: Yeah, I think that’s probably the only explanation, yeah.
Strogatz: That, that, that clocks are an artifact of northern development.
Tokieda: I, I think it is hemisphere, hemisphere dependent. So if the, the clocks were developed and popularized on the southern, the — sundial, that is — on the southern hemisphere, it probably would have had the opposite convention.
Strogatz: Oh, that’s an interesting idea.
Tokieda: By the way, until the end of the 19th century, you know, clockwise and anticlockwise was still … Mathematicians like to call anticlockwise or counterclockwise positive and clockwise negative, because they’re perverse. But there were two conventions, and they were kind of coexisting.
Strogatz: Mm-hmm. It is, it is a little surprising that we use positive for counterclockwise; that’s the one we all learned.
Tokieda: Uh, it is a bit con– confusing, isn’t it?
Tokieda: And there is a, another sort of story in support of why clockwise should be positive, and this is really an extraordinary story. And as you said, I’m a classical philologist, and I think when you came to Cambridge and I was — did I show you and your daughters? — I think, the Renaissance library of my college.
Strogatz: Absolutely, yes.
Tokieda: Yeah, I, I … So I showed you one document which told the story, but I’d like to tell it —
Strogatz: Yeah, please.
Tokieda: —in public, as well. So Herodotus, who flourished around 500 B.C., he is the father of history, as he’s known, because, you know, he … Before then, Western civilization had no history at all. And so he’s the first person who wrote something that we can recognize as history. Of course, he was followed by Thucydides later on, and so on. But Herodotus had no documents, written documents, to work with in order to write history. So he traveled very, very widely, in particular, went to Egypt, and then wrote many, many things that he brought back from there. It’s very wonderful.
Now, in the fourth book of Herodotus — Egypt is the second book, but this story happens, appears on the fourth book. He tells the story of a pharaoh, Egyptian pharaoh, called Necho, about, about 700 B.C., so this is 200 years before Herodotus’ writing. And Necho, pharaoh of Egypt, wanted to find out what shape Africa was.
Tokieda: Very sensible question.
Tokieda: In other words, more concretely, whether it was entirely surrounded by sea — in other words, Africa would be an island, a big island — or whether land, land continued south forever. Very reasonable question.
Tokieda: And rather admirably, instead of speculation, Necho decided to find this, find out by observation. And to this end he sent out, he commanded the Phoenicians — the ancestors of today’s Lebanese and the best sailors of the time, and they founded Carthage, and so on — to sail off from the Red Sea, what we call the Red Sea now, and start going south along the African coast, you know, along the … where now Kenya is and so forth.
Tokieda: And they just keep going and, and see what happens. So they sailed off, and the Egyptians didn’t hear back from them for quite a long time [LAUGHTER], and so they were scratching their heads, “Well, what happened? You know, they just sailed off and no, no news.” Well, there was no, no, no, no transmitter for news, in those days. But they were surprised that, several years later, those Phoenicians came back through the Pillars of Hercules — and that’s the classical expression for Gibraltar —
Strogatz: Oh —
Tokieda: — in other words, from the other side —
Tokieda: — and came back to Egypt.
Tokieda: This caused an enormous amount of controversy, because the Phoenicians came back in a boat different from the boat on which they sailed off. So the Phoenicians said, “Yes, yes, we circumnavigated this entire continent, and it is an island, it’s entirely surrounded by, by sea.” But the Egyptians said, “Oh, we should have never trusted those Phoenicians, you know, look at the way they conduct business, they cheat all the time. They probably, you know, went off for a certain distance, and abandoned their boat, and cut across on foot, and built another boat and came back.”
Then the Phoenicians said, “No, no, no, it took a long — us a long time, because of course our boat got damaged in the course of the journey, and we had to land and sow and harvest to supplement our food, food stock, and then build another boat, and then keep going. And so, and then, and so, and that’s why, you know, we had to keep going along the entire coast, and that’s why it took us a long time.” And Herodotus, at the end of this remarkable passage, says that as far as he’s concerned, he doesn’t know whom to believe.
Tokieda: Yeah, and very, very good historian, but, he says, “There is some story that the Phoenicians are supposed to have been telling, which was told to me, later on, which is so strange that I have to write it down for posterity.”
Tokieda: He says that the — according to the Phoenicians, while they were going around this continent, for a period of time, they had the midday sun on their righthand side.
Tokieda: So, if you can picture going around the current map of Africa —
Tokieda: — uh, from the Red Sea, and going around clockwise, speaking of clockwise —
Strogatz: Yeah, yeah.
Tokieda: — and going, rounding the Cape of Good Hope —
Tokieda: — righthand side means that they had the midday sun not to the south, as we assume on the southern hemisphere, but to the north.
Tokieda: And, of course, that’s what happens on the southern hemisphere.
Strogatz: Yeah, yeah, yeah.
Tokieda: Yeah. [SPEAKS GREEK] is the original phrase. They, they had the midday sun to their righthand side. Now, this phrase that Herodotus reports 200 years after the Phoenicians said it, and the — to us, now 2,500 years later — proves to us that the Phoenicians must have made it. Because nobody on the northern hemisphere had the imagination to invent such a lie, in those days.
Strogatz: Maybe so. I mean, I don’t know about proof, but certainly very plausible, given that. That’s — you —
Tokieda: Oh, oh, I don’t think, I don’t think anybody —
Strogatz: You really don’t think —
Tokieda: — anybody had the imagination that there was a part of the, part of the universe where you would have the sun to the north, midday sun to the north.
Strogatz: Hm, but —
Tokieda: All the experiences of all the surrounding cultures of the northern hemispheric people had the midday sun to the south.
Tokieda: So mathematicians’ experiments — and in fact, I wanted to complain about, well, so lovingly complain about mathematicians. You know, I often —
Tokieda: — show various experiments and tabletop demos and so on to people, and hopefully they have some surprise element.
Tokieda: But in order to enhance the surprise, I usually ask my, my spectators or my, my friends to guess what’s going to happen.
Tokieda: Yeah? Of course, that’s very nice, because, you know, by guessing —
Strogatz: Of course.
Tokieda: By the way, guessing is extremely important, because if you guess right, you can be very proud.
Tokieda: And if it is wrong, of course, that’s an opportunity to become more intelligent.
Tokieda: And so, it’s really — but guessing also commits you to the problem, and so —
Strogatz: Yeah, yeah.
Tokieda: — really, every time you try to solve a problem, you should first guess. And if, if it’s a complete, you know, potshot, that’s fine. You, you should guess to the best of your ability. Anyway, I, for example, I tried to drop some — unfortunately, I can’t show you the experiment — next time we meet, I’ll do this. I tried to drop some coins and so on, and then, you know, they are going to do something funny in midair, and so on. But I ask them, “Well, guess what’s going to happen,” and usual people guess, “Oh, this is going to happen,” “No, no, it can’t happen,” um, “I think this is going to happen.” And of course they, many of them resort to the psychological approach, “Ah, if Tadashi’s asking, then I, I guess this way. But I have to guess the opposite because he’s a mean person,” and so on.
Strogatz: [LAUGHS] Yeah.
Tokieda: Okay. But the one common response from mathematicians is really delightful. When I ask them and I’m about to, you know, say, drop some coins in front of them and so on, and I ask them, “What’s, what’s going to happen,” their response is, “What are we assuming?”
Strogatz: Oh — [LAUGHTER] Wow.
Tokieda: It’s really wonderful. It’s what the French call deformation professionnelle, but it, it really says something about, I think, the mathematician’s culture, that we —
Strogatz: Let me try to translate that: professional deformation?
Tokieda: Yeah, professional deformation, yes, that’s right.
Strogatz: [LAUGHS] So, so, that your profession warps you in, in such a way that you’re —
Strogatz: — scarcely recognizable to the rest of humanity. [LAUGHS]
Tokieda: Yeah, that’s, yeah —
Strogatz: So that’s interesting, so they say, “What are we assuming?” You — then, when you — so they — will they refuse to guess?
Tokieda: So, well, well, I, I, what can I say? You know, I can say, “Well, please assume that you live in this universe, and [LAUGHTER] I’m about to do something actually real in front of you, and —”
Strogatz: Well, I would like to speculate what’s going on there, which is —
Strogatz: — because I — and I think it brings us into other interesting territory, here —
Strogatz: — about education and, you know — well, okay, so let me just come out with it. That, that there’s a very high premium on not being wrong in mathematics.
Strogatz: For reasons that we talked about earlier that, you know, one counterexample is enough to sink a whole theory.
Tokieda: Yes, yes.
Strogatz: So, so for that reason, I think we put such extreme emphasis on not being wrong —
Tokieda: That’s true, yes.
Strogatz: — that — which, which I think is very stultifying for our students. You know, because as you frequently emphasize in your teaching and your toys and so on that, that, being wrong, as, as you just said a minute ago, being wrong is a chance to learn something.
Strogatz: And the best teachers appreciate that, that they, they don’t punish students for being wrong —
Strogatz: — although, of course, you know, that’s, that’s a difficult issue. We have to start talking about —
Tokieda: Yes, mm-hmm.
Strogatz: — grading and what do you grade on. But still, if, if your main goal in life is to never — and notice I’m not saying it’s important to be right. I’m saying it’s important not to be wrong.
Strogatz: It’s a little bit different.
Strogatz: Because not being wrong is a very conservative attitude.
Strogatz: Being right is a more adventurous attitude.
Tokieda: Mm. Mm-hmm.
Strogatz: You have to take a chance. Anyway, I, so I, I feel like that’s an … You’re, you’re bringing up with that story an example of not just the deformation of mathematicians through their training —
Tokieda: Yes, I agree, I agree that mathematicians, really more than any other people, are afraid of being wrong, saying something wrong.
Tokieda: So they try to say something correct, but sometimes at the expense of, sort of, guessing something right.
Tokieda: But it is also a necessary part of our profession, because we really have to guard against those leaks, right? And it’s, it’s kind of a curse and, if you like …
Strogatz: Oh, you mean because we’re building this edifice that needs to stand for all time?
Tokieda: That’s right, I mean, yeah. And in, in other words, the other side of the coin — there’s always the other side —
Tokieda: — is that the, I mean, this thing that we are building is really something surreal, that it’s actually reliable. That the, you know, it’s, it’s going to hold up —
Tokieda: — no matter what. This — Probably what you’re saying is reflected in, in the cultural difference between … I, I think about physics a lot, nowadays, and most of my work’s in physics. But between physicist and mathematicians, when … Going back to the implication, “If then, if a, then b,” when a physicist says, “Well, if a, then b,” what they mean is … uh, let’s start with the mathematician. So, when the mathematician says, “If a, then b,” what they mean is, “Well, if you give me the condition a, then whatever else happens, it doesn’t matter, anything else can happen, I can guarantee to you at least b.”
Tokieda: That’s what the mathematician says.
Tokieda: That’s what “a implies b” means for a mathematician. But often, when a physicist says, “If a, then b,” what the physicist means is that, “Well, if you give me the condition a, and if nothing else happens, if it’s a sanitized situation and if you can allow me to idealize, then I can, I can tell you that b happens. But if you start bringing in some other stuff, well, then all bets off.” That’s the attitude. In, in other words, the … So, this logical implication means some, almost opposite things to a mathematician and a physicist.
Strogatz: Hm —
Tokieda: Yeah. And this, I think, is kind of the, connected with whether you, you wanted to avoid saying something wrong, or you wanted to sort of go out, stick out your neck and say something right. For example, let’s start adding up odd numbers, for example, 1 —
Strogatz: [LAUGHS] Okay.
Tokieda: — plus 3, that’s 4 — oh, that’s interesting 4 is 2 squared, right? Okay. How about 1 plus 3 plus 5? Well, that’s 9 — ah, that’s, 9 is 3 squared, it’s still a square. And then, 1 plus 3 plus 5 plus 7, jeez, that’s 16, again, a square. 1 plus 3 plus 5 plus 7 plus, now, 9, that’s 25, again a square. And so, it turns out that, as you know, this pattern continues, if you keep adding odd numbers and stop somewhere, it’s always a square.
Strogatz: Yeah, yeah.
Tokieda: Okay, so, the mathematical statement is: If you add up the first odd numbers, and then stop somewhere, the conclusion — so, let’s assume that you do this — the conclusion is that the answer is always a square.
Tokieda: Now, this is absolutely true, in whichever country you are, under whichever political regime you live, and whatever mood you happen to be in, you can be in a foul mood, you can be in a cheerful mood, you can be, have, have any kind of orientation in your life, or you can have, you can be living in any period of history — and in fact, we believe if we — if extraterrestrial intelligence exists and then we get in touch with them, and then, if they are sort of clever enough, they will agree, “Yes, yes, that’s absolutely true.” It doesn’t really matter, and in fact, if … Laws of gravity have nothing to do with this. Whatever, you can change any, everything else in the universe, but it’s true, that if you add up the odd numbers and stop somewhere, you always get the, a square.
Strogatz: Very good, yeah.
Tokieda: So that’s the kind of thing that the mathematicians will say, and believe, and I think it’s true. In fact, time and again, you know, the applications of mathematics which rely on this absolute sort of unshakable reliability.
On the other hand, a physical experiment — you know, I do this kind of thing all the time, for example, this Slinky experiment. Well, you know, you have to be careful. You have to actually stand high enough, and then, when you suspend the Slinky, that I mentioned, I have to let the Slinky settle down and quiet down, and then, you know, come to a stop. If it’s still shaking when I release, then you won’t see the effect. So — and also, if while I’m doing the experiment, somebody barges into the room and bangs into the Slinky, of course the experiment wouldn’t work as well.
Tokieda: Right? So, you know, there are all sorts of things that you have to remove in a physical experiment, and then, isolate, idealize and, if you like, sanitize the phenomenon, before you can say, if you, before you can make a pure conclusion.
Tokieda: Pure conclusion, which may be idealized, but which is suggestive and somehow captures the essence well enough, that even with … So, the “dirty noises” that you might actually in practice … it somehow tells you what’s essential about the phenomenon and what’s interesting about the phenomenon. That’s kind of the physical implication.
Strogatz: Right, right, right. So, yeah, it’s the example you’re mentioning reminds me of a little episode from the history of science, which is that Galileo, when he was discovering the laws of falling bodies, did experiments where he took a ball and put it on a, a ramp, which actually wasn’t —
Strogatz: — a ramp; it was more like a groove in a long narrow piece of molding.
Tokieda: That’s it.
Strogatz: And then he would roll ball down. Okay. So, as you know, he, he says in Two New Sciences that he, he made the ball very round, and he made the groove very smooth and very straight.
Strogatz: So he’s doing all these things that you just described, trying to remove —
Strogatz: — all the dirt —
Strogatz: — all the contamination —
Strogatz: — all the things that could deviate from the ideal case.
Strogatz: And actually —
Strogatz: — he stated the law of falling, using exactly what you just said about odd numbers. He called it the law of odd numbers.
Tokieda: Mm-hmm. Yeah, that’s right, that’s it, because the acceleration — that’s right, that’s right, yeah.
Strogatz: Yeah, so the square, you know, distance traveled is proportional to the time squared. He, he said —
Tokieda: Yeah, very, very —
Strogatz: — in the first instant, we roll one distant, one unit —
Strogatz: — then in the next instant — or actually, it wasn’t instant.
Tokieda: Yeah, yeah.
Strogatz: After one unit of time, you go some distance; in the second unit of time, you go three times that distance; and then in the third, five.
Tokieda: That’s right, that’s right, yeah, yeah, yeah, very, very good.
Strogatz: Isn’t that nice?
Tokieda: Yeah, very, very good.
Strogatz: Yeah, so it’s … so the thing you just mentioned is — and, and also, there’s two tendencies: the, the tendency to idealize —
Strogatz: — which led him to discover the law. But then the need to eliminate all the details that could corrupt [LAUGHS] or, or ruin the experiment.
Tokieda: That’s right.
Strogatz: Hm, interesting.
Tokieda: That’s right. So [NAME] said that, you know, contrary to what people think, you know, mathematicians are in much more intimate contact with reality than natural scientists.
Tokieda: And the reason is —
Strogatz: How is that?
Tokieda: Okay, a physicist can talk about an electron, but the closer you look at the concept x —
Tokieda: — the fuzzier it becomes, you start not understanding what it means. And if you are looking at it from far away and, and so on, you know, it’s a very useful concept, but the — and what is a chair, for example? Well, it’s — chair is — I’m sitting on a chair while talking to you.
But then, if you look really closely, it consists of molecules, and then, but they are not of course in a crystal structure and, you know, what is it? I mean, we don’t quite understand, right? Whereas a mathematician’s object, the closer you look, the clearer it becomes.
Strogatz: Oh, the statement is the objects of mathematics, not — I mean, mathematicians are not more connected to real reality — they’re connected to their own reality?
Tokieda: Right, so the real reality, not the physical reality, but the, somehow, the, the, the universe with which mathematicians tend to deal is much … in a really sharp focus.
Strogatz: I see.
Tokieda: And it is really nothing fuzzy.
Strogatz: I see. So the point being, we know, we mathematicians know our world better than the physicists know their world.
Tokieda: Well, in, in, well, in some sense. But it’s really in the sense of, you know, the — again, when we say an implication, we say, “Well, whatever else happens, we, I, I don’t care, I mean, I know that this is true.” Whereas, the physicist has to idealize and, and so they extract the essential feature, which might be masked by noise and so forth and so forth.
Strogatz: To a mathematician, the world of mathematical ideas is a world of perfect information. There is no noise, there is no fuzz, there is no contamination. Everything is as clear as it can be. Whereas for a physicist or a scientist, there’s always, you know, the intrusion of, of complication, friction, noise, things that have to be cleaned off before you can see the real phenomena of interest. And so that’s why he says that physicists are always going for the essential phenomenon, because the essence is the part that’s beautiful. You have to clean off the fuzz and the gunk to get to it. At the end of our conversation, I found myself wondering how do audiences react to Tadashi’s world, the world of playthings and toys.
Strogatz: Besides making toys and thinking about the — ’cause some people may be wondering — okay, I don’t want to be obnoxious —
Strogatz: — but, but just on behalf of any skeptical listeners who are thinking, “Oh, you know, this is a lot of cute whimsical gameplaying” —what’s the point of all this, really?
Tokieda: Ah, what’s the point of all this? That’s, that’s an interesting question. And, you know, the — and of course, in this conversation, I didn’t demonstrate anything, because the — you know, as I said, many of the things that we do depend on vision, depend on sight, and over the telephone conversation, we cannot do this. But, you know, when we meet we can, and then I, I can show people and so forth. And when I give, for example, public lectures, I do quite a few of these, and at the end there is in fact a question which is every more direct, shall I say, more, more in your face. Which is, somebody asks, “Well, you know, you’ve been showing all those, you know, counterintuitive, surprising phenomena — tabletop demos which we witnessed. We guessed and we guessed wrong, but we understood how it worked and then how it’s connected to the rest of the universe, and so forth, that’s all good. But does this have any practical applications?”
And this question is very, very common, and in fact almost canonical, as mathematicians like to say, “Does this have any practical application?” Six words, six exact words.
Tokieda: It’s as if somebody taught all these people to ask this question in exactly those words.
Tokieda: Yeah. And it’s very interesting. Now, the, my response — and by the way, I do many, many other things, beside designing and sharing toys and so forth, but let’s focus on this aspect of what I like to do, and so on. My response is, well, suppose that my answer was, “Well, this allows us to make the trunk of an elephant longer.
Tokieda: “Well, would you accept that’s a, as a practical application?” And the answer is probably no.
Strogatz: No, maybe not.
Tokieda: Yeah? So this shows that we have to agree a priori, ahead of time what constitutes what a satisfactory practical application —
Strogatz: Good, yes.
Tokieda: What do we accept as a practical application —
Tokieda: — a priori, because, you know, apparently, you are not going to accept the prolonging the, the trunk of an elephant as a practical application.
Tokieda: “Well, too bad, I thought it was a practical application,” “No, I don’t think it’s a — “ and so on and so on. We have to agree on something. So it’s very important, then, to search our souls, so to speak, and, and decide or start glimpsing, what we think is a practical application, and what would satisfy us. And then, we sometimes go into a psychoanalytic session with, with my audience and, and, and now it’s my turn to probe them and so on. And it’s very, very interesting that often the answer — not always, but very often — the answer converges to two things: Something is a practical application if, for example, it allows me to make a billion dollars very quickly.
Strogatz: Mm, okay.
Tokieda: Or if it allows me to kill a million people very quickly.
Strogatz: Mm, awful.
Tokieda: One or the other. Yeah.
Tokieda: And people who arrived at this con– conclusion through, if you like, my– my– myopic and rather sort of mean-spirited sort of guidance are very surprised by their answers, surprised —
Strogatz: Not medical applications?
Tokieda: So sometimes medical comes up, but very often it doesn’t, which is very interesting.
Tokieda: And so, and — anyway, so, I’m just reporting. And then they are really shocked by their own answers, and then start revisiting what they think is practicality, and so on and so on. And then, you know, my — the real answer is, well, you know, “I don’t know what’s practical application for you, but I have one practical application of all this. When I show this to children, they seem to be happy.”
Strogatz: [LAUGHS] Beautiful. Well, thank you. That’s a very sweet note. I think we should, we should leave it there.
Tokieda: Well, and if that’s not a practical application, what is a practical application?
Strogatz: Next time on “The Joy of x,” Cori Bargmann takes us into the brain of her comma-sized scientific muse, the worm.
“The Joy of x” is a podcast project of Quanta Magazine. We’re produced by Story Mechanics. Our producers are Dana Bialek and Camille Petersen. Our music is composed by Yuri Weber and Charles Michelet. Ellen Horne is our executive producer. From Quanta Magazine, our editorial advisors are Thomas Lin and John Rennie. Our sound engineers are Charles Michelet, and at the Cornell University Broadcast Studio, Glen Palmer and Bertrand Odom-Reed, though I know him as Bert. I’m Steve Strogatz. Thanks for listening.
[End of audio]