Thank you. Thank you, John. Thank you. Thank you all for coming out here on this November evening.
Thank you to the Massive Institute. I mean, it's a tremendous privilege to be able to talk to an event like this in a prestigious venue like this,
but a particular thank you to the Mass Institute. And I've never been here before, but I do want to say thank you to them for this,
which is my daughter Fiona studied maths at this institute in the old building a few years ago.
She went on and she did a PhD and she now teaches maths there 400 kilometres from anywhere else and she's spreading the joy of maths.
And I really just want to thank everyone from the last institute for their part in such a fabulous positive outcome.
So thank you. I'm not a mathematician, as John said, this is a math store.
But I am going to talk about a mathematician.
I'm going to talk about this chap here when I tell him he won the Fields medal in 1958 for his doctoral thesis.
He finished that in his twenties. But what I'm going to talk about is the work he did later in life, which is sort of so he wrote up in this book.
He essentially invented catastrophe theory, a whole new branch of mathematics.
There were other people involved. Vladimir Arnold And so forth.
But let's keep it simple. The hero of my story today is, is René Thom.
He's the protagonist now. It's a kind of wonderful, crazy book.
This it's full of wonderfully deep mathematics, but some really crazy sort of musings on magic.
The military society reading tea leaves the shape of the human genitals.
There's a whole section on that in the book on the stability theorists.
As John said, I teach structural stability at Cambridge and it's called structural stability and morphogenesis.
And structural stability is really important in my field. This is structural instabilities.
We do not want these to happen. Many good people were killed in these accidents and many of my students go out and design things here.
It's not my students, but the students from my department. His Pete did the velodrome roof.
Chris has just finished this bridge in Turkey Bay Bridge.
The main span is about a mile without touching the ground, a phenomenal feat.
Yohannes has just finished the Mersey Gateway Bridge, which connects Runcorn, my hometown.
We witness up near Liverpool and Suzanne is there on the oil rigs in the North Sea, which she saw the busy decommissioning.
We do not want any instabilities in those, so we need to understand structural stability theory.
And here is really Tom's book in my library.
It's there with methods of structural analysis, design of steel structures, and I think that's a good thing.
But actually it's in totally the wrong place because it did not mean structural stability in the way that I mean, structural stability.
He meant something completely different. This is a few sentences from my book.
As Children we taught a geometrical sequence beginning triangle, square, Pentagon, hexagon.
It's an alphabet applicable to straight line. You see it in geometry.
Ronnie Tom, however, pointed out there's another alphabet appropriate for curved geometry and it contains the sequence fold,
cross, swallowtail, butterfly. And these are all across the top there.
The fourth shapes the fold swallowtail butterfly.
There's an infinite sequence of those across the top. The cross boys.
And there's also this.
There's a whole infinite sequence of old buildings, but these are the seven simplest, the three umbilicus at the bottom, the lowest one there.
The parabolic immobility was of great interest to Salvador Dali because that's the catastrophe that Rene Thom thought explained.
The shape of the human genitals or the genesis of any animal is Salvador Dali thought this was quite interesting now.
John sort of already told you that that. I teach stability theory.
I might be designing a straight column with straight lines. You think engineers can only draw straight lines with a ruler?
But so understand why a column, a straight column, doesn't collapse.
I have to think about the energy in that column. That energy's got lots of curves.
It's usually a sort of curved surface. And in my lectures, I've drawn these curves, surfaces.
I'm looking at them for different angles. My lecture notes are full of curves.
Even if the column is completely straight. And then, as John said, I introduce life drawing, lots of positive reasons to department.
And then one day I've been out in the morning drawing the curves of my energy surfaces.
And then in the afternoon I went to the life class and I was drawing the same curves.
These are the same curves, I thought. And the funny thing is, I speak a language of power.
I've got this language of folds, cusps and swallowtail,
etc. and although the artist to draw a thousand times better than I could, he didn't have this language.
And I thought, Well, I should teach everyone these words. Lives will be enriched if everybody just sort of knew these words, knew these shapes.
We know triangles, square Pentagon.
We just don't know these. And I think these are sort of fundamental. So.
The shapes that I see, these curves that I draw are they speak really tones, language, the stability, boundaries that we must not cross.
I've got foals, cusps and Swallowtail on so that book.
What he meant by structural stability was these shapes you will see in these shapes.
These shapes are robust. These are important shapes. That's what he meant by structural stability.
It was a sort of theory of shape, because I see those shapes.
I actually need them in my lectures on the stability of structures because those shapes are there.
It was a lovely ambiguity, but it's actually the perfect place to be that library book.
So there are the beautiful shapes.
So here's another sentence or two from my book. Many previous books have sought to build links between mathematics and art,
exploring topics such as symmetry, proportion, perspective, tessellations and polyhedra.
The series of almost invariably been built using straight line geometry with some even seeking to explain beauty,
using the aspect ratio of a particular rectangle, a rectangle they call golden.
No matter what colour it is. It does not matter how much you say about one plus the square root of five, all divided by two.
To me it is just a rectangle. And I think there are far more beautiful shapes in the world.
And there is some of the sort of shapes I think, of beautiful shapes. And actually catastrophe theory.
I think he's got something to say about every one of these shapes. And so catastrophe theory is a theory of curved shape.
So maybe it's got something to say about beauty after the life drawing class.
When I sort of I thought, I'll take five weeks off and I locked myself away to write this book.
The link between stability,
theory and catastrophe and life drawing was carefully considered engineering and then much freer sort of expression of, of, of the life choice.
But it just exploded.
The more I thought about it, the more I learned about it, the more it explode into lots of different fields, into physics, into optics,
into gravitational cosmology, into psychology, biology, and then over into architecture and art and even into the theory of aesthetics.
Possibly it just kept going. And I think there's a beautiful thread all linked by curves, that links all of those subjects together.
And that's what ended up as the book. So.
Let's have a look at these shapes again. The purpose of my lecture really today is to teach these shapes.
To change the way you see the world. Really? This will change what you see.
You will see these shapes everywhere. I can see them all over your faces.
You just haven't been told them yet. You go out there in the world, you see rectangles everywhere.
You will see thousands of rectangles on every one of your faces.
There is an infinite number of curves. An infinity beats a thousand in my book.
So thank you for bringing your faces and all their curves tonight.
Let me set you these. We're going to start at the very beginning with the fold.
So that's the folder top left. Your left.
It looks a bit like a parabola sort of tilted over on its side.
Does it matter which way you draw it? There's these two surfaces coming together, these two lines coming together, a solid one and a dotted one.
So how does this work in the live class? Well, so you were drawing me.
So you would draw my arm. You will draw my arm. You probably draw a line along the top.
On a line along the bottom. Well, you wouldn't actually draw a line here.
You draw it on your sketch pads. You draw a line, two lines there for me, I see something totally different.
I would draw a line here on the line here on on mine.
So actually, every one of you, the line on your sketch card corresponds to a different line along the surface of my arm.
It depends where you're sitting and your perspective.
The line you've drawn separates the visible so that you can see from the invisible over on this side.
So you've got the invisible on the visible come up and they meet at the full catastrophe.
So where is the fold catastrophe? In a sense, it's not on my arm.
It's in the whole set of of your act of perception, of my arm.
It's between you and your sketchpad and you looking at my arm and drawing on the sketchpad that creates the full catastrophe,
because you will each try different, fold catastrophes, depending why you're looking at them.
So that's the two fold. So you can actually say that's a life drawing.
A simple outline is a catastrophe map.
It is the outline. The outline of the figure is the locus of the visual fold.
Catastrophe is defined by the viewer's position, by the act of perception of it.
So that's a concrete out go to a Brusca and I can honestly say without being rude to every line on that drawing,
it's the catastrophe, the folds, the lines, the outlines, all the focus is created by the way God has been looking at that drawing.
This is the pioneer as well. The pioneer plant that was sent into outer space of one day mate meets aliens.
All of the lines on that drawing are fold catastrophes as well.
This will actually tell something to the aliens, not just about what our body shapes are,
but about how we perceive body shapes and how we look at them and how our acts of perception.
Because maybe the aliens have got 50 eyes that surround things and have a totally different view of the world.
Somehow, I don't know. But aliens will learn things that we think that that's an important thing to tell them.
So the next step, if this is a sort of alphabet of shape or a periodic table of shape, we've got the fold.
It's a bit like the hydrogen atom.
So now we can make a hydrogen molecule, we can stick two folds back to back, which as I've drawn over here to this s shape.
Now, the S has a very important place in our history, in the theory of beauty from Hogarth and so forth, the serpentine line.
And this is the icon of catastrophe theory back in the 1970s.
The s sort of emblematic of what catastrophe theory was all about.
On the weather, the catastrophe is, if you imagine the system of equations describing an oil rig or a column is appear and it smoothly changes.
It's a bit like a ski slope.
You reach this point and then you drop to there and it might be a nice landing and you might be able to continue or that might be natural catastrophe,
like those instabilities that I showed you earlier. You can do this yourself.
You can feel these catastrophes. If you hold a train ticket between your fingers and you press on it and suddenly go, Pop, pop.
There's a sort of place where it suddenly pops. I can do it. Actually, with this.
This is a model of a column. You get this sudden large change.
So if I hold it here and I make this column shorter now, I've got to rotate my arm.
That's what I'm going to take my arm slowly so slow. Rotate my arm that and slowly, slowly, slowly.
Slowly release a small change of my arm is a small change of a vat, and I will reach the catastrophe, which is about there.
Our small change suddenly flops over there. So you've gone through a catastrophe, a small change.
So that's actually sort of going over the falls, dropping off there, like so.
So that was a letter. A letter B, if you like. The number two is the cost.
The cost is the spiky shape. You may say, hang on a minute.
I thought this was a sort of series of smooth, curved shapes.
Why have we got something infinitely spiky as the sort of second letter?
Why wasn't into a spiral or something nice and smooth? Why have we got this spike?
We do see spikes like this costs instability theory over at the lowest site that this is ruled as arch.
If you vary where you apply the load on the train ticket and plot what load creates, it's a pop through.
You get this spiky shape with a two. So it's parallel like so.
So I'm familiar with the shape for my stability lectures. How do you see it?
Well, this ski slope explains how to see it. I've got a ski slope there where you can ski down one side nice and smoothly.
No trouble at all. But on this side here, I've got the shape of folded it over sort of pleated.
If you tried to ski down that side,
you drop from the tip of that precipice to there and you continue possibly happily or maybe the act of dropping is is a real calamity.
So if I were to draw that, as I have done where these two fold lines meet of the axis,
you give you this infinitely sharp spike, all the cusp catastrophe, and it's infinitely sharp.
It's not like a church steeple as a triangle. And it really has this sort of infinite spike in us.
But there is no infinite spike in there is no spike in it at all on the ski slope.
The ski slope is perfectly smooth everywhere. You can run your hand over it under it.
It's the act of perception that it's a drawing.
It has created this shape, this infinite spike in this is the act of perception that sort of created that.
And this pops up in my stability lectures as well.
So this is my sort of first lecture on stability to my students at Cambridge, to the engineers.
So say I've got a column like that, what, over there? And it's a straight line, but I'm interested in how it buckles.
So this orange red track is the energy stored in that column.
It's a sort of U-shape. When there's no load on it, there's only one solution on the car.
Well, the column sits at the bottom and it's only one solution.
But as I increase the load, the energy function changes.
If I work out the mathematics of it, it changes, it gets brought on flatter, and then it goes like that.
And what you can now see is, look, there are three solutions. There's one over here, one over here, and an unstable solution in the middle.
We've come from this oneness. So this.
So this. So this trainer's okay.
And I can repeat that experiment with a slice. Let's put an imperfection in the colon now.
So before I start the calls, the column has got a bit of a biased to this direction, but there's still only one solution.
And as I now increase the load. The column just starts moving to the right.
Moving to the right. Moving to the right. Over here to the to the left.
This other one, this other solution has form, but we never actually see it.
It's over there. But you can see it if I do it here.
So we start off with a short column. Has a bias. It's got a bias.
There's only one solution if I make this longer and longer. It keeps moving over and over and over and over and over.
And it really, really long like that. Really long.
Now, the other solution is appeared over there and there is a solution over there.
You said, Well, why would I be interested in that in my columns buckling over there?
Why would I want to know if I could take the column and put it over here?
Well, the answer is you don't need to know that in less than one, but in less than two.
When we do the book of oil rig lengths and shells, it's the one over there that kills you.
So we will need to know about it. So if I join all those together, going from the U-shape to the W shape,
if I join them all together, I've got this beautiful curved surface called the energy surface.
And you can see they go for 1 to 3 inches along the valley floor at the bottom there with this sort of pitchfork bifurcation.
That was the thing with the over on the far side, we've got the what happens if I did it with a bias.
I now get a different sort of equilibrium path where it just grows and grows.
Now, if I take all these blue lines for different imperfections and join them all together
and stack them up and create a second surface called the equilibrium surface.
And if you look at that equilibrium stuff, it is nothing other than the ski slope, it is the ski slope.
And we're just looking down on the ski slope.
We sort of see the cost. So this is just looking at this surface. So what you notice is I'm always looking at curve surfaces.
You might just see a straight column, but in my head there's loads of beautiful curved surfaces that I'm looking at on these.
This is the language of these surfaces that is the cusp.
It's the sort of edge of the precipice is this cusp here.
Now, you could easily explain this two thirds parallel, and this is the hardest bit of maths in this whole lecture, actually.
So don't worry if it's, if you don't get it, but you possibly might, might.
So I might just pick up a general piece of why and honestly, this works with the wisteria in your garden.
This is just a general piece of wire. And if you pick a points on a look along, it's pick a point at the top and look along it.
And you should all, as I move past, you see a cusp.
Yeah, you should all see it. Yeah, there's a cusp at the top of that and it's just a general Warren general position and
that's got the two thirds power law and here's the maths you'll look at along it.
So that's the tangent to the curve at that point. So that's the linear terms.
And by look it along that you've thrown away the linear terms, does the best fit quadratic, which is this part of the curve.
So anything left in this direction must be higher than quadratic, so it's cubic in higher.
So you've basically got rid of the linear things you quadratic those cubic to three three upon two parallel.
Don't worry if you didn't get it, but it's really that simple. It's just the property of look it along the line.
So there they are that we look at along the line, at the top.
We've got the two thirds power looking down on the cusp. So we go back to Rhoda's Arch where we we saw this cusp.
You might have thought that to explain this, calculated the principles of mechanics, of arches.
What really some would say is all you do is look at along the line that two thirds power law.
There's just a line, a smooth line. And you're looking along. It's of course, it's two thirds power law.
Don't know anything about the properties, the mechanics. It's the sort of property of the symmetries, etc.
That's really Tom's take on it. I think you're just looking a longer term.
This is where you would find them on the body. So the outline suddenly ends.
If you join those parts of the body, you would have to take your pencil off the paper at that point,
because the outline ends because you can't see the other fold, the hidden fold that's hidden behind the skin, but it is there.
And so there is a two thirds parallel at that point on the body.
And at that point, though, and it's not it's not on the body.
It's on the act of perception of it is on the taking of the photo that's got the two thirds parallel to it.
Number three is the swallowtail. It's my favourite of all the catastrophes.
The most beautiful. I think it's got two back to back costs like so, but we slightly sort of short of time already.
So I'm going to skip over the swallowtail and move on because I've got more interesting things to tell you.
But dodge breaker if if that model was a bubble woman rather than having skin,
he could draw the whole outline without taking his pencil off the paper.
And he would have to have swallow tails almost everywhere.
He's got the costs, wet outlines and but you would have to have swallow tails almost everywhere.
And if we were bubble people, I would not need to give this lecture because when you were a child,
you would have said, Mummy, what's that funny shape under your arm? And she said, That's a swallowtail, dear.
Instead she teaches you about triangles and squares and I have to give this lecture.
Curvature, Einstein said. If you want to understand the world, the universe, you should understand about curvature.
Now, everything that I have to say, it works in four dimensions, 13 dimensions.
But let's keep it simple. So we got a three dimensional room on a two dimensional surface in that room.
And there are two types of surface, two types of two dimensional surfaces,
elliptic and hyperbolic elliptic is sort of dome shaped or both shape and hyperbolic.
It's subtle shape with sort of opposing curvatures like so.
So there's a rather neat thing you can do. You can make a thing called the curvature detector.
You've got a little coin shaped, transparent thing and you put some black lines across it, leaving clear gaps between them.
And you place the answer in this case of place it onto a sculpture.
And if you place it onto an elliptic part that's dome shaped, you get elliptic moiré fringes,
you get this moire interferometry, it shouts out, Look, ellipsis, I'm elliptic.
If you place it here or place it on the waste hyperbolic region, you get this crosshairs,
you get this cross with hyperbole in the quadrants and it's shouting out, I'm hyperbolic here, so let's scale this up.
That was just a little points shaped so I could do this with an A4 sheet of paper.
You could all do this. You go to Microsoft Word, you print black lines with gaps in between, and you print it on acetate, which is what I did here.
I then put this on my wife's back, who was reading a biology book, but she said, What you're doing, it sounds so vast.
Don't worry about it. And I took this photo. I took this photo and I thought it was so beautiful.
This was the first what I ever took on is the morphology on the back there with the hyperbolic and the elliptic bits.
It's exactly the same as on the front of my Catastrophe Theory book.
So I know the something in catastrophe theory is linked to the beauty of my wife's back.
These are all the ones of my wife's back. In fact, some of them are actually floorboards, a cattle shard only to my wife's back.
And I think the floorboards is where the tree is.
The last vestige.
She'll cry of the sort of natural tree which shouts out, I was once a beautiful thing and you sold me in the tyranny of the rectilinear.
You made me into rectangles. But look, I've just the woodgrain is sort of shouting out that it's it was once a beautiful thing.
Now we can scale it up yet again. Now we go to the stripe, the curvature to the size of a door.
So you get a large Perspex strain and you put stripes on it similarly to stripes to moisture clear.
And then you get a life model to walk behind it. Now, I did say this.
Talk to change nudity. This is the nudity part. If the side of the human body offends you, look away.
Now, here's the model, Becky. She's not behind the screen.
She's about to walk behind the screen. Oop, that didn't work.
So she's about to walk behind the screen now.
And as she walks behind the screen, her body, you're looking through the stripes on the screen at the shadows of the stripes on her body.
And those two sets of stripes talk to each other, and they create this moiré interference.
This is Joe again. We just start to now see the sort of elliptic regions.
It's slightly more complicated on the male body here.
Maybe we should.
You can see the elliptic regions sort of all over the top, the breast, the hyperbolic regions up by the waist, etc., and then go back to Joel again.
We can see beautiful, hyperbolic regions on the waist, elliptic regions, hyperbolic regions between the muscles there, etc.
So. These screens are the fun at parties, I must admit.
So these are small factors. So again, we've now understand that these are the sort of elliptic regions,
hyperbolic regions, quite complicated hyperbolic regions though, elliptic, hyperbolic, etc.
And what's important about this is the three things I've listed there across as a sub tells you the surface is hyperbolic.
Not only that, you only see cusps. Swallowtail is the higher order catastrophes on the hyperbolic regions, the elliptic region.
All you'll ever see is a boring old fold.
Amazingly, the cross from your little curvature detector actually tells you where to put your eye if you want to see a cross.
So looking at it from here, you now know, if I put my eye over there, I will see a cusp at that point on the body there.
So that's called the indicator tricks of do pan of the cross.
So here's something I did on it. I made this sort of idealised, I don't know, doorknob or something.
It's got two elliptic regions connected by a hyperbolic region.
Over the hyperbolic Rachel. I calculated all the little crosses for the intricate tricks, and I extended them to infinity, to the celestial sphere.
And what you can see is that these sort of two arches. So if you're in those two arches, the sort of the second bit's there.
If you're there, you will see a cusp on the sort of waist of this object that if you are not of your sort of look at all,
it's straight and more of the sort of traditional full frontal, dare I say it.
So the position you won't see a cross on the waist at all.
So this actually transfers. It's a life drawing to us. What it says is that the waist of the bottle does these sort of arches and.
They sort of defined who could see what from where. If you're within one of the arches, you will see it.
If you're not, if you look straight on, you won't see it. So between the arches and the non arch, but there's this sort of surface.
On this surface it's this sort of invisible sheet of nothingness that sort of permeates the life class.
I once I sort of alluded to it, you can become aware of it and you say, okay, this is the swallowtail surface over here.
There's no crossover here. There are costs and there's the swallowtail.
And the whole life class is sort of divided up by these surfaces of sort of infinitesimal nothingness that extend to infinity.
When I've told this to life models,
I've been very interested in this because it so takes the act of being a life model from being a sort of passive thing to a much more active thing,
whether controlled and dominating the space with turn round completely.
The act of perception for what does one person see?
So what does everybody see? This is sort of a map of what everybody sees and people don't really use this.
I think architects should use this. I think landscape gardeners should use that.
They want to know, what does everybody see? Not just what does one person see?
If I was an artist, this would be my genre. Everyone thinks I'm an artist.
I'm going to be making things that are concrete. I would fill the Tate modern with sheets of infinitesimal nothingness.
So I would then alert you to and then you go, Oh yeah, it's there.
You're right. I did write the time. I'll tell you about that later.
And so this is what you see if you walk round a round a life model who's sort of reclining, you look at the waist,
you get this transition from the from the from the swallowtail through the soles to the swallowtail going the other way.
Number four. I still need to teach these things to the butterfly.
You'll see what we have. There is a swallowtail. One cost.
Two cost. But we've added an extra coast. So we now got three coasts.
There's a drawing of a surface. Again, the surface is perfectly smooth.
You can run your hand over that surface. Well, complicated surface.
But there are three infinitely sharp cusps. If you try to draw it or take a photograph of it.
The act of perception creates the costs. Here's an example of a butterfly.
Every year, my students students in my department go to Norfolk to build some structures.
In the first year, they build about a metre long.
In the third year, up to ten metres long, a year after they've left, they might design something 100 metres long.
Five years later, something a kilometre long like tree. So here they are, about the ten metre stage.
In their third year they have to build an oil rig. A model of an oil rig is basically a cuboid.
They're going to build it and it drives up. They've got to fill it. They've got to fill the dry dock.
This thing is going to float out across the lake and then they're going to fill it with water and it's going to sink in supplies.
That's the plan. I say to them before they go, Have you checked?
It will float. They say, Yes, of course, we checked it to float. You know, we can even draw a line on his Archimedes.
It's really simple. I say you have you check which way I hope it will float.
They say, oh, I don't we don't know. We haven't what you read.
All right. Let's just start with a simple problem.
So let's start with a simple problem. Let's try this on. On you.
Yeah. If I throw a chair leg into a pond, which way will it float?
This is what I asked my students. Let's start with a simple. What is a chair like?
There's a pond I'm going to throw it in now, which helps because the flow is going to float like that now it's going to flow like that?
Yeah. Is it going to float like that or like that? The first one I want for that one is a hand for that one there for the 45 degrees or the square on.
Well, let's see, shall we? Let's just throw it in. Oh.
Can you all see that? It's a funny angle. It was. You were both wrong.
It's a funny angle. It's about 15, maybe 17 degrees.
That's just a piece of maple. I can rotate over there and it's also stable.
I could spin it and it's also stable. There's no weight in there that is causing this.
It is there are eight equilibrium solutions that it floats at about 70 degrees.
I've got all the chair legs here. What if I throw that wanted? Oh, you right?
Yeah. What if I throw that one in? Oh, what else? You all right?
So we're all right. We're all all sort of winners. What I need is a chair leg with a variable density, because it depends on the density.
My son and I invented this. It is a infinitely variable density machine we put it in, so the water on it does that.
You push it down. It's also over the 45.
But there is a position, if I could take my time where it's sort of fully angled, all sorts of funny angles there.
You might think this is not the most interested or most excited experiment you ever see.
Nothing exploded. There were no bags.
But honestly, if this was an oil rig the size of a department store and it started doing that, you would be really interested in what it was doing.
So you can freeze the video later and see how to calculate the potentiality for this to work out the
energy surface yourself and you see that it has this pattern for very light material like polystyrene.
You can increase the density of that. You can see what are these?
Pitchfork Bifurcation is the variable angles and then it's stable in that position, etc. But there's a more beautiful way of doing it,
which was invented by Sir Christopher Zeeman, who was a master, one of the colleges here mathematician,
went off to Warwick and he said, What you should do is draw the locus of buoyancy of your vessel,
and then you get this curve and then you draw the normals to it and you start to make these rather beautiful patterns.
And as long as the centre of gravity stays below these curves, the ship is stable.
This is for a square sided vessel. At the top of it. There is actually a butterfly catastrophe there if you do this yourself.
So I get my students to do it. Well, the final thing I say to my students is.
Which way does this float? This is a cube. Okay, which.
Where does the cube float? Hands up for that.
Hands up for that. Hands up for the pyramid shape.
Yeah. Okay. There you go. One. So yeah, it is that pyramid sort of shape.
But then I say to them, What shape was your oil rig? Oh, okay.
Now they get a bit worried about this. Their problems are more complicated because they have water inside it.
And the water, as it sips the water, moves and starts to tip it over further.
So. What they work out is that as they float it out, it will be stable as they start to fill it with water.
It will reach a critical position where it will start to tilt, start to heal.
And then as they add more water, it will lose stability and it will turn turtle.
And I say, What are you going to do about it? They said, We'll just hold on to ropes and stabilise.
I said, This is a 20 tall butterfly on the end of that rope.
They said, Well, we can't do anything else. We've got to build it. The old Electra says, We have to build it.
And I say, So you are going to deliver on time, on budget, a total disaster.
They say we have to you know, we get marks for delivering on time and on budget.
So let's see what happens, shall we? Ah.
Ah. Ah. Oh. So I ask you when I get back, did it work?
So, yeah, it was brilliant. It was it was fine. No, it did. It was a shallow lake.
It went down and then they dug in and then down it.
When they cheated, they dodged the bullet. And I know we did.
I said no.
If that was in the real ocean with £1,000,000,000 with the oil rig, you would have watched as that went to the bottom on time and on budget.
I did a marvellous project with a complex numbers.
When they're doing the mathematics, they worked out this, this butterfly catastrophe.
They work out the mathematics is really rather beautiful. So I say, Right, I'll give you marks for realism, a march for imagination.
You can submit your get your maths right, instability right,
but submit your answer in any genre of your choosing and you can score some imaginary marks.
And I will then give you a mark, you know, sic 80 plus 60 I is the top, it's a three, four, five triangle that's 100%.
So these are some of the artworks that they've submitted. There's Michael Thompson's catastrophe machine, which explains the stability of the oil rig.
It's basically rolling on its buoyancy because these are the student artworks over here.
The Flickr book was gorgeous, but this one is a masterpiece.
This one should be in the Tate Modern. I will sell this as is priceless, but I will settle on several hundred thousand pounds for this one.
It is an homage to the Marcel Duchamp readymade.
Yes, you all know that. Marcel Duchamp, what? Reports of urinal from a shopper.
He puts it in an art gallery and he said it is art. Well, my students went to late and they bought a sieve and some skewers and the sieve is the
buoyancy locus and the skewers create the évolue to the buoyancy locus so the rig is stable,
provided the central of gravity lies with underneath those skewers.
And what's beautiful about it is they've recognised there's a radial symmetry to this problem even though there's a square cross-section.
So I think that's a masterpiece there.
The next one teacher of the umbilicus.
Well as the elliptic the hyperbolic and the parabolic there these shapes and you can make them it explains in the book how
to sort of visualise these you create a thing called the Whitney umbrella and you bend it one way to get bend it down.
So you get the hyperbolic Brenda up to get the elliptic or you put a kink in the ridge.
So get the parabolic. These crop of stability theory in the stability of shells and plates and oil rig likes these.
And some pictures from the book by Michael Thompson and Giles Hunt showing the various umbilicus.
When you get various different modes interacting, you can start to get these more sophisticated to actresses.
So more beautiful example of where to find his umbilicus.
When it's here, it's on the swimming pool floor on a sunny day.
It's all it's on the side of boats in a mediterranean harbour on a sunny day.
You've all seen these. Shapes on the side of boats, haven't you?
Yes. Put your hands if you don't like them. It's a human.
Universal. I've yet to meet a person who does not like those shapes.
And if I did, I think I'd stay really well.
Clear them. And actually, these shapes.
So these are the swimming pool floor here. I'm not just talking round.
It's almost language. What you call what you can think of it is. So a light comes down as a sheet hits the waves and then it gets bent and folded.
And you can see on this diagram I've sort of zoomed in. So in this position over here, the light sheet has been bent to be three layers thick,
which is why it's brighter there at the edges of the bright fold catastrophes.
These then start to speak the high road languages. There's a hyperbolic and they're hyperbolic and they're on a butterfly.
They're these things are just sitting there shouting out some language into the world, but they tend to go too fast for us to see them.
Now, light is not a sheet. It's actually got a wave nature.
And if you zoom in on the cusp, as Michael Berry has done for Bristol University and you take accounts of the wave nature,
the coast actually breaks up for being infinitely sharp. It's got a sort of Ripley wave in nature, and it's a rather beautiful theory.
This is Michael's theory. These are his experiments. I agree beautifully.
The precedent that applies also in gravitational lensing because that we were talking about the wave surface bending or bouncing the light.
If you've got a heavy object out in space, it bends the light, you know,
this gravitational lens, and so it bends the light and you start to get overlaps.
You start to get the folds and the costs. And it's exactly the same.
You can explain. So here we've got a heavy object in the middle and these blue ish bits are all the same galaxy in the background.
And it's like looking through a base of a wine glass at a light.
You'll see multiple images of it. I drive my wife mad with this in a restaurant.
She's talking to me. I'm just playing with these patterns on the tablecloth. I love them all.
This is an asteroid. And people say, Well, what's the use of it?
Well, I say, you could weight a supercluster of galaxies using this method.
You can detect planets around distant earth, like planets around distant stars using this.
This is the fold. This is the real data from the ogle experiment showing the thickness, the triple thickness,
the brightness, etc. So the gravitational astronomers are interested in this.
There are other ways of weighing these things and discovering distant planets.
The rainbow is a fold catastrophe.
Essentially, a flat sheet goes in to the raindrops are what comes out is shaped like a French beret, and around the edge is the fold.
And because the light is refracted differently, depending on its wavelength around the beret, it's the rainbow.
So I explained in the book how many twinkles on a sunlit.
See, you can notice these are all beautiful things the human body, the rainbow, the twinkles on its own.
We see the catastrophe theory. He talks to all of them and basically a light sheet, because then it hits the surface of the the ripples.
It bounces off the light. She gets folded. And depending on how many folds there are, that's how many songs you see.
So there is actually a formula worked out by Michael Higgins and Michael Berry, John Hannay,
Jill Nye, etc. A rather beautiful formula, but the one is the sun that you just see.
It's flat. That's the one. And then for all the extra folds,
you get these extra layers that hopefully you can sort of see in that top diagram there is sort of explain more in the book.
Really, Tom's book was called Structural Stability of Morphogenesis.
He wasn't really interested in how do we see things in the act of visual perception.
He was interested in everything, so of course he was interested in that, but he really wanted to know why are things shaped the way they are?
It's a much deeper, more profound question. And so, as I say, there is a section, this book, there's a whole chapters of chapters on Morphogenesis,
the emergence of form, how the embryo changes its shape,
etc. But there's this there's a one chapter which has sort of shown that to do with it tries to explain the shape of the genitals,
how they go through very symmetry, break your by fixations whilst returning,
retain in the left right symmetry but bifurcating into the male and female directions.
And he says the simplest catastrophe that could possibly do that has got to be the parabolic ability.
But read very Thom's book. Read my book. Biologist by large, don't believe any of this.
They're not really having any of this.
It was his great wish in the 1970s to say something about morphogenesis, but biologists haven't really picked it up.
But I think one day they might. If you think about protein folding, the proteins hook up multi stability.
It could be stable in the shape or it could be stable in that shape and it
could flip between the two and there would be catastrophes on stable things,
but they haven't really done much of that.
They've made fabulous progress in all sorts of other ways, of course, and it explodes and of skipping lots of different things here.
The psychology dimensions, there's all sorts of dashing out to finish on time for art and architecture.
And I think once you've got these catastrophe aware eyes, what you can see these shapes.
Once you understand the process of perception a little bit better than you previously did,
you could go back and revisit the whole canon of Western art or of any art and sort of look at it through to new eyes.
And I do that in the book very quickly, not as quickly as this, but put it in the book.
But what I end up with is now Garbo, who deserves special mention, Russian constructivist artist, team member of this entire school down in Cornwall.
And he made these beautiful string models, you know, really fantastic.
And we could see the focus on that sculpture.
Those are aren't they are not on that scale today on that photograph of that school to react to perception.
There is no cost on the sculpture. The sculpture is perfectly smooth.
It's the act of perception. And you may say, well, that's not very interesting.
Okay. So he basically spun it. No, he was responding to exactly the same stimuli is really tall.
Now, Garbo had looked at the patterns that light makes when it bounces off irregular blobs of water.
And this is his response to is trying to capture the shapes that he saw and really time responded by.
Mathematics and these two never met. And it's a real tragedy because they were contemporaries.
Rarely time did actually meet. Salvador Dali.
They got on famously. These are Dali's last words on art, I believe.
So this catastrophe theory is the best beautiful aesthetic theory, the world.
Since I first learned of it, it has bewitched all my atoms.
And this is Salvador Dali, his last painting. This is a photograph from my book.
This is Salvador Dali, sat next to his left pet, his last painting.
So the last painting is is over here. It is called the Swallowtail.
It is an homage to Red Eye Tom, etc. It's sits in his castle.
And people people walk past it. They have no idea what it's all about.
Most of them. Occasionally a mathematician walks past it, says, okay, Dali painted a swallowtail, and so on.
As I recognise it, you see the mathematical formula and he's just painted it.
No, Dali really understood this. I think this is an absolute masterpiece.
I could write a book about this painting and in a sense I actually have done because
in seven brush strokes he sort of says everything I said with all my worldly words,
he's captured the downfall of catastrophe, the beauty.
You'd have to read the book when I've got more time to explain it better.
But I think this is a profound painting, a masterpiece, unappreciated, and it should be sort of taken out.
And more should be made of this painting here. But my book doesn't end there.
My book does end with a chapter on Dali. My book ends at the deathbed we are.
The setting for the final scene is at the side of the deathbed of Salvador Dali.
I think that's not bad for a book that started off with some columns on a bit of life drawing.
But we end up in this sort of rather.
Interesting place and. I really like the ending of my book and I don't want to reveal it now.
It will be a spoiler. It was sort of spoil it would spoil the book.
So I'm really pleased with the ending. So if you do buy the book or borrow the book, if you do buy the book or borrow the book, you know,
read the first three chapters if you are interested in the stability,
but do read the last few chapters where we get to the, the sort of the, the ending there.
So I do hope you like the ending of the book, and I do hope you've liked this talk.
So thank you all very much for listening.