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16| Gábor Domokos — The Gömböc, a shape at the limit of possibility
120 Plays1 year ago
The Gomboc is a curious shape. So curious many mathematicians thought it could not exist. And even to the untrained eye, it looks alien: neither the product of human or natural processes.
This week Gábor Domokos relates his decade-long quest to prove the existence of a (convex, homogenous) shape with only two balance points.
The Gömböc is not just a mathematical curio, its discovery led to a theory of how "things fall apart", of the processes of abrasion that — whether on Earth, mars, or deep space — ineluctably reduce the number of balance points of objects.
The Gomboc is the shape all pebbles want to be, but can never reach.
Show notes at multiverses.xyz
(00:00) Intro
(2:40) Start of conversation — what is a Gomboc?
(4:30) The Gomboc is the “ultimate shape” it has only two balance points
(5:30) The four vertex theorem: why a 2D shape must have 4 balance points
(6:30) (almost) nobody thought a Gomboc existed
(8:30) Vladimir Ilych Arnold’s conjecture
(9:00) Hamburg 1995, the beginning of a quest
(10:30) “Mathematics is a part of physics where experiments are cheap”
(11:50) A hungry scholar sits next to a mathematical superstar
(13:00) Ten years of searching
(15:00) Domokos and Varkonyi's gift for Arnold
(15:30) Arnold’s response: “good, but now do something serious”
(16:50) We cannot easily speak about shapes.
(18:00) A system for naming shapes
(21:00) “The evolution of shapes is imprinted in these numbers”
(21:50) Pebbles evolve towards the Gomboc, but never get there
(24:50) How to find the balance points of shapes by hand
(30:00) Physical intuition and empirical exploration can inform mathematics
(30:30) A beach holiday (and a marital bifurcation point)
(34:00) “No this was not fun, it was a markov process”
(36:40) Working with NASA to understand the age of martian pebbles
(38:20) An asteroid, or a spaceship?
(43:00) The mechanisms of abrasion
(45:50) The isoperimetric ratio — does not evolve monotonically …
(47:50) … But the drift to less balance points is monotonic
(49:00) The process of abrasion is a process of simplifying
(50:00) We can name the shape of Oumuamua because it is so simple
(51:00) Relationship between Gomboc and (one way of thinking about) entropy
(55:00) Abrasion and the heat equation — curvature is “like” heat and gets smoothed out
(58:00) The soap bar model — why pointy bits become smooth
(1:00:00) Richard Hamilton, the Poincaré conjecture and pebbles
(1:04:00) The connection between the Ricci flow and pebble evolution
(1:09:00) Turning the lights on in a darkened labyrinth
(1:12:00) The importance of geometric objects in physics (string theory)
(1:13:30) Another way of naming natural shapes: the average number of faces and vertices
(1:15:00) “Earth is made of cubes” — it turns out Plato was right
(1:16:30) Could Plato’s claim have been empirically inspired?
(1:17:50) “Everything happens between 20 and 6”
(1:18:30) The Cube and the Gomboc are the bookends of natural shapes
(1:19:30) The Obelisk in 2001 — an unnatural, but almost natural shape
(1:22:00) Poincaré on dreaming: genius taps the subconscious
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Transcript
Mathematical Discovery Without Advanced Tools
00:00:00
Speaker
What does it take to discover something? Perhaps you need a powerful telescope to look for distant planets, or a high-energy and very expensive collider to look for new kinds of particles. But how far can you get with no tools? How far can you get with a mixture of mathematical intuition, a lot of perseverance, and a little bit of luck?
The Gombots: A Revolutionary Shape
00:00:22
Speaker
This week's guest is the Hungarian mathematician Gábor Demorkosz.
00:00:27
Speaker
Gabbel discovered an entirely new shape. And you might think, well, that's not that impressive. I can just draw a squiggle on a piece of paper and say, that's a new shape. And you might be right, that might be a shape that's never been seen before.
00:00:41
Speaker
the shape that Gabor and his collaborators discovered, the gombots, has an unusual property. In fact, a property so unusual that many mathematicians thought that this shape simply couldn't exist, neither in nature or even in the abstract space of mathematics.
00:00:57
Speaker
For me, there are many fascinating aspects to this story. One is the shape itself, which we'll discuss. One, a shape that doesn't seem to exist naturally in nature, although now it's been manufactured.
Gábor Demorkosz's Journey and Challenges
00:01:11
Speaker
But then there's the quest to find it, which took over a decade. And in relating it, Gabor gives just, I don't know, just beautiful anecdotes, but also one gets a feel for the frustration that he felt at times.
00:01:25
Speaker
But then there's a whole program of research which flourished from the discovery of this shape and from the kind of tools that it employed, particularly the way of categorizing shapes by the number of balance points that they have. And this is just something which, if you listen to this podcast, you'll learn how you can name or categorize the trillions of pebbles that you can find on a beach. And not only that, how you can pick up a pebble in your hand
00:01:52
Speaker
and feel its age, just by feeling for the number of balance points it has. And this is something universal, something geometric. So this has been applied to understanding rivers on Mars and the pebbles there, and also understanding putative alien spacecraft, which turn out to be more likely to be a comet, not to mention turtles. So there's a few loose threads right there.
Properties and Theorems of Gombots
00:02:19
Speaker
Listen on and I promise these will get tied up.
00:02:21
Speaker
I'm James Robinson and this is multiverses. Welcome to multiverses. Thanks for joining me. Thank you for having me. It's a pleasure.
00:02:48
Speaker
So there's many interesting strands to your research, but it seems that there's a kind of central idea that they've flown from, which is this notion of a shape which many thought was impossible. So many mathematicians thought that there was
00:03:10
Speaker
a three-dimensional shape that had properties that just couldn't hang together. You couldn't have this shape, not only in nature, but it just didn't work as a concept. And so there were mathematicians who were trying to prove as a theorem that this shape couldn't exist. But then the story turned out quite differently. So perhaps you could tell us a bit about what that shape is, what the properties were, and
00:03:39
Speaker
that made it seem that this thing couldn't exist. That's very abstract. Hopefully you can make it a bit more concrete. Well, the properties of the shape are rather simply described. If this shape, this, the name of the shape is the gumbet. And if you put such a shape on a table, then it will always get back to the same stable position like a weebl.
00:04:07
Speaker
And the other equally important or very even more important property is that it only has one unstable balance point. So one stable to which it returns and one unstable from which it runs, so to say. The stable point is very easily visible in an experiment. The single unstable point is much harder to see, but it can be seen. But that is not a tabletop experiment.
00:04:37
Speaker
Now, this shape, it is the ultimate shape because we know that a smaller number, so this has two balance points, as I said, stable and unstable, and a smaller number cannot exist. That is a fact.
Vladimir Arnold's Influence and Insights
00:04:53
Speaker
And in the plane, so in two dimensions, it was known that the minimal number is four. It is always an even number. And the minimal number in the plane is four,
00:05:06
Speaker
And this fact, so if you take a planar disk and let it, a homogeneous disk, and let it roll along its perimeter, then like an ellipse, then it will always come to rest at two places at least. And between these two places, there will be other two which are the unstable points, like the ends of the long axis.
00:05:33
Speaker
Now, this was known, and the 3D version, and this is connected to another theorem, which is called the four vertex theorem, which says that a planar smooth curve has at least four extrema of its curvature. So this seemed like a well-established mathematical fact, and people were looking for a generalization in 3D,
00:06:03
Speaker
And one possible generalization was that also in 3D, the minimal number of balance points would be 4. And this was a big temptation. So as you said, people tried to prove it. It looked like a beautiful serum. And there was no counterexample known. So nobody has ever seen an object which had less than 4.
00:06:31
Speaker
Well, when you are kind of fumbling around and this mathematics is not just about the proof, it is about intuition. And that was a strong intuition that for the past few thousand years, nobody has seen anything like this. It looked like a fundamental thing. And if nobody has seen it, that probably doesn't exist.
00:06:58
Speaker
So people were putting their bets on the non-existence, mostly. We should say here as well, I was just going to say, I think one thing people might be wondering as well, what about the weebl, which you mentioned very early, but the reason that has... The weebl has this property, as you point out, yes, it has the property of having just two balanced points, but
00:07:20
Speaker
it has another property, it has an added weight, which is sitting at the bottom. So if you are admitted to put in weights, then it is very easy to do this. And until now, I forgot to mention, oh, I think I mentioned once that we are talking about homogeneous objects. So the material is completely uniform. And then
00:07:42
Speaker
Everything is controlled just by the shape. So the geometry is translated directly into mechanics. There is no additional information. And so people were trying to prove this. And with confidence, I can say the people who were trying to prove this were very much more versed in mathematics than I am, probably brighter than I am.
00:08:13
Speaker
In mathematics, no matter how bright you are, if you are trying to prove something which is not true, then you are bound to fail. And the story took a turn when in 1995, Vladimir Arnold, who was at that point one of the most celebrated mathematicians and a little good reason,
00:08:38
Speaker
He just offered the conjecture that there could be a counterexample. Now, if anyone else would have offered this conjecture, it would not have been taken seriously. But since he was the man who had the best chance to prove anything about this problem, this was a very, very strong fact.
00:09:05
Speaker
And I was a person who got lucky and he just told me this conjecture.
Applications of Gombots in Science
00:09:11
Speaker
So that was kind of a personal luck.
00:09:19
Speaker
Yeah, I was going to say, I've heard you tell this story, and there's an element of luck, but there's also some persistence on your side to try to talk to Arnold, who was, as you say, just this mathematical superstar at the time. And I think you had to... Yeah, he was.
00:09:38
Speaker
Is it right you had to sort of, you kind of had to pay to access him, as it were, at the conference? Yeah, yes, indeed, indeed. So, it was in Germany. It was still before the UWELL, so they had the German mark. And I was a young postdoc or something like that.
00:10:01
Speaker
I got like 60 German marks from the Germans as a stipend to come for one week to a conference to which I only went to see Arnold live. And he was one of my main heroes at that point. So I saw that he's
00:10:20
Speaker
As a young man, I saw that he's a kind of scientist whom I really admired because he was not just a mathematician. He was a much broader kind of thinking person. He told that mathematics is part of physics where experiments are cheap. And that was his approach to science. So I really wanted to see him. And when I got there,
00:10:49
Speaker
went to his talk, which was given to a huge crowd, and nobody understood the talk, but everybody understood there was a recurrent theme that some quantity was bigger or equal than four. And the number four was something everybody could understand. But then I realized that
00:11:12
Speaker
This might be the same number four I have been thinking about. So I was with my friend Andy Ruina and Jim Papadopoulos. We proved the case, the 2D case earlier. And then we started to think about the 3D, but we just couldn't get a hold on that problem. And I saw that this might be related. So this number four could be the same number four because Arnold was talking about many subjects.
00:11:41
Speaker
But then I could not access him because everybody wanted to talk to him. And as I said, the Germans advertised that if you pay 30 marks, which was 50% of my budget, you can be seated with a celebrity of your choice at lunch. And that was this lunch, which I still remember. I paid the money.
00:12:06
Speaker
I went on a diet, a very slim diet for the rest of the week. And I was seated at the same table, but as due to the efficiency of the Germans, there were 15 other people. And every single person wanted to show something to Arnold, so he couldn't eat. And then he just asked at the end of the lunch, what is your result, young man? And I told him there is no result.
00:12:34
Speaker
It was so awkward that I wanted to get out. And then finally I got lucky and I still could talk to him and then he knew I came from Hungary because of my badge and he knew I paid the 30 marks so he knew I had to have a very good reason to go to that lunch. So he basically insisted that I tell him what it was about and told him and then
00:13:00
Speaker
It turned out I did not understand much of this problem. And this conversation was a little bit awakening for me because it turned out that I did not understand the problem well. And he told me that probably there could be a shape out there. And then he left. And if anybody else would have told me in any other circumstances, I wouldn't have paid so much attention. But then I saw that
00:13:27
Speaker
this could be really interesting and I was digging in for 10 years and I found many properties of this shape which I don't didn't know whether it existed but I gradually I realized that if it existed it is really a miraculous shape and then I was again lucky because I got a brilliant grad student Peter Varkoni and
00:13:55
Speaker
Then the problem started to converge on a really nice mass problem. Originally, it was more like a philosophical problem, whether it existed or not. And how to put, it was very hard to grab this problem because there is no algorithmic approach to find such chips, even today. With computer, you cannot do it.
00:14:21
Speaker
And I was even searching among pebbles to find this shape, and I couldn't. So it was a desperate thing. And in these 10 years, there were ups and downs, and in the downs, sometimes I started to doubt whether Arnold was right, sometimes he was wrong, most of the time he was right.
00:14:41
Speaker
But then we could come up first with a theoretical construction proving that it existed, but that construction could not be manufactured. And then another year later, we could produce something which could be manufactured. And the first copy, which we gave the number number one, we brought to his birthday to Moscow.
00:15:08
Speaker
and that was a very interesting event so we in many ways and we gave it to him and we got a lot of feedback and part of the feed it was in his style the feedback was definitely in his style so he was he was
00:15:26
Speaker
Basically, he told us it was a good job, but now we should start to do something serious. That was the main message. But otherwise, he immediately pointed out that this finding
Naming and Categorizing Natural Shapes
00:15:40
Speaker
is a mathematical thing, but its significance is probably not in pure mass. It's probably in natural sciences. That was his basic intuition. Immediately, he
00:15:54
Speaker
told us to look for pebbles and to shapes and at that point I did not understand again what he was talking about. It took me again a couple of years and unfortunately he died in 2010 so he couldn't see the second wave of ideas coming from his intuition
00:16:17
Speaker
getting into being realized, but that proved to be probably more fruitful than... I mean, it reached many more people, because the Gomberts were very popular in many ways, but it was not clear at the beginning where this would lead us. And his suggestion that
00:16:43
Speaker
So the number two is somehow hardwired into this shape. And it is a philosophic, it's not a philosophical problem, it's a more physiological problem, how we think about shapes, because the brain is divided into left and right. And the part of the brain which has the linguistic skills and the numbers and so on,
00:17:11
Speaker
is different from the part which has this kind of geometric intuition. And as a result of this, we cannot speak about shapes. We have no words for shapes. When mankind is conquering anything, like an island, the first thing is to give it a name. Or we discover anything, then we give it a name.
00:17:40
Speaker
But if you look at the shapes, very few of them have names. We have trillions of shapes around us, literally trillions of shapes. And we have, if I ask anyone, how many names can you give, you can give a dozen names. And that is a big, big handicap. Because if you cannot talk about something, that is
00:18:08
Speaker
Even in real life, if you are not admitted to talk about something or you are unable to talk about something, it is very hard to think about it. And then despite landing on Mars and all that kind of achievements and having the iPhones, we are still severely handicapped by talking about shapes.
00:18:30
Speaker
And this idea is not changing this, but a little bit, a little bit, because this gives a way to assign numbers to shapes. Once you get numbers, it is like a name. So numbers and words are basically the same. And you can assign names
00:18:52
Speaker
In many ways, you can assign names in an arbitrary way. You go to the beach and give a name to each pebble. But that is meaningless. We would like to have names which are encoded by nature. So we want to discover what is the name of that shape. We don't want to give it a name. And we have to acknowledge that there are many more shapes than we have names.
00:19:17
Speaker
But still, if we can give a meaningful name to a collection of shapes, that is already a big step forward from where
Philosophical and Practical Shape Studies
00:19:29
Speaker
we are now. We have names for some idealized shapes, which I would call degenerate, like the sphere or the cube.
00:19:37
Speaker
physically you cannot have a sphere because if you have an arrow by an atom it's not a sphere anymore you cannot have an exact cube there are ideal shapes like platonic kind of ideas for real shapes we have no names and
00:19:54
Speaker
And this gives you a chance, this kind of idea. This started an avalanche, because it turned out that you can give it a number. The gumber has the number 2, and then you can talk about other shapes having other numbers. And actually, it has two numbers, one and one. So we have a table.
00:20:16
Speaker
And then it turns out in each field of this table, say 2-2, which is the ellipsoid, you can do subdivisions. And you can go in and you can do a catalog which is more detailed. And then you can dig in even more and you can do an even more detailed catalog. This catalog will be never as fine as many shapes you have. It will always give you categories.
00:20:45
Speaker
But because these categories are defined by nature, so we are not inventing these numbers, we are taking a pebble, taking an asteroid, taking whatever shape you give me, and we determine these numbers which have been inscribed by nature.
00:21:03
Speaker
Therefore, this is a natural catalog. And these numbers are telling. They tell us where this shape came from, how it evolved, because the evolution of shapes is imprinted in these numbers. These numbers evolve with the shape. And there are very simple rules. So one very simple rule is that, for example, in an aberration process,
00:21:31
Speaker
the general tendency is to get rid of this balance point. So this number is getting reduced, no matter how you do it. If you do it on a large scale, the statistics is it starts from somewhere, which I will explain later, and then it gets reduced. So they move in some sense, they move towards this combat shape, which has the minimum number, but they never get there. So this gives you a kind of a good view.
00:22:01
Speaker
But I view from a very complicated process, which is universal in space and time. It has been going on for billions of years. So it's a useful thing. So Arnold sparked this whole thing. He was the mastermind. And yes, we were the lucky guys at the right time at the right place. And we tried to make the best of it.
00:22:30
Speaker
Well, I think you understate the persistence. We can touch in on something. We were persistent. I think it's been told to me, you know, everyone has luck, but it's really how many times you roll the dice, right? That's what separates success from pain. I acknowledge that, yes, you need to have persistence. Yes, I take credit for
00:22:59
Speaker
believing are not for an extended period of time. And then yes, there was a piece of luck. Yeah. Yes. I want to. Yeah. So I want to read back some of the things you just said, because I think this is really, like you say, this is one of the. This is perhaps the bigger idea. Many people will be familiar with the with the Gombok or the Gombots.
00:23:21
Speaker
because it is such a fascinating shape. It's a wonderful desk toy, the way it sort of rolls in such an unusual way, unlike anything else in nature. But the bigger idea is how do you talk about shapes? And as you mentioned, we cannot speak thereof. We must remain silent. If we don't have a way of talking about pebbles or other natural shapes, we can't think about them.
00:23:52
Speaker
And this idea of classifying them by the number of stable and unstable equilibria, which you say gives you this kind of infinite chessboard, if you like, with at the origin the gombots at one, one. And then from that you have shapes with
00:24:13
Speaker
one stable, two unstable, or two stable, one unstable, and in fact there's a way of showing that as long as you have the gombots, all those other shapes are possible because there is a way of changing, you know, knocking a corner off or what have you, knocking a bit of kernovich off to create every other shape from that. So
00:24:35
Speaker
But it is an immensely, I think I want to ensure that every listener, when they finish hearing this, is able to go to the beach and actually have a way of thinking, a new way of thinking about pebbles, at least we should say the the convex pebbles, so the ones that don't have these sort of inward curves.
00:24:54
Speaker
which are probably about half of pebbles are convex. And all you need to do is pick up a pebble and you can quite easily find the stable balance points by just finding where it rests down and just looking at a pebble. You'll have a very good intuition of where those balance points are, which way you need to turn it to find a balance point. And then to find the unstable ones,
00:25:21
Speaker
If one thinks of an ellipsoid pebble where, like you were saying, it's got this long edge, it's rather long and thin as it were, often you can see again. But if you can't see, you kind of pick it up in your hands and you put it on
00:25:43
Speaker
you support it so it doesn't wobble in one plane and let it within another plane kind of find its stable point. Is that a good procedure, have I? Or maybe that's better for saddle points, which are another form of equilibrium. But probably this is easier if people actually try it. Yeah. So if you want to do this with a hand experiment,
00:26:13
Speaker
it is very easy to find the stable ones, and if the pebble is sufficiently flat, then it is very easy to find the unstable ones either also, because you stabilize it in the plane of this flatness, where it is very flat, and you roll it around, and in that roll, you will feel there are stable points constrained to that plane and unstable points in
Abrasion Processes and Shape Evolution
00:26:41
Speaker
that plane.
00:26:42
Speaker
Now, the points which are stable in that plane, but otherwise they are not stable, they are the settles, and which are still unstable, even constrained to the plane, they are the two unstable points. So this is not a complicated experiment, and some geologists are already doing it. On the other hand, of course, for a complicated shape, this doesn't work.
00:27:09
Speaker
And I mentioned that there are finer subdivisions, not just the number of these points, but also their arrangement can be found. These are identified by graphs. There are several types of graphs which tell you how these shapes are arranged on the surface of the pebble. And those are really not forehand experiments. But now we have computer programs which
00:27:38
Speaker
If we have a scanned pebble in 3D, then we can tell these numbers. But it is fun to go to the beach and have a look. I think it is for maybe 100 pebbles, but I know you spent a whole holiday with your wife looking at things. I spent a holiday with my wife. I still didn't know whether there was a gumburt.
00:28:04
Speaker
You know, this is a mathematical problem, seemingly. But if you go to the beach and you find a good shape, then you solve the problem. Assuming the shape is homogeneous, but most pebbles are, well, then you can reconstruct it and check it. But you have a good chance to solve the problem by finding a suitable pebble.
00:28:28
Speaker
But I didn't find them. Then we did 2000 pebbles. But since we have done many more pebbles with my students and...
00:28:39
Speaker
it is really, you are gaining a lot of intuition. And it is not just about the mathematics, you learn. So we spent days and weeks and months, I can tell you a story. So not just here in Budapest, student of mine went to Southwest Australia,
00:29:07
Speaker
to measure the pebbles. Another two of them went to New Zealand, and then somebody went to Puerto Rico. So we participated in expeditions, basically, with geologists to get a feel on this. Because it is not just about the mathematics. You want to be sure that the data you are talking about is robust. And that is not
00:29:35
Speaker
You cannot get any kind of intuition about this without wading into the rivers and creeks and picking up the pebbles and measuring them and comparing the numbers. Now, after some years, we have a pretty good understanding and we feel that this is robust data. I think that there is a really important point here that actually we often find that
00:30:05
Speaker
things are sort of discovered in mathematics and then find some application within physics later and someone comes up with some, I don't know, imaginary numbers or something and oh actually this turns out to be really useful in quantum mechanics and all over the place.
00:30:22
Speaker
But there's almost an element of this story, which is the reverse, even though you didn't find the gom box on that beach with your wife, who we have to give a lot of credit to as well here for accompanying you in this quest. She didn't find a divorce. Some people might have done this. I mean, we went for a holiday and I suggested to the social program that
00:30:46
Speaker
Each evening, we pick up, say, 40 kilograms of petals at the beach, bring to the hotel and then classify them.
00:30:56
Speaker
based on the Poincare-Hoff theorem. This is the Poincare-Hoff theorem, which gives you the numbers, the relationship between these numbers. So some marriages. So this is a bifurcation point for any marriage, because it can go two ways. But yeah, so yeah, by the way, the Poincare-Hoff theorem, I guess that says that relates. I mean, many people will think of that about
00:31:23
Speaker
in terms of hairy spheres and not being able to comb a sphere. But actually it also shows that there is a relationship between stable, unstable and number of saddle points. Which sort of means if you know
00:31:38
Speaker
you can have this plane categorized in any two of those three variables and then you can derive the third one. But the point, yeah, I think one thing I've heard you mention is that when you went, presented this, the first Gomberg to Arnold and showed him the work that you'd done on it, you were surprised to see that he was so interested in what you'd done at the beach because for you that had seemed like a kind of fruitless,
00:32:05
Speaker
in Denver. But I think it was, I think there was maybe a slide where you showed sort of, kind of like a, almost like a set of boxes, which represents that plane, a part of that plane. So you had a box for one, one empty, because you didn't find a gone box. But then you had a box for, you know, one stable or two unstable and two stable equilibria.
00:32:33
Speaker
And that was filled with pebbles. There was a lot in that box, which is weird because it's so close to the Gombok box. And then to the right and below that box, as it were. So if you think about going to three stable equilibrium and three unstable equilibrium, moving away from that point, you had boxes with fewer pebbles. And here's some
00:33:00
Speaker
Arnold saw that as kind of empirical data, I guess. But he also recognized in it, perhaps, that there was an echo of some kind of fundamental process going on. You mean that we saw it, which I didn't see. So I showed this slide as to illustrate a fun story and to illustrate a major failure. So I saw that we failed.
00:33:25
Speaker
And after the talk, I met him in the elevator, I think, or somewhere. And, oh, after the talk, he made a comment about the pebbles, which I didn't understand. And then after the next day, I met him in the elevator and he said, did you understand my comment? And I said, sorry, I didn't. And then he said,
00:33:49
Speaker
But I enjoyed your talk. And do you know which was your best slide? And I was guessing and guessing and guessing. And then I guess this one with the pebbles. And he says yes. And I said, it was it was fun. I said, he said, no, that was not fun. That was the Markov process.
00:34:07
Speaker
And so he immediately saw two layers behind this. And he was right again. So these pebbles are migrating in this table based on a very strict rule, which is a stochastic process. But stochastic processes are very well-defined things.
00:34:32
Speaker
And they are migrating in this table. And these process are bringing them randomly. So there is some randomness. It is like Brownian motion. Brownian motion with a drift.
00:34:45
Speaker
Brownian motion would not bring them anywhere, they would just wander around. But Brownian motion with a drift, it brings them into this corner, but almost never does it bring them beyond the boundary of one and one. They don't get there. And we know why. But the whole process is strictly controlled.
00:35:05
Speaker
by this motion. And what Arnold saw, what his eye saw, my eye saw it only three years later, that if you apply this process to anything, then you get a very, very defined distribution in the end, which we call a stationary distribution. So you put down million pebbles and you play this game,
00:35:30
Speaker
And you wait a million years.
Shape Evolution in Space
00:35:32
Speaker
And then you will have a stationary distribution of bubbles in these boxes. And he saw that what we measured was very, very close to that distribution. I, of course, I did not. But then after some thinking and some learning, because then he told us, now you have to learn mathematics.
00:35:53
Speaker
Of course, I didn't know what was, how did such a Markov process really work, so I had to learn. But after learning, after doing computations, we saw that, yes, this is the governing process. And again,
00:36:13
Speaker
Well, I wish I wish I wish I could have shown you some later results, shown him some later results as well, because this led to some interesting discoveries. So it is not just it's a good feeling to give names, as you pointed out, it makes you more comfortable. But nowadays, people in science asking for results. Philosophical advantage is not counted as a result. But
00:36:43
Speaker
we could translate this into some tangible results, which the scientific community recognizes as a result. So we were working with some NASA people to unlock the history of patterns, which have been photographed by curiosity on Mars. And based on their current shape, we could backtrack
00:37:08
Speaker
the pebbles and that we could tell, based on the current picture, we could tell how big the pebble originally was. Now, if you tell that to a geologist, he becomes a happy person or she and tells you how long the river was. So this sounds like an outlandish question, but they know the gravity, they know the kind of frog, they know everything, they know the slope,
00:37:36
Speaker
If you tell this pebble was originally this big, they tell you how long the river was. And that was the question NASA was actually looking at. So that is something which you can sell on the scientific market. You cannot sell this kind of philosophical jazz that you give names to shapes and then this is a cognitive advantage. And so this is not something you can
00:38:02
Speaker
actually market as a scientific result. For me, this is very important, but then you have to convert it into things which society fears it's useful. And we had moderate success with, we had some such achievements, mostly in planetary science. This was one example, and other case was this asteroid, this ooh-ah-moo-ah-moo-ah.
00:38:32
Speaker
It came in, and this is the first intergalactic object ever tracked and observed by humanity. And the first thing they noticed, it is either very flat or very thin. That was for sure because of the reflection of light. And almost immediately, Avilog, who is a very famous astrophysicist at the Harvard Smithsonian,
00:39:01
Speaker
he concluded that this probably a spaceship. And even... It looks like a pencil, right? Yeah, it looks either like a pencil or a disk. We don't know which of those, but either a pencil or a disk. And you don't see these shapes in space normally. And every single radio telescope on Earth was directed onto this thing, and they were trying to find the little green guys running around.
00:39:28
Speaker
and giving the signals. And there was no signal? I don't know. I mean, I was not there. There is no evidence supporting or either in favor against this claim. So there is no scientific conclusion. However, our theory told us that if you take any shape and you let it travel for 500 million years straight line in the intergalactic space,
00:39:59
Speaker
than probability one, it will be either very flat or very thin. So this is Occam's razor. So it's a very simple explanation, which does not mean that it is true, but it has some appeal compared to the spaceship.
00:40:19
Speaker
Again, I have no firm standing here. I cannot claim that it was not a spaceship. I just claimed that there exists an alternative explanation which is very simple and appealing, and it is connected to a conversation because if you have something which is very flat, it's bound to have exactly two stable points.
00:40:43
Speaker
And if you have something which is very thin, it's bound to have exactly two unstable points. So we are in the second row or in the second column of the table. And as I just explained, this migration of pebbles in this table ends up in the second row and in the second column. So this is some ultimate shape. So what we just saw is something which
00:41:13
Speaker
has undergone evasion for a very long time. And in the solar system, we cannot see these guys because if you get very thin or you get very flat and you undergo a slightly bigger collision, then you break. And while the solar system appears to be kind of empty,
00:41:34
Speaker
It's not empty compared to intergalactic space. Intergalactic space is much, much, much, much emptier than the solar system. So the chances of getting to this ultimate geometrist, which are very thin or very flat, are very higher if you are traveling out there. So in the solar system, if you are just going around in the solar system, the probability is very slim that anything could evolve to this kind of very matured
00:42:05
Speaker
state but in the intergalactic space yes it is a reasonable thing and we think that was one explanation and again returning back so a philosophical theory about giving numbers and names
00:42:21
Speaker
turned into something which catch the interest of very good astrophysicists. People have been reading this kind of note and they got some feedback. So I think it became something which you can call useful, right? Explaining nature to some extent or giving an idea about nature, whatever.
00:42:43
Speaker
I mean, I think it's fascinating that you can look at the shape of rocks on a beach or the shape of rocks on Mars and say something about the history of those rocks and how, you know, how, what their original size was, how much they'd been abraded. I think we should make this, we should probably just clarify, yeah, how this, you know, the mechanisms at work here, because we said everything sort of wants to move towards that 2-2 square.
00:43:12
Speaker
And it is, you know, it's not particularly fancy. It's abrasion.
00:43:18
Speaker
So just things rubbing against each other. It's simple in a kind of more or less random way. I wanted to add as well, I guess the exact form of the process of abrasion determines whether you end up with these sort of ellipsoid type shapes in that two-two box, all these long thin sort of, I'm going to say this wrongly, uma, uma, uma shapes.
00:43:45
Speaker
Yeah. Okay. One more. Say that one more. Okay. You're well-practiced at saying that. And it's, as you say, the long, thin sort of pencil type shapes and discs are quite susceptible to being broken. So if you think about pebbles, they're fairly, fairly large. And so if your, if it's pebbles are braiding amongst each other, you tend to not to get those shapes because a collision can fracture.
00:44:15
Speaker
sort of a long thin thing. Whereas if it's wind abrasion, then you do get these and you can kind of imagine that you see this in deserts where I suppose you see these quite long or quite sharp pieces of rock. Is that a kind of right, more or less correct explanation?
00:44:38
Speaker
Yes, this is very good. I did not talk about this yet. So it's a size-dependent thing. And the curious, so if something is being operated by objects which are much larger than itself, then the shape would become round there. If it is being operated by things which are much smaller than the shape, then it will become more angular, more elongated.
00:45:08
Speaker
What sounds like a miracle though, so these are totally opposite evolutions. What sounds like a miracle though that the number of balance points appears to be decreasing in both cases. Although the shapes, the way the shapes evolve doesn't even resemble each other.
00:45:31
Speaker
So in one case, things become more spherical, and in the other case, become like, you can get a tetrahedron, or you can get a flat disk, or you can get thin shapes. And still, so if you, there is a number called the isoparametric ratio, or the isoparametric quotient, which measures the sphericity of an object.
00:45:59
Speaker
And if something is being operated by large objects and this isoparametric ratio is going up, its maximum is one, and that is the perfect sphere. And if it is being operated by small objects, like as you mentioned in the desert, I hope is being operated by sand, or you are an asteroid in outer space and you are being operated on metal rights, then the isoparametric ratio is going down.
Mathematical Analogies for Shape Transformation
00:46:27
Speaker
So it is completely opposite.
00:46:30
Speaker
But in both cases, the number of balance points is going down. So this is really, and that makes it the more interesting. So if you are on a beach and you are a newcomer to a beach, you are a big rock being thrown onto a beach, then initially you are being operated by things which are much smaller than you are.
00:46:56
Speaker
then you become smaller, you become a peer to them because you are about the same size, and then you become smaller than your neighbors. So this whole thing makes a big U-turn. And anything which is related to classical geological understanding of shapes, like the other parametric ratio is more or less classical geology.
00:47:23
Speaker
is non-monotonic. So even if it is monotonic, it could go up or it could go down, but in many cases it is not even monotonic. It goes up and down in the life of one single pedal. And if you have anything which is non-monotonic or it is monotonic but you don't know the direction of monotonicity, then
00:47:50
Speaker
from the current value of that function you have no way to deduce its past. But if a function is monotonic then in mathematics we say it is invertible. That means that from the current state, so you can basically backtrack things if you have a monotonic evolution. And
00:48:13
Speaker
we have no full mathematical proof for this for the monotonicity because it is not a deterministic monotonicity it is a statistical monotonicity for special cases you can prove deterministic and we have good understanding where the source of randomness is coming from
00:48:34
Speaker
But if you look at data, data tells you that this is an overwhelming thing. It is going on. So if you measure these numbers, they are just going down. And one way to think about this is that if you look at the shape, you explain very well that if you take something in your hand, immediately you see.
00:48:57
Speaker
unless it has many. So it has many points, you don't see it, and a common way to say it is a complicated shape. And if you have few points than it is, you would tend to say in a common phrase would be a simple shape. So this whole operation process is simplifying and taking out
00:49:22
Speaker
what was the original character of that shape. It is redressing the shape with new features, which are simple features. And therefore, because this kind of simplicity, achieving simplicity, is an overbearing principle, when you catch a shape and you see how simple it is, you can tell how mature it is. It tells you
00:49:49
Speaker
on what stage it is in this evolution. And Uemuamua happened to be a very measured, very simple shape. It is not a coincidence that in a podcast, we are in a podcast, but we can talk about the shape of Uemuamua because it had such a simple shape. It was either like a disc or like a pen. If it had been like a very complex shape, we could not talk about it. Right. Right. Yeah.
00:50:19
Speaker
As you say, it is striking that these very different, they're shapes that we can talk about, but discs and ellipsoids look to us very different or pencils, discs and ellipsoids. And yet they all fall within this same two-two box. And yeah, it seems that was a revelation to geologists, geomorphologists who before would have
00:50:49
Speaker
using this isoparametric ratio and other things to classify shapes and would have given very different classifications to the various shapes falling within that box. But as soon as you apply this other lens, this other language of talking about things, you actually see that they're all following the same process when talked about in the language of equilibrium points. So yeah, that is really
00:51:15
Speaker
striking. The other thing I wanted to mention was, yeah, this idea of shapes becoming more simple. And I was curious if you've ever thought about this in terms of kind of entropic terms and how each box seems to be like a kind of macro state. So one of the ways of thinking about entropy is obviously you have how many different sort of microstates
00:51:45
Speaker
fit within a particular macro state. And if you have lots of different ways of making something that sort of in that coarse grained way of talking about things is the same. So the classic example is lots of different ways you can arrange something to have the same volume and temperature and so on. Then that is a high entropy thing. And if it's very hard to make that,
00:52:11
Speaker
macro state, that high-level object. If there's only one way you can arrange things, then it's very low entropy. So I suppose we could say, in some ways, the sphere seems very low entropy, because there's only one way you can arrange stuff. There's only one thing in that box, if we even think of it as something you can put on this plane. And the Gombox as well seems to be quite
00:52:40
Speaker
low entropy. Although, as you say, there's no kind of computer way of generating all the different GOM boxes. Is that something that can be categorized? I don't have a strong intuition on this. Yes, yes. I mentioned earlier that there are these boxes which you mentioned.
00:53:05
Speaker
And inside each box, the small boxes, so subcategories. So they're like the microstates and the microstates. But it is somehow, the further you go from the Gumbok, the more microstates you'll see. Yes. And that is a fact. So that we know. And in fact, it is exponential. So if you go out in
00:53:36
Speaker
In 2-2, you just have two microstates. And then as you go out and increase the numbers, and you get more and more microstates, I don't want to say bad numbers, but I kind of remember one number. So if we say that s plus u equals nine, so we are on a diagonal, right?
00:54:07
Speaker
Then I think all together we have over 70,000 microstates already. And that, so in this sense, it's, and yes, if you converge to the corner, this number of microstates is dropping radically. Next to the Gumbok in two, one, and one, two, there are only one microstate, which I would call a microstate.
00:54:36
Speaker
And these states, these microstates, they are characterized by graphs, very simple graphs, by the way. And the way when things get updated, these graphs are also evolving. And the way they evolve, they evolve by things which mathematicians love. They evolve by phase contraction.
00:55:05
Speaker
So there is a graph and it loses its face as it converges. So yes, one can think of this in this manner. And there is another manner in which you can bring in entropy. If you just look at the aberration where you are being hit by large objects, then the shape is evolving towards the sphere.
00:55:32
Speaker
And what happens is it is like the heat equation, but in a geometric way. In the heat equation, you have an object, and you insert heat sources, and then you let it dissipate. And then heat, after some while, it converges to uniform heat state.
00:55:52
Speaker
Now, the same is going on here, but heat, and heat is a conserved quantity. So the total amount of heat in the object remains constant, but it becomes more and more uniform. So it is an end of piecing. Now here, curvature is replacing heat. The Gaussian curvature of any object which is simply connected is constant. It is 4 pi.
00:56:18
Speaker
And if you look at a polyhedron, then it has no Gaussian curvature anywhere except at the vertices.
The Poincare Conjecture and Shape Insights
00:56:26
Speaker
So this would correspond to an object where you have point heat sources. And the equation what it does, it is smearing out curvature. And when curvature is uniform, this corresponds to uniform heat, it is a sphere.
00:56:45
Speaker
Yes, it is called curvature entropy. And for some special case, curvature entropy would evolve also in a monotonic manner. And people have written about this. So this is probably along the lines which you suggested. So this is one way to think about entropy. And the other thing is about graphs. But in both cases, we come back to the concept of simplicity or complexity.
00:57:15
Speaker
Yes. And it is uniform. It is uniform, at least in the statistical sense. In some ways, it's striking, though, that things evolve to a lower entropy state. That's sort of the opposite of a physicist's intuition. But of course, there's probably other ways of describing the entropy here. It's a kind of slippery thing. Yes, you can look at it like this or this, but
00:57:44
Speaker
I think what we should notice, there is an anthropic concept and there is a monotonicity. Now, whether you look for entropy or negative entropy, then this is just a binary question. We have an anthropic concept and we have a monotonicity. So your question was spot on, yes. Yeah.
00:58:09
Speaker
I wanted to make sure that people, the heat equation is really interesting here and so maybe we can come back to that, but I wanted to make sure that everyone has a, there's one model which you've given for this process which I really like as I think
00:58:25
Speaker
whether you're a mathematician or just someone who uses soap, you're going to know this model. And it's in terms of, I mean, I think you know the one I'm referring to. So I'll let you talk about it in terms of a kind of a cubic or a rectangular bar of soap. Yeah. So if you take, if you take a rectangular soap bar, which is like a cube,
00:58:55
Speaker
and you wash your hands, which you should do, then you will notice that the shape of the bar will change. And it will not change in a self-similar manner, so it will not remain a rectangular shape, it will become rounded. And as in this process, as we just mentioned, things change. The volume changes, the mass changes,
00:59:23
Speaker
the angularity changes, but there is one thing which stays constant. And that is the integral of the Gaussian curvature on the whole surface. So that remains constant. And that means that initially all this curvature was concentrated at the vertices, at the sharp vertices.
00:59:50
Speaker
it was absolutely non-uniform. In mathematical language, we call this Dirac deltas, where are the vertices for curvature. All the curvature was there in eight Dirac deltas. And then afterwards, it gets smeared out. And if you are doing it very carefully, the hand washing, you are lucky you get to a kind of a spherical singing.
01:00:15
Speaker
Now, this is very interesting analogy, and it is not just a folklore thing. It's more than that. You can actually write down these equations so you get something which is very similar to the heat equation. And just to tell you a curiosity, we were talking about the mass over curiosity, but now I tell you some different curiosity. In the heat equation,
01:00:43
Speaker
One can prove that if you insert a heat source, the heat will change in the whole object instantaneously. And this is not physical. So heat is propagating faster than light in this model. And this is, of course, an artifact because this is a linear partial differential equation. Linear models tend to have kind of funny aspects.
01:01:13
Speaker
The curvature dissipation model we are talking about, the abrasion model, is a highly nonlinear equation, which can be linearized. But it's very difficult stuff. I don't know much about this. So this is very technical, hardcore, geometric PD stuff. There are a handful of people on the planet who work on this. Now, one of them is Richard Hamilton.
01:01:42
Speaker
He's very famous. He's a great mathematician. And he got credit, partial credit, for the solution of the Poincare conjecture. So Gregory Perlman came up with a solution, but the solution was based on something which Hamilton has invented, which was called the Ricci flow. And Perlman refused the money and the prize and everything,
01:02:12
Speaker
Hamilton got some credit, very well deserved credit. So he is one of the great geometers and PD people now. Believe it or not, when he invented the Ricci flow, they were in four dimensions, of course, they were out there.
01:02:30
Speaker
But to flex muscle, they did some experiments or they did some kind of exercises in lower dimensions, in 2D and in 3D. And Richard Hamilton himself wrote a paper about the shape of warm stones.
01:02:51
Speaker
And this is, needless to say, it's a sheer librarians paper. But it tells you something which is really interesting if you are diligent enough to dig into that paper. It is not a bad time reading, but it is something exceptional. So Richard Hamilton tells you that while the heat equation has this property of infinite propagation speed,
01:03:18
Speaker
The curvature flow does not have it. It has a finite propagation speed. And back to the soap. If you do it carefully, buy your soap, start to wash your hands, and you will notice that after finite time, maybe small time, the middle part of the flat sides will be still completely flat. So curvature does not propagate immediately to the total surface. It takes a little bit of time.
01:03:47
Speaker
And that was an evaluation for physicists, because now it resolved this puzzle. What does the heat equation, why does the heat equation tell us of infinite propagation speed? This was the answer, because it's linear, and this is the non-linear equation, and now we have a finite propagation speed. I think this is a beautiful, small bit, a beautiful hermathematics.
01:04:10
Speaker
is contributing to physical understandings. And it is about the soap analogy, which you mentioned, anybody can test it. Yeah. And I think it is, I mean, I don't understand enough about the Poincare conjecture to know if this is a good intuition or not, but it does seem like there is, you know, a reason why.
01:04:34
Speaker
There's an interesting analogy, at least, between the Pankare conjecture and pebbles, right? Well, the Pankare conjecture is in a higher dimensional space, as you say.
Role of Intuition in Mathematical Discovery
01:04:42
Speaker
It's essentially saying, and this was an unproven conjecture until Parraman, using some maths from Hamilton, proved it a few years ago. 10 years ago, more than 10 years ago. OK. So it'd gone 100 years unproved after Pankare had said, OK, if I get this right,
01:05:04
Speaker
every kind of surface in a four-dimensional space where you can take any loop and collapse it onto a point. If you have that property, you can morph that surface into a three-sphere, so a sphere in that four-dimensional space. So in fact, the Poincare conjecture in general sense was
01:05:29
Speaker
living in several dimensions. I am not an expert on that, but what I can tell you that it was proven in steps. And in each dimension, the person who proved it got the fields medal. One of these people was Stephen Smale.
01:05:51
Speaker
and it went on and on and on. And Perlman was the last. So it was the most difficult part. And then he finally nailed it. And to do that, he needed a tool. And basically, Hamilton invented the tool. And the tool was a horribly complicated geometric partial differential equation. That was the tool, the hammer, which Perlman needed.
01:06:19
Speaker
And that equation, if you write it down in three dimensions, it is the governing equation for the abrasion for pebbles. It is not something analogous. It is the equation. Yeah. And that makes sense, right? Because in the high, your morphing, the whole proof is to try to morph something onto a sphere. And that's exactly what's happening
01:06:44
Speaker
You know, with the pebbles they're trying to get to. Exactly. So it is exactly the equation in the case when the pebble is being hit by large pebbles. Then the Ricci flow restricted to three dimensions, which is a Gauss curvature flow. This is a Gauss curvature flow. So we have an object.
01:07:06
Speaker
and each point is moving along the surface normal inward with a speed proportional to the Gaussian curvature of that point. So that is the Gauss curvature flow. Now that is the governing equation for abrasion by large operators. This was found
01:07:25
Speaker
So the general case, it was not clear how this fitted into this huge picture until Hamilton and Perelman came about. But people were trying to do this. I mean, the first person to wrote down this equation was fiery. And then it was generalized to different kinds of apparatus.
01:07:47
Speaker
But the big picture, the geometric picture became apparent when these guys started to work on the Poincare conjecture, because then they started to prove miraculous theorems about the curvature flowing two dimensions. And then they didn't do much in 3D. One exception is this paper of Hamilton, when there is a hand-drawn sketch of an abraded bubble.
01:08:14
Speaker
and Hamilton writes, I wish somebody would do this on their computer or in a lab, because then I could see it. We often claim, tend to say that mathematicians are just out there, but this is not always quite true. So sometimes they have a good intuition about the physics as well. Yeah. Yeah. And it's, yeah, I guess I was really, I loved that quote from Arnold that
01:08:44
Speaker
maths is a part of physics where experiments are cheap. I'd never heard that before, but it sort of runs in reverse as well, you know. Clearly maths can learn something from physics and I know you've got a lot of puzzles and drums. It goes both ways, yes. And yeah, I mean, the heat equation. If you look at these people, Arnold
01:09:07
Speaker
maybe Hamilton, I don't know Hamilton, but in case of Arnold, I know, I mean, I have talked to him three times. And I know that his mind was racing back and forth between physics and maths. And he picked the bits from here and wrote it over there and going back and forth. If you go to an average mathematician,
01:09:27
Speaker
He expects you to bring a problem which can be solved by the path of mathematics he knows. That is a natural thing. If you go to somebody like Arnold, you bring a problem and he creates the mathematics on the fly to solve the problem. So it is a very different view.
01:09:49
Speaker
As you pointed out, mathematics can learn from physics and the other way around. That is a very healthy kind of dialogue. I know you're fond of the quote from Galileo, which, by the way, Julian Barber, who was a guest on this, loves this quote as well. The language of nature is mathematics and its symbols are, I think, triangles and circles.
01:10:15
Speaker
And then the final piece of that quote, which is just really sealed the deal, is like, and if you don't have those, you're in a darkened labyrinth. You're not just in a labyrinth, but someone's turned out the lights, so you're really lost.
Classifying Shapes: Methods and Philosophies
01:10:29
Speaker
But again, there's those two aspects, like that quote has the two ideas in it. The one being that you can learn about mathematics by reading off nature, right? You can uncover the symbols of that book by looking at nature.
01:10:44
Speaker
But then, of course, you can play it the other way, and you can do pure mathematics and discover some symbols that you then go out and find in nature. But I'm so curious that the Gombok, for instance, is something that doesn't seem to... It's one of those things where it was inspired by nature, but actually doesn't exist in nature.
01:11:07
Speaker
Yeah, that is nicely put. That is nicely put. So let me make a few comments. So yes, I do like this quote, and it goes back to an earlier part of our conversation because Galileo says a lot in this quote because he not only says that nature is written in the language of mathematics, the book of nature, but he specifically says that this is about shapes.
01:11:32
Speaker
And as we started off in this conversation, we discussed that we don't know. I mean, we have a very limited understanding of these shapes and that may be a global handicap. I mean, if this is true, if this hypothesis is true and nature is constructed in a language which relies on shapes, then we have a very limited ability to understand it.
01:12:00
Speaker
There are many directions in which you can see these, that geometric objects pop up at string theory. So geometric objects or Plato's theory of the elements, which could be discarded as like a myth. But as you know, Heisenberg and
01:12:30
Speaker
or contemporary Frank Wilczek. They were both kind of advertising this theory saying that it has some vision because it says that matter is, ultimately matter consists of objects which have symmetries and which are identical. So at the core of the things, again, we see a geometric idea.
01:12:57
Speaker
I don't claim that everything is geometry. But what I feel is that geometric ideas pop up in a very broad range of subjects. And what I definitely know that our ability to grasp geometric objects is limited. And this might be a reason why mathematicians, part of mathematicians prefer algebra.
01:13:22
Speaker
Yeah, it's interesting. I think possibly Julian Barber might claim that everything is geometry, which I think he probably would. We're sort of running low on time, but there's one just on this idea of how do you classify, you know, nature seems very, very detailed and hard for us to give names to. And so, yeah, we just have these these platonic shapes.
01:13:48
Speaker
We've discussed one of the ways that you can talk about shapes, but I know that you've worked. There's another thing, there's another idea I want to introduce of yours, which is around counting sort of faces and the number of faces per for each vertex. Yeah. Could you kind of outline that even if we don't have time to go into all of the details of it? I think it's just such a nice idea and that we'll give listeners one more tool in their box.
01:14:18
Speaker
It's a simple idea. So you go to any park or your yard, front yard, backyard, and you grab a handful of crashed rock, which can be seen all over the place. And you notice that these shapes are angular shapes. They're not smooth. They have edges and vertices. They look like polyhedra. They're not quite polyhedra, but they're kind of polyhedra.
01:14:47
Speaker
And then you kind of approximately count these numbers, the number of faces, the edges, and vertices. And if you do this again persistently, we have done it persistently, and you take a big average, then you find that the average number of faces is six, the average number of vertices is eight, and the average number of edges is 12.
01:15:12
Speaker
which is exactly the numbers of the cube. And this was the element Earth in Plato's theory. So he said Earth is consisting of cubes. But what we found is that this is too in a Platonic sense. Plato was always looking behind things. And he said that there is this analogy
01:15:39
Speaker
of the cave where the prisoners see the shadows, but they cannot see the real object. So in the platonic sense, these shapes, these curved rocks are just distorted images. And if you look at with the naked eye, you just see many different shapes. But if you look with the eye of your mind, you take the average, you see the ideal shape, which is a cube.
01:16:06
Speaker
And this is a pretty universal thing. There are exceptions. There are processes which produce fragments which don't abide by this rule. But on Earth and on other planets, in many cases, this is what we see. So you have to be able to fit them together. And the way they fit together, on average, they give you these cubic averages. Now, again, we have
01:16:36
Speaker
a multitude of shapes, zillions of shapes, but we can make some relatively simple statement. Whether or not Plato knew this, it is way beyond my pay grade, so I don't know whether he knew it. But let us look at it like this.
01:16:56
Speaker
If you go to a psychologist, she or he will tell you that your mind is averaging out every five seconds something. So this is a very standard thing what we are doing. We are looking around and we are averaging. It is beyond doubt that Plato's mind was probably slightly better than the average mind. I think this is a safe thing to say. Now he was walking these beaches, he was looking at these things
01:17:25
Speaker
Is it possible that his averaging brought him some intuition? I don't know. But it is certainly tempting to believe that maybe the theory has some quantitative and even it was not, he didn't know this, he suddenly helped us to find this simple fact. This is a very simple fact, but it helps to
01:17:54
Speaker
Again, to talk about numbers. So the cube has 6 stable, 8 unstable and 12 saddle points, so 26 balance points. If you look at crushed rock, on a broad average, it would have 20 odd balance points. Not as much as the cube, but almost in the same range. And if you go to the beach, you find that
01:18:21
Speaker
Most bubbles have six, like the ellipsoid. So nature on Earth, on other planets, now,
01:18:29
Speaker
And for the past few billion years, everything was happening between the numbers 20, 0, and 6. That was the process. Things were climbing down from the numbers. So this is the start of the process. Nothing has more than that. So we start in that range at 20. So the cube and the Gombok are standing like two brackets.
01:18:55
Speaker
which we don't see actually in this process. We don't see perfect cubes unless you look at crystals, but in crash rock you don't see the perfect cube. And you don't see the Gombok either. But these are the outermost things. And although we don't know individually the fate of average shape,
01:19:15
Speaker
for the mind, it gives a little bit of support to know that this whole process, which is a very, very broad thing, it is happening within these limits, within these brackets.
Imagination in Mathematical Boundaries
01:19:27
Speaker
And this helps to, maybe to talk about it. Yeah, I think this is, it certainly doesn't.
01:19:38
Speaker
I'm reminded of something that Harvey Brown, philosopher of physics once said about the film 2001. He said, what's so strange about that obelisk? And it's because it's so it's like a perfect obelisk, right? It's got these perfectly flat platonic sides, if you like. And so that's, you know,
01:19:58
Speaker
it's an elongated cube, right? It's one end of the bracket and that's why it just doesn't look natural. And I do wonder if Kubrick was to remake, you know, come back from the dead and make a new 2001, would it have gone back in? Yeah, I looked at this movie, of course, not just once, several times. And yes, this occurred to me that the fact that a cubic object was featured there,
01:20:27
Speaker
maybe it was not a coincidence. So the cube has a central role in our thinking. And this is several reasons. And this might be one of the reasons that it is in front of our eyes in nature in a kind of a non-immediate or not non-obvious way, but it is on display. And this is, in some sense, it's an ultimate shape.
01:20:56
Speaker
It's more ultimate than other shapes. So maybe Stanley Kubrick had this intuition or maybe he was reading Plato. I don't know, but I certainly like the movie. Yeah. And yeah, it's just so, it's immediately recognisable that this is something that's alien, right? This couldn't have been produced by nature. Yeah. Yeah. So it's because it's... Yes, but it is not, it could have been like as a head-on, but it being a cube,
01:21:24
Speaker
is being something which is just outside nature. And again, we can argue whether Plato knew or whether Kubrick knew or they were just having this intuition. So I don't think it matters actually.
01:21:45
Speaker
It just matters. So the mind is working in a very complicated way. And Poincare, of whom we talked today, he was not just a great mathematician, he was also a very good writer. He wrote some real, he was eloquent writer. And he wrote some good books. And in one of his books, he's talking about how the mind is working. And his understanding is that
01:22:15
Speaker
the geniuses are not basically different from the smart people by having more brain power or something like that, but they have a better kind of channel of communication with their subconscious. And he thinks that in the subconscious, the large amount of thinking is done and some people get these pops
01:22:44
Speaker
And this is evidence because many people have experienced that while going to sleep or waking up, you get a flash. And that is where your subconscious is still there and then this communicating with the conscious mind.
01:22:59
Speaker
And Po and Carrie is telling us stories from his life. And it was obvious that it's some sudden flash. And he says, it is not a heavenly insight. It is just from the subconscious. And we tend to know that brilliant people tend to be sometimes a little bit unstable. And if it is true what Po and Carrie thinks and their subconscious is freely communicating with their conscious mind, then it is not a miracle.
01:23:29
Speaker
that they are a little bit less stable. So we don't know about Kubrick. We don't know about Plato, but we know that both of them were attracted to the cube. And they were not the only ones. Picasso was also attracted to the cube. And Marcel Breyer was also attracted to the cube. So there were some people attracted to the cube. And maybe there is something to it.
01:23:56
Speaker
Yeah, we dream of these impossible things, or almost, you know, possible, but beyond what we see. So yeah, I think that's a good place to, that's a good point to conclude on. We need to dream more. Yes, yes, dreaming is a good thing.
01:24:17
Speaker
And thank you for your questions. I mean, you wrote in some cases more than I did. In particular, thanks for bringing in this movie.
01:24:29
Speaker
I have been thinking about this, but not now. It was in my subconscious. But you brought it in, and I think this is a very good touch. I mean, some people can connect to this much better than to partial differential equations. Yeah, I think that may be many people. But yeah, thank you so much. It's been a real pleasure. Thank you, James.
01:25:05
Speaker
So,