The Infinite Monkey Cage - Nature's Shapes - Dave Gorman, Sarah Hart and Thomas Woolley

Episode Date: March 26, 2025

Brian Cox and Robin Ince unpick the hidden codes behind the shapes we see in nature with Mathematicians Sarah Hart & Thomas Woolley and comedian Dave Gorman. The panel marvel at how evolution so often beats mathematicians to finding the most elegant solutions, after all, it’s had millennia to experiment. How do trees achieve the optimal distribution of leaves and why tortoise shells are so geometrically exciting?Plus we learn why the cheetah got its spots, thanks to the work of Thomas Wolley’s mathematical hero, Alan Turing, how numbers can be more or less irrational, and why Dave Gorman has a vendetta against oblongs. Producer: Melanie Brown Exec Producer: Alexandra Feachem Assistant Producer: Olivia Jani

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happened. The German Wings plane crashed in the French Alps. Find out what the pilot's voice recorder revealed. Search and subscribe to Witness History wherever you get your BBC podcasts. Hello, I'm Brian Cox. I'm Robin Ince and this is the Infinite Monkey Cage. And today we are going to ask what shape should the infinite monkey cage be if indeed something infinite can be a shape

so I was kind of wondering about that because you've already told me that annoys audiences a lot about the fact there's not just one infinity there's bigger infinities and bigger infinities and yet even little infinities they are infinite at the same time so I'm just wondering can an infinite monkey cage have a shape geometry, right? Okay universes got a geometry, right? What's what's the most symbolic geometry hyperbolic geometry? Yeah, and it can be curved or it can be flat But so it can be like everything is flat or it can be curved. I'm gonna be like a saddle. Yeah Saddle because the flat earth is that's entirely everyone's monetized that as much as they can

But if I leave the saddler theory, I could imagine that could be quite a success for me. Yeah, yeah, I'll give that a go. Today we're looking at shapes in the natural world. Why are planets and bacterium cells spherical? Why are honeycombs and the giants caused by hexagonal? Why do birds suddenly appear every time you're near? Just like me, they long to be close to you, but not too close.

Just here's fine. Carry on. What is the origin of the symmetries and shapes that we observe in the natural world? Taking us through the round window and the square window and the octagonal window are a mathematician, a mathematical biologist, and a comedian whose wild optimism has led him to declare that modern life is goodish and they are? Hello my name is Dr. Thomas

Woolley and I'm a reader at Cardiff University and the shape that we don't see enough of in my opinion is the 11-sided polygon. The reason is because I was once asked at a wedding what an 11-sided polygon is called, and to my detriment, I didn't know then, and I still don't know now. So if there were more of them, we would all know what an 11-sided polygon was called. It's a funny thing to be asked at a wedding.

Is it just, did they know you were a mathematician? I was on the table where no one knew anyone, and I was asked, oh, so what do you do? And he was like, I'm a mathematician. And you're, oh, oh, knew anyone. And I was asked, oh, so what do you do? I'm a mathematician. And oh, oh, my son wants to talk to you. He's like, he really doesn't. Wait there, I'll get my son.

And this 10-year-old boy came over, and he said, what's an 11-sided shape called? I don't know. And the fire of passion of science died in his eyes. So I'm sorry to that boy. It's a shame I wasn't there. I could have told you it's a hen decagon. And an 11-sided shape with a little bit sticking down is called a stag decagon. The mathematicians may leave, you are no longer required.

It turns out the cheek turn to my side has other skills. LAUGHTER Hello, my name is Professor Sarah Hart. I'm a mathematician and author and a fellow of Gresham College and Birkbeck University of London. And the shape that I think we don't see enough of is a curve called a cycloid, and it's my favourite curve of all time. It's really simple

to make. You just get a wheel and roll it along a road and you follow a point on the rim of that wheel as it moves and you get these sort of arch-like curves being produced. But that doesn't tell us even the beginning of the story because this curve has so many fabulous wonderful exciting properties and it enthused mathematicians from Galileo to Newton, Descartes, Fermat, they all wrote about this curve because it's so beautiful. It goes into all different areas of mathematics and in physics, it's got applications everywhere. You do get occasionally cycloids in nature but not enough. I think nature's missed a trick here because they're fabulous. My favorite naturally occurring

Cycloids it's very cool. They occur in two different ways in ice sheets Some near the poles on planet Earth and some on Europa the moon of Jupiter But the reasons are totally different why these things occur. So it's fabulous My name is Dave Gorman. I'm a reader at bedtime. And the shape I think we don't see enough of is a tippy hedron, which is a 3D rendering of the star of the bird's tippy hedron. And this is our panel. And this is our panel. Now this main show is going to be about tippy headroom,

so much stronger on that than polygons. So you mentioned there about this pure mathematics, essentially the description of curves appearing in nature. So the first question, I suppose, has to be just why? Because I suppose when many of us think of mathematics, you think of something that's rather abstract, and products of the human mind, maybe. And yet, mathematics describes nature.

Yeah. It's wonderful. And if you're a mathematician, it just brings even more joy to the excitement of studying mathematics and geometry. For me, I think the reason we like beautiful shapes and symmetrical shapes is the same reason that nature likes them. That appeal comes from them being the best solutions to the problems that nature throws up. So why is symmetry so important in nature? Well, if you've got a solution, some efficient mechanism that's worked

and works in one direction, then why not use it in all directions? And so tiny organisms can be spherical, right? The sphere is the most symmetrical shape. And that's because they don't have any constraints that are stopping them being spherical. But when you get to be a bigger creature like us humans, we have annoying things like gravity to cope with,

we have like a top and a bottom to us and we want to move forwards and backwards, we want to look forwards if we're hunting or wanting to see. And so we can't be symmetrical front to back, we can't be symmetrical top to bottom, but we can be symmetrical left to right, and we are. And that's like, why do something more complicated when a simpler solution,

i.e. a symmetrical solution, will work? So that's why symmetry is everywhere for me. And I think that idea of symmetry, maybe you could expand on that, maybe tell us you talk about symmetry. Because I suppose it has a colloquial meaning, symmetry, to what I might think of a snowflake. But you talked about symmetry in the context,

for example, of a sphere. So can you carefully define what you mean by symmetry? So symmetry is the things you can do to a shape that leave it looking the same. So snowflakes, I mean, nature is full of hexagons, but snowflake's an example of this. They're hexagonal, right? So that means if you rotate a snowflake, like one sixth of the way around a circle, it's going to basically look the same and you won't be able to tell.

So if I close my eyes and you, for some reason, are holding a snowflake and you rotate it and then I open my eyes again, I won't be able to tell you've done it, right? So that's a symmetry. Now, if we think about a circle, actually you can rotate a circle about its center any amount and it'll still look the same. So the circle is infinitely symmetrical. And you can also do things like mirror lines, right? You can put a diameter of a circle, reflect it in a mirror there, you won't be able to tell. So the circle has infinitely much symmetry. It's the most symmetrical shape, two-dimensional shape you can draw. Other shapes like the lovely regular hexagon that bees use,

snowflakes use, the hexagon there, it's pretty symmetrical. It's got 12 symmetries. So that's quite a lot. It's more than some amorphous blob. But the circle is like the winner of this. And the sphere, that's the same in three dimensions. You know, you can rotate it in any direction in three dimensions and it still looks the same. So that's how we can kind of count symmetry, really. Have you ever read Flatland? In Flatland, you know, the circle is the... If you are circular, you are the best shape.

They're like the high priests of the circles. And all of the men are polygons, and all the women are just lines. But then the polygons would have bits cut off them to become circular. Yeah, it's this weird... It's like cosmetic. Exactly. They go to the, like, circulotherapeutic gymnasium to be like, you know, do workouts every day to make them more regular

It's the gym to become more spherical I think an interesting point from your very first question there is why do we see Patterns and curves in nature's we like to think mathematics is separate from nature's it's a game of the mind and that's certainly true At some point. But many, many times, mathematics has been influenced and motivated by nature. You see the pattern. Why is that there?

Here is the mathematical framework. I'll use to understand it. Then I'll play with the mathematics. And at some point, you come all the way back to realize that biology is beating you to it. Biology has had millennia to get good at what it's doing. And so it usually beats us to the answer. Bees, honeycombs, best packing, how did they know?

The bee's brain is the grain of sand. How did it know to pack it as a hexagon? It didn't. It just had millennia to get it right. I mean, I have a shape here called the Gombock. This was motivated by some mathematicians. They were looking for a shape that only had one stable point. They did the mathematics, they found it,

and then they realized that, again, biology had found it before them. Do you have an idea what that looks like, what biological system? Dave, you're an artist, so could you describe to the listeners at home who can't see this, what does that look like? From certain angles, it looks like an acorn.

And then as it's rotated, you go, oh, no, it's not. It's a nice idea. But it's a little bit like a Cylon baby. You remember the Cylons from Battlestar Galactica? I would imagine they would look a little bit like that. It's like a Roman helmet on a Subbuteo base. That's very strong. Conquistador helmet. I can like that. It's like a Roman helmet on a sub-Utio base. That's very strong. Conquistador helmet, I can see that.

But no, it's the shape of a tortoise shell. So the Indian star tortoise has a very similar shell. So the idea of the shape is it's a self-writing shape. And no matter where you put it, it will always end up on its bottom. And you can imagine why a tortoise would hopefully want a shape like that. Why would a tortoise?

Because doesn't that mean it ends up on its back? Hopefully not often. But when you do, you want to make sure you can imagine why a tortoise would hopefully want to shape like that. Why would a tortoise? Because doesn't that mean it ends up on its back? Hopefully not often. But when you do, you want to make sure you can get back up. There's two kinds of tortoises. Ones that are very flat, and they usually have long necks. And they can wiggle their neck around. But an Indian star tortoise has a very big shell. And when that flips over, it uses its self-righting properties

to pop itself back up. So this is essentially, in both cases, the hexagonal packing of a honeycomb and the toy toy shell, is evolution finding the solution. Just finding its way, yeah. You know, it's just trying all the solutions and it finds a local maxima and it just turns out, oh yeah, that was the equation that solves that we didn't know and nature just found it better than us.

So just to say about this shape, so it is a shape that if you roll it on the, I don't think you're allowed to roll it on this beautiful floor. Yeah, roll it on the floor of the Royal Institution. What's the worst? But it is a shape that will always eventually So you've ever played with a weeble it's not like that So if you ever played with a weeble you it was a little egg-shaped person you'd knock it over and it would always write Itself, but the thing with those is that they have a heavy bottom, and so that they're not uniform. What the Gabor Domokos, the person behind this discovery,

he wanted to say, can I just find a shape that would do this? So it's not heavier on the bottom because I make it heavier. It's the shape that drives the self-writing mechanism. So it's absolutely beautiful. So is there ever a worry, it's something we've mentioned before on this show, Brian Green, he once talked about the fact that he would have little panicked dreams occasionally, where he imagined the aliens coming to Earth and saying,

oh, we used to think mathematics was the language of the universe too. And do you ever? It's nice that that was his panic dream. Some people have dreams where the aliens come down and eviscerate us with lasers. But his was more of a mathematical conundrum. And I just wonder, do you ever think that, are we imposing, is there any doubt that you

ever have in terms of seeing the mathematical language that we see within the shapes of nature? So I think there's certainly a risk of sometimes over-diagnosing patterns, like we love spotting patterns as humans, and sometimes we can read more into things than is there, but in terms of the basic way we look at the universe, for me, mathematics is more of an approach than a collection of facts, like sciences. So what does a mathematician do?

They look at a situation, they try and see where the patterns are and they then try and explain what is it that's governing these patterns and can I put some theory behind it and maybe make predictions. So that isn't something that is, you know, you can then say this shell proves this is wrong because it doesn't, you know, it's just like, can I put these patterns together and try and understand them? And as I say, sometimes we go too far and perhaps we like to see the golden ratio everywhere when it's only in quite a lot of places,

but the basic underlying principle, it still holds. I'm glad you say it. We look for it, therefore we put it there. So it is an art because we put it there. It is reason, there is motivation for it to be there, but is it truly there in the petals of a flower? Maybe, I've not yet found a good reason for it to be there or not.

I'm not saying it's not there, but I'm yet to be convinced. It's a golden ratio, so maybe you could expand. So the golden ratio, this is one of the most, it's got some really good marketing, the golden ratio. Everyone's heard of it. Yeah, everyone's heard of it. It's got this really good like marketing. The golden ratio. Everyone's heard of it. Yeah, everyone's heard of it. It's got this great name and like it's got the Greek letter phi is the symbol for this number or ratio. So what it is in essence is if you take a line and you divide it up sort of part way

along. So now you've got it divided into a longer bit and a shorter bit. The golden ratio is the exact ratio that you can divide that line up into so that the kind of the short bit compared to the long bit is the same as the long bit compared to the whole. So you've got these three quantities and they are related in this lovely way. You see it in geometry, so in a regular pentagon, and the diagonal is in that relationship with the side. You can fit, for example, a dodecahedron, you can fit inside a sphere with that radius.

If you get an icosahedron, which is one of these platonic solids, it's made of 20 equilateral triangles. It's got 12 vertices, 12 vertices, and they can be grouped into three fours which make rectangles that are at right angles and are this golden ratio rectangle. So lovely, lovely properties. Phi comes from, this is the first bit of myth making, it's called Phi because Phidias was the architect of the Parthenon and in the 19th century there was a writer who said the Parthenon. And in the 19th century, there was a writer who said,

the Parthenon temple is all based on the golden ratio. It isn't. And then Vitruvius, the great kind of Roman architect, you know, wrote a lot about proportion and how things are all in harmony and the human body and all of this. Lots of ratios there. None of them are the golden ratio. Leonardo da Vinci did not

use the golden ratio in a Vitruvian man. The famous book, there's a very famous book by a mathematician called Luca Pacioli called the Divine Proportion, De Divina Proportioni, about the golden ratio. Never says it has anything to do with the human body or anything like that. And this just sort of came along later. He talks about the golden ratio. He doesn't call it that. He calls it divine proportion. It's got this threefold, a bit like, you know, the Holy Trinity. It goes on forever.

It's an irrational number like the immortals and all of this. But yeah, there's a lot of myths around. But there are some real places it happens in nature, like in the spiral patterns of seeds in some flowers and some plants. I know you love Kepler and I was just thinking, is the golden ratio, would that have been the reason, you know, when he tried to look at the pattern of the planets and he used various different perfect solids. That's a different thing, the platonic solids.

And I just wondered whether that would have, again, this kind of way of finding that pattern would have some similarity to the kind of, you know, the trying to create the perfection that we see in golden ratio. So at the time Kepler wanted to understand why it was the planets were the particular distance they are from the sun. So he took the six planets that he knew about and he looked at the distances from the Sun and he realized that to quite a good approximation, if you nested the five platonic solids like in between those

radii of the planetary orbits, it gave you the right sort of answer. So there's these beautiful, beautiful diagrams with all these nested solids inside them. So these are the sphere, the square... So the platonic solids, the tetrahedrons, that's like four triangles, like a triangular pyramid. Then you've got the cube. You've got an octahedron, which is eight triangles.

Let Dave see if he can guess the next one. Yes. You've been very strong so far. You're very good at hen. Have you ever played with Dungeons and Dragons? No. Ah, there you are. He does look like he plays Dungeons and Dragons.

My oldest brother played Dungeons and Dragons and that was enough to put me off. Yeah. So the dice in Dungeons and Dragons often are these platonic solids because they're very symmetrical and you get an equal chance of hitting each number like cube dice. So the other ones have this similar property. So dodecahedron, that's 12-sided. Plato thought that was the shape of the universe. And then finally the icosahedron, 20 sides.

So Kepler thought if he nested them all in the right way, it would explain why the planets are where they are. This isn't true. I mean, now we know it can't be right because there are exactly five platonic solids and there are more than six planets, right? So it's not going to work. So that was an example of Kepler who was, we all love Kepler, but he had a bit of a mysticism thing going. But the genius was that then he changed his mind based on the data. Yeah so he's like this is what I want but it's not that. So then okay we try again and that's the mark of a good scientist, a good mathematician, they can have a theory but

if it doesn't fit what's there then they throw it away. This is interesting. What for you Dave I'd just like to find out in terms of as someone who you know you did train as a mathematician Have you found a moment where a theory that you've been very you know heartfelt theory and you still find the mathematical mind means That it no longer works for you No, my only Confrontation with this sort of thing would be and I say this is a father of a nine-year-old

So I dropped out of university, I didn't complete my math degree, didn't think about it forever, and now I'm bumping into a nine-year-old's homework. I would like to remove the word oblong from the language. I think it's a pointless word. I thought it was a word that we used when I was a child and then grew out of in the same way that Burgundy used to be maroon. But it turns out we're still using the word oblong and that has annoyed me a great deal. I mean the big problem with the maths homework occasionally is that,

so I get slightly exercised because you get to say, which of these things is a rectangle and which is a square? And I say, well, the square is of course just a special kind of rectangle. And then the child, poor child will say, mum, you know very well that that's not what we're supposed to say. Maybe I should say no matter what troubles you have with your maths homework, my daughter's is in Welsh. What's the challenge of Welsh Math? I was in Thelanethyla just yesterday, what's the challenge of Welsh Math? When they lay it out with numbers and equations, I'm very happy, but it's just the language I don't know what integer is in Welsh.

I don't know what oblong is in Welsh. It's a rectangle. It's a rectangle. It's an interesting... You mentioned the idea of looking for the golden ratio then, so we go and look at it. You said plant leaves So are we in danger? It's a double-edged sword in a way because as we've explained there's there's there seems to be an underlying beauty to nature

Which is described by mathematics, but at the same time this quest For perfection has led us astray many times if you couldn't justify it so just to clarify the point about the leaves is that there is this golden spiral in the leaves because you can define what a leaf wants to do. And mathematicians love defining things. A leaf wants to do two things. It wants to grow such that it blocks as little sunlight from the leaves below it and it gets as much sunlight as it can from above it.

And if you optimize over those two ideas, you get the idea that it should be a golden spiral and your leaves form So by golden spiral, how do you get the golden ratio from that? The ratio of one petal to the next comes back to this idea of you breaking up the stick such that each petal is the same ratio to the whole as to the next. It's 1.6-ish. 1.6-ish, yeah. But it's 1.6 of the way around the circle. Yes. Yeah, so it's the angle that's being defined there in that. And the reason...

So if you think, as the plant's growing, it's kind of the place where the direction is rotating where the next leaf is going to grow. So you think, what's the perfect angle for that? If you did say 90 degrees, every 90 degrees, I'll have a leaf, then after four leaves, you'd get overlap. So if you have things that are like a fraction, like a quarter or a third of the way around,

then you start overlapping really soon. Waste of a leaf, though. Waste of a leaf, you're not filling the space, you're not getting the most light. And so what you really want is an angle that isn't a fraction. You want an irrational angle, something that can't be written as a fraction. And it turns out that the golden ratio, that angle that you get, that is, in some kind of strict mathematical sense, the most irrational number there is. So numbers that you can have different kinds of infinity, you can have better and worse kinds of irrational.

Please expand on that. There's joy in his face. Because Robin, for years, for about 10 years, we've talked about different kinds of infinity, which we won't go into now. Just to show you, there are different kinds of numbers, the pi, there's the e, exponential. So, rational is a decimal that goes unfairly. Exactly, so there are many, many types of irrational number,

but as we said, phi is the most irrational. So, if you use pi for your sunflower seeds or your leaves, then after, like, seven revolutions, you'd nearly be back overlapping. So, even though it's irrational, because you've got a really good small number approximation, it sort of acts a bit like a rational thing. Now, phi, the golden ratio, it turns out that when you try and make these approximations,

you have to go to really, really big numbers before it's close to being the golden ratio. And you can prove this mathematically using a technique called continued fractions that actually there's no other number that's worse in this respect. Has an architect ever made a circular block of flats with balconies arranged on this principle? Almost certainly.

If not, why not? That's a great idea. It's a great idea. Almost certainly. That would be a copyright. I wonder if Corbusier ever did. I mean, I said it. You did.

But you didn't say copyright. Typical, typical scientists. Try and take the credit. Copyright isn't like dibs. They're Gorman flats. Do you remember in Singapore where there's that flat which at a certain angle is deliberately being designed that looks two-dimensional?

That's in biology because we think that's what a leaf is trying to do. There's a reason for it to be there. That's what I care about. That number, we can say, is there because of these reasons. We all can all pretty much agree that leaves want to maximize the sunlight.

But there are other cases where some people say, well, the number of petals that a flower has is a Fibonacci number. And there I'm a little bit more hesitant, a little bit more skeptical. The Fibonacci sequence is 1, 2, 3, 5, 8, 13, and so on. And 50% of the numbers less than 10 are Fibonacci. So you've got just a good chance of getting a Fibonacci number

anyway. But people have used this to say, well, you don't get a four-leaf clover, because four is not a Fibonacci number anyway. But people have used this to say, well, you don't get a four-leaf clover because four is not a Fibonacci number. It's like, I don't yet see a reason why that's true. So that's the two kind of differences there. Some reasons, yes, I can see that there's a golden ratio in the spiral of petals. But is there a Fibonacci number of petals?

This week on Witness History, we mark the 10th anniversary of a plane disaster that there are a number of petals. the pilot's voice recorder revealed. Search and subscribe to Witness History wherever you get your BBC podcasts. I don't know. I don't know. Thomas that you worked on them. So not patterns in leaves, but also patterns that you find, for example, on the cheetah or the leopard, which is a very famous problem. I think Alan Turing was the first to begin to look at that problem. So the patterning that you find on a cheetah, let's say. Yeah, Alan Turing, I mean, my absolute mathematical hero, he worked on computers before computers

existed. He broke the Enigma code. I mean, just stop there. You've done enough. But then one of his little pieces, one of his little known pieces of work he did just before he died was he was never one for small questions and he asked, why aren't we spherical? You know, we were looking at this question earlier, you know, small animals can be spherical, why aren't humans?

What makes us go from an egg, which is spherical, into this weird spindly shape you see before you? spherical into this weird spindly shape you see before you. And he wrote down this beautiful theory called the chemical theory of morphogenesis. And we still work on it. Turing was ahead of his time, always was. And we are still unpacking his ideas today to try and understand how it works.

And you'll see my fine waistcoat I have here, which I have to thank my wife for making. So Dave again for the radio audience. Describe that waistcoat. If you can imagine an impossible maze. The pattern is called Labyrinthine. In purple and orange. It's basically a costumier who took some LSD and made a waistcoat. That's pretty close. So Turing wrote down these equations, these reaction

diffusion equations. And he showed that they could generate patterns without a hand of God. And what I mean by that is that some people have worked on, where does pattern generate? How does it come about? But they'd always have something that kicks you off, something that starts the pattern,

some asymmetry in the system. Whereas Turing's system didn't need any of that. The asymmetry arose from the interactions of the components themselves. And like I say, it's absolutely beautiful from a theoretical point of view, as well from a visual point of view. And what's even better, to come back to the cheaters,

is it's a testable theory. But as I say, biology makes fools of us all. One thing we can derive from this theory is that patterns should get simpler as skin gets smaller. So in a wide part of the skin, on a big part, on the body mainly, you should be able to get spots. Then as you go towards the legs or the tail, it should simplify to stripes. And then towards the end, there should be no patterns at all, whether there are nubs. Does that work? Well,

it works for the cheetah. Spots on the body, stripes on the tail, nothing on the end. Beautiful. Does it work everywhere? No. There are fish out there. Spots on the body, stripes on the body, spots on the tail, stripes on the tail. The tapir hate tapirs. They make mockery of the whole thing. They have stripes on the body, spots on the legs, and then as they get older, they lose everything. So what we find there is just that our mathematics

can tell us a certain amount. We know it can't explain everything. We don't try to explain everything. We just know, okay, this works for cheaters, but it doesn't work for fish. There's a great example, a great story that goes around that mathematicians prove that bees can't fly. And people take that as, oh, mathematicians, you don't know everything. We don't.

But it's not that we prove bees can't fly. It's that we prove that that's the wrong type of mathematics to work on bees. Now, I wanted to ask you actually about, but we were told that you had a fantastic analogy involving a sweating grasshopper. Sweating grasshopper, yeah. And you know when you hear that someone's

got an analogy involving a sweating grasshopper, it's very hard not to open that Pandora's box. So the original analogy, which we'll skirt over, came from Turing, and he used missionaries and cannibals. So we'll move past that colonialism into sweating grasshoppers. So the analogy of where does these patterns come from? How does it work?

And this came from a mathematician called Jim Murray. And imagine a field full of grasshoppers. And when these grasshoppers get hot, they sweat. Okay? But if you just release some grasshoppers in the field, they jump around, they're happy, there's no pattern there, they're randomly jumping. Okay, remove the grasshoppers, put them to one side. Now burn that field. At the end of your burning, you'll have a completely black field. Again, no pattern. Now let's put them both together. You put your grasshoppers in the field, and you set fire to one corner.

The fire will start to spread, your grasshoppers get hot, they start to sweat, they make parts of the grass wet, so they can't burn. So when the fire is burnt out, you will get patches that are burned, and patches that aren't. And that was the genius of Turing. He saw that two mechanisms that don't produce patterns, the fire and the grasshoppers, separately, when combined,

can produce patterns. And that's without the hand of God. I have to get to you that the grasshoppers would run away from the fire. And that would make them hot, and that's what they sweat. Talking about patterns, and you talked about the different patterns and different scales of patterns. The classic example of a pattern, I suppose,

that's really surprisingly the same, whichever way you look at it, is fractal. So can we talk about fractals and what makes a fractal where we find them in nature? Fractals are almost the perfect design strategy for things like plants, but lots of natural phenomena. And I say design, obviously, these things evolve. And the reason fractals are useful for nature

is because, as you say, they have this kind of self-similarity property. In other words, whatever scale you zoom into, the mathematical fractal will just keep looking the same. And in nature, that's useful because imagine, you know, you're a fern that's growing or a plant. It's a very simple recipe that can be evolved that, okay, I grow a little bit in a straight line and then I split, I branch.

And then each of those branches, it doesn't need new instructions, it does exactly that same thing, grows a bit, split, grows a bit, split. And so what you get is these kind of smaller and smaller and smaller branchings, which, you know, with a tree, you get big branches and smaller and smaller and then little twigs, but it's the same basic structure that's being created at all different scales. And they're great for transport as well. Yeah. In the tree, they're transporting water through these things. In our body, our venous system, transporting blood. Fractals are a great way to transport things.

But even another, we don't tend to call these things fractals, but the same property of you can make it as big or as small as you like and when you zoom in you get the same shape is another reason why we see spirals in nature at every scale, right? Galaxies, but also little shells and everything in between. We get these kinds of spirals that are called logarithmic spirals. So there's two spirals that mainly we think about. One which looks maybe like a coil of rope right that's called an Archimedean spiral that every kind of layer of that is sort of constant just going up in constant distances. The

logarithmic spiral it's called that because there's a logarithmic there's a power relationships between how far around you've gone and the angle you've gone. That the the parts of the are getting, as you get further away from the centre, they're getting wider and wider and wider. So it's what happens when something is growing and rotating, like shells and when galaxies spin. When something's spinning, you get this kind of pattern. But they look the same at every scale. If you zoom in, it still looks the same, so it's perfect for like a shell that is going to grow,

you know, an nautilus, for example, that grows and wants to stay the same basic shape. It's a perfect solution. But there's a really sad fact about these logarithmic spirals. So the mathematician Jacob Bernoulli loved them, wrote about them. He called this kind of spiral the miraculous or marvelous spiral. And he loved them so much that he wanted one of these spirals carved onto his gravestone. And so when he died, they were like, okay, do this spiral. And then the person who did it did the wrong kind of spiral. They did an Archimedean spiral. Again, these ideas, those are places where the spirals are known to be,

because mathematicians have created them. But there are places where they can be used to describe phenomena. So there has been work done on looking at tissues. And the dimension of the tissue, the fractal dimension, correlates with whether it's cancerous or not. So whether that's a useful diagnostic tool, I'm not going to comment on,

but we can still use these mathematical objects as ways of describing features that otherwise we wouldn't be able to. Yeah, thinking of these kinds of structures. So people might have heard of the Koch snowflake curve, which is another kind of fractal. You start with a triangle and then you, every time you have a line, you take the middle third out and put another triangle and you keep going and going.

Creates this nice sort of snowflake-like fractal curve. Now that was first introduced in what, 1904? Very early in the year. Something like that. Okay, you can sort of do that and I used to often do it in lessons if I was thinking very hard about other things. You can draw these things, but you can't really take it very far. You certainly wouldn't be able to do something like the Magelbrot set or the other kinds of fractals just really by hand.

So it's only when computers became powerful enough to start drawing these things that we could really start exploring the possibilities and doing these wonderful things where, yeah, you zoom in and you zoom in and you zoom in and you say oh my goodness there's another one of those those beetle shapes or those cardioids. And I love that you use the word explore because I remember coming across some of the great book by Roger Penrose that I love a lot called The Emperor's New Mind where he talks about that. One of the great mathematicians.

That may be a mathematician that book. Yeah and he talks about that you're exploring the mathematics as if it's there. It's a world, a universe. We should because we pretty much run out of time, but we haven't done the latest shape news. I love that. Discovering new shapes. Yeah, so should we go for soft sell or should we... That is what it's called by the way. Oh, brilliant, I'll commentate.

No, he's not. I mean, you would think, wouldn't you, that all shapes were known. So could you talk about these new shapes that have been discovered, let's say, soft cell. The gombok you see before you, that was discovered by Gabor Domokosch just by asking this question of,

can I find a shape that will always write itself? Guess who is behind these new shapes? Gabor Domikos, just by asking this question of, can I find a shape that will always write itself? Guess who is behind these new shapes? Gable Domikos. He just asked, can I find a shape that will tessellate? And by tessellation, I mean that they fit together so that there's no gaps. What's the minimum number of corners a shape can have and tessellate?

So squares. Have you asked that, haven't you, before? Oh, yeah. Yeah. Oh. What were your findings? Three. Triangles, triangles, test lathes.

If you've got an infinite floor. Yeah. And that's the minimum? I beg your pardon? Is that the minimum? I thought so. And you're certainly right. If you have straight edges, triangles

are the minimum, the three. But if you start allowing curves, you can get down to two. And that is the minimum. You can prove that that's the minimum now. But then the question you start allowing curves, you can get down to two. And that is the minimum. You can prove that that's the minimum now. But then the question is, well, what about three dimensional shapes? If we start having curves, we don't just have to put blocks

together or tetrahedra. What happens when we put curve shapes together? What's the minimum number of corners there? And through this, they found whole families of shapes, which again, biology had found first. Biology hates a corner. We spoke about beehives. found whole families of shapes, which again, biology had found first. Biology hates a corner. We spoke about beehives, but really what a bee is doing there is not creating a hexagon,

it's creating the wax around its body, it's using its body as a template. And if you pack circles together, you get a hexagonal structure for free. So biology hates corners, because it needs a lot of energy to keep a corner rigid. And so we would much rather have curves. And so soft cells are found in biology everywhere. Biology hates a corner, but then it's never had to put storage into a one bedroom flat. Thank you to our guests, Dave Gorman, Sarah Hart and Thomas Woolley. We asked the audience questions today. We asked them the one shape that they didn't think is used enough in the world. What have you got Brian?

One that tapers smoothly from a fat flat. I hate tapiers. I hate them. So tapers. Tapers smoothly. If an animal, the thing about spots in the big parts of the body and then stripes as they get thin and then nothing, so if you imagine an animal as just a simple cone, it would have spots at the fat bit of the cone. So what you're saying is a tapir tapers incorrectly? Absolutely.

Mike, thought my blood pressure rising. That sounds like the beginning of a Flanders and Swann tribute act by the way. This one is. So you said it's a smooth tape, a smooth tape from a flat circular base at the ape to an apex, not contained in the base because things cone only get better. More people today. I've got one here from Elena Mills, I think, who says, we asked chat GPT and it said a trapezoid. It's like a rectangle that just gave up halfway. It's the shape no one asked for, but it deserves a comeback just for the sheer chaos it would bring to everyday items.

A little hobby of mine, actually, is going to to IKEA and making lots of trapezoids just by loosening some allen keys. I'm gonna hand this one over to a mathematician. Oh! That is the whole of the answer that was placed there. So this is a shape we all know and love right the rhombi are cosy dodecahedron You know that shape isn't that in wet north Wales No, it's it's it's a very very interesting shape It's it's not quite as regular as the platonic solids, but it's in that list of like the next kind of rank. And but Jeff here says it would look good on a Scrabble board.

I mean, it would get you a good score on a Scrabble board. It's a rhombi. Rhombi icosydoidecahedron. As a crossword compiler, Dave, I hope that we'll find at least two of the words you've heard today somewhere in whatever crossword you're working on next. Hendecagun maybe. Yeah, so also I would just say that this is the end of this series and the next series starts with a very big celebratory episode because it's our

201st episode and we're doing a big celebration but we wanted to throw out to the mathematicians if any of you know why we chose 201 to be the celebratory episode, which seems abnormal. I think it's because you booked better guests than us. For your 201st episode. You two mathematicians, why do you think? I mean, is it the start of your third century? No, not quite that. It is technically, yeah. No, not quite that.

It is technically, yeah. No, no, no, that's not why we chose it, so that is still wrong as an answer. 201, I mean, if you came up to me and asked me what a 201-sided shape was, I'd have to disappoint you again. Dave was actually closest, but the reason that we didn't do the 200th episode as a special celebration of 201st, is because we haven't counted out how many episodes we've done We've recorded them all and

We were hoping you'd come up with something that would make it look like we had a cast-iron alibi Thanks very much everyone. Bye-bye In the infinite honky cage Turned out nice again. Hello, Greg Jenner here. I am the host of You're Dead to Me from BBC Radio 4. We are the comedy show that takes history seriously and then laughs at it. And we're back for a brand new series, Series 9, where we're covering all sorts of things from Aristotle to the legends of King Arthur to the history of coffee to the reign of Catherine of Medici of France. We're looking at the arts and crafts movement and the life of Sojourner Truth and how cuneiform

writing systems worked in the Bronze Age. Loads of different stuff. It's a fantastic series. It's funny. We get great historians. We get great comedians. So if you want to listen to Your Dead to Me, listen first on BBC Sounds. This week on Witness History, we mark the 10th anniversary of a plane disaster that cost the lives of 150 people and left many struggling to understand the truth of what happened. The German wing's plane crashed in the French Alps. Find out what the pilot's voice recorder revealed. Search and subscribe to Witness History wherever you get your BBC podcasts.