The Joy of x

Tadashi Tokieda’s Special Kind of Magic

The mathematician Tadashi Tokieda and host Steven Strogatz explore what we can learn about the world from simple “toys” with remarkable physical or mathematical properties.

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 Production and original music by Story Mechanics.

Listen on Apple PodcastsSpotifyAndroidTuneInStitcherGoogle Podcasts, or your favorite podcasting app, or you can stream it from Quanta.


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 —

Strogatz: [LAUGHS]

Tokieda: — but I don’t remember the last time I felt hungry, since early childhood.

Strogatz: Really?

Tokieda: But —

Strogatz: Wow.

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.

Strogatz: [LAUGHS]

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: [LAUGHS]

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 —

Tokieda: Mm-hmm?

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 —

Tokieda: Mm-hmm.

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 —

Tokieda: Mm-hmm.

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.

Strogatz: Uh-huh.

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 —

Strogatz: Yeah.

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.

Tokieda: Mm-hmm.

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 —

Tokieda: Uh-huh.

Strogatz: — jump over the top —

Tokieda: Yes.

Strogatz: — and climb down to the next stair —

Tokieda: Yeah.

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

Strogatz: Mm-hmm.

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 —

Strogatz: Yeah.

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.

Strogatz: Yeah.

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 —

Tokieda: Yes.

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 —

Tokieda: Yeah.

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.

Strogatz: Mm-hmm.

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 —

Strogatz: Yeah.

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 —

Strogatz: Mm-hmm.

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.

Strogatz: Yeah.

Tokieda: In other words, the intelligence or the information content of the magic trick is in the hands of the magician.

Strogatz: Ah —

Tokieda: Yeah?

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.

Strogatz: [LAUGHS]

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.

Strogatz: [LAUGHS]

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 —

Strogatz: Yeah.

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.

Strogatz: Mm-hmm.

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 —

Tokieda: Yeah.

Strogatz: Causerie

Tokieda: Yeah.

Strogatz: — but tell us what you — what, what, how, how do you understand it?

Tokieda: Well, causerie is obviously a —

Strogatz: Yeah.

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.

Strogatz: [LAUGHS]

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.


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