The Science of Symmetry

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Live in the Bay Area long enough, and you know that this region is made up of many communities, each with its own people, stories, and local realities.

I'm Erica Cruz-Guevara, host of KQED's podcast, The Bay.

I sit down with reporters and the people who know this place best to connect the dots on why these stories matter to all of us.

Listen to The Bay, new episodes every Monday, Wednesday, and Friday, wherever you get your podcasts.

This is a download from the BBC.

To find out more, visit bbc.co.uk slash radio4.

In only 4.6 billion years, the sun will swell into a red giant engulfing our own planet Earth, Mercury, and Venus.

So let's enjoy these last few moments we have together.

This is the Infinite Monkey Cage.

Hello, on my left is the scientific advisor for the film Sunshine, in which a group of astronauts go into space to send a bomb into the middle of the sun to restart it.

Yeah, and yet somehow he's a professor.

I take most things he says with a pinch of salt.

It is Brian Cox.

And on my right is a man who recently failed his Turing test.

It's Robin Ince.

Now,

occasionally, it's said that monkey cage is unnecessarily elitist.

So to show that that isn't the case, if you're not familiar with the Turing test, it's prefigured in Descartes' great 1668 treatise Discourse on the Method.

So, just dig your copy out and have a look.

So, today we're going to be looking at the tail symmetry of the barn swallow and the visualising differences between honeybees and bumblebees.

Hey, honeybees like pentagonal symmetry, but bumblebees like mirror symmetry.

What's all that about?

Honeybees?

Hey, just shtick.

So,

in other words, we'll be looking at the importance of symmetry.

Why does symmetry underlie everything from mating to the fundamental laws that govern the universe?

As usual, we're joined by three guests.

Our first guest remains our only guest who is a self-declared wizard.

Now, for the audience in the studio, which one could it be?

Clean-shaped man number one, clean-shaven man number two, or the man with your enormous beard?

His work has included the creation of Watchmen, V.

Fendetta, Tom Strong, and Maxwell the Magic Cat.

It is Alan Moore.

And we're joined by a man we like to think of as the Indiana Jones of Symmetry.

Nothing will stop him from his quest to view the symmetries concealed within the great buildings of antiquity.

The only thing that stops him is if a man in a hat goes, oh I'm sorry, I can't actually go in there.

That's the only thing.

He's a mathematician in the Charles Simony Chair for the Public Understanding of Science at Oxford University.

It's Marcus de Sortoy.

And just as Brian has been described as the sexy face of particle physics, I am pleased to say, yeah,

it goes to someone different every year.

It's like Pipe Smoker of the Year, you know.

Our next guest has been described as the handsome head of ocular genetics.

A man who can look deeply into your eyes on a first date and say, you know, as I look into your eyes, what I see is the potential for a detached retina.

I know someone who can deal with that.

Yeah, but I think you will know him better for this.

This is really true.

This is a paper that he authored.

I know it's very, very well known.

It's male sexual ornament size, but not asymmetry reflects condition in stalk-eyed flies.

Who could it be?

Of course, it's our favourite stalk-eyed fly expert, Adam Rutherford.

There we are, that is our panel.

So,

proceedings of the Royal Society, it's really true this.

Do you remember what the conclusion was?

We conclude that ornament size is likely to play a great role in sexual selection.

That's the conclusion.

Man, over there going, I've got a lovely set of Toby jugs, just you wait.

By the way, can you stop saying it really is true?

Because that feeds into all those people who write to us and go, I see you've made up some science again.

This bit really is true.

The rest of it, we're just guessing triangles are predominantly magic because Marcus DeSoto.

We will start with you, Marcus, actually, because in a show where we are going to be dealing with symmetry, symmetry in nature, symmetry in the universe, let's first of all have a definition.

What is symmetry?

Well, it's actually quite a hard question to answer that one because we've taken about 2,000 years to try and pin down what we really mean by symmetry.

And I suppose most people's first impression is symmetry is something to do with left-right reflectional symmetry.

Our faces tend to be symmetrical.

But actually it's something much more than that.

And if you've got an object like a dice, what can you do to that object such you can turn it and put it down and it looks like it did before you moved it?

So in some sense I like to call symmetries like the magic trick moves.

What can you do to a structure, change it in some way such that when you put it back down again, it looks like it did before you moved it.

And symmetries are physical objects, but they can be also symmetries of more abstract things.

Like, say, take a pack of cards and shuffle those.

Then a shuffle can be regarded as a symmetry of the pack of cards, but that's a much more abstract sort of symmetry.

And it took us really until the beginning of the 19th century to formulate what we really mean by symmetry.

There was this great guy called Evaris Galois who came up with a language to actually articulate what we mean by symmetry when two objects have the same sort of symmetries, when we can say we found all the possible symmetries that there are in nature.

And Galois was a great figure, he's one of our most romantic figures in the whole of mathematics because he died in a duel over love and politics at the age of 20.

And already by 18, he'd cracked this problem about creating a language to understand the world of symmetry.

And I know that your research is looking for symmetries.

So what's the object with the most known number of symmetries?

Well, that's quite simple.

It's a circle, because the circle has an infinite amount of symmetries.

So if you think about what is a symmetry of a circle, if I take a circle and move it, well, there are infinitely many ways that I can move that circle so actually it looks like it did before I moved it.

So, actually, there are things, and I spend a lot of my time dealing with objects with infinitely many symmetries.

So, that's the sort of the most symmetrical, or a sphere, for example.

But then you can sort of get a breaking of symmetry.

So, for example, if you have something like a dodecahedron, which can sit inside a sphere, so a dodecahedron is a shape with 12 pentagonal faces, that's got less symmetry because all of the moves of the sphere don't align the dodecahedron inside it.

So if you look, you find there are 60 rotations that I can make of the dodecahedron which make it sort of sit back inside its kind of outline.

And that seems to be at the heart of the way our universe evolved, actually.

That it's about something which started with a lot of symmetry, and those symmetries started to break, which gave rise to the kind of interesting world we live in.

Adam, we think of symmetry predominantly, well, mathematically, but in terms of when we look at the slightly more skewif world of biology and living beings that we have,

where do we see symmetry within nature, within biology?

The skew-if world of biology.

See, it's a good show.

You don't know about the skew if world of biology?

It's very good, it's on Dave at the moment.

Notice that he

normally get this from Brian.

He said that.

I was going to say it wasn't me.

Yeah.

Well, actually, it's skew.

It's not, you know, we look around at the moment, don't we?

And, you know, to actually see the patterns in nature, sometimes you have to go slightly deeper than just initially what we can see with our own senses.

True, but symmetry is obviously almost omnipresent in nature.

Animals are categorized as being bilateria, which means that we are symmetrical via the axis of our nose.

Almost all animals are like that.

Some aren't.

Some are radiata, which have rotational symmetry, like starfish or other round animals.

There are very few.

That's exactly the kind of description I'm looking for on my skew if world of nature.

That's that you are the perfect expert.

You tried David Attenborough far too specific with your other round things.

It's a huge research interest, other round things.

There are very few examples of asymmetric animals.

Sponges is one.

But it's a basic facet of almost all animals that we have this axis of symmetry.

And it shows quite how efficient evolution is, making sure that their energy is conserved and that we produce things which are easy to make.

And you can see it through

600 million years through evolution.

Alan, in terms of because we've dealt, obviously with the experts there, we've looked at mathematics and biology, so I thought now for you we'd look at mythical beasts.

And

the ideas of symmetry within storytelling and then within ideas of biology, where I think it was Plato who came up with a challenge to say why did we require love, why did we require a partner?

And I think Aristophanes came up with this idea that

we were basically a beast that had been split in two.

We were a four-legged beast, and we were constantly trying to find the person that would then make ourselves one again.

So, you know, when in terms of storytelling, ideas of symmetry seem to be played around with a lot in ancient times.

Why do you think that is that it's something you can conjure with?

Well, it's very pleasing, symmetry, isn't it?

I mean, I've heard that in the field of the arts, apparently an obsession with symmetry is a marker for psychosis.

Apparently, now I'm not dismissing the entire

obviously,

I'm just making a comment.

But I think that with symmetry, it's so pleasing, but in terms of in narrative terms, I mean, the simplest form of symmetry has probably got to be a palindrome.

You know, Abel was I, ere I saw Albert, or something like that.

Yes, that's a palindrome.

I once tried to write, well, I actually succeeded, but nobody knew because I tried to write a piece of prose where the words were symmetrical and it all hinged around the.

I just put a full stop in the middle of the page

and then put the same word on either side of it and then just worked out from there.

But unfortunately, by the time you get to the end of the piece, you can't remember the beginning.

I did a similar thing in Watchmen where I'd got a long sequence where I'd laboriously laid out all the panels.

So overall, the entire issue, the panel layout, is completely symmetrical.

But you don't really notice that.

I can't remember why I did it, to be perfectly honest.

Because you could,

because I could, because I could.

So, I think what's interesting is artists very often use symmetry to set up expectations about where, because you think you know the pattern, which they then break.

And I think that is that kind of tension between symmetry giving you structure, which then they play off.

Bach is a great example.

The Goldberg variations are just a song to symmetry.

You've got all of these, each variation is somehow using symmetry to vary the theme in some way.

But when you get to the 30th variation, it's called a quad libbet.

It's a musical joke.

It has nothing to do with the rest of the structure of the piece.

But it's the breaking of the symmetry there which makes you realize how much structure Bach's put into the piece up to that point.

But that works exactly for DNA as well, because the process where DNA replicates, it's better described as duplicating, because you've got the information for both strands encoded on both strands of the double helix.

And when it splits in two, you're actually getting a duplication of two symmetrical molecules.

But in exactly the same way as what Marcus just said, evolution is about when you get deviations from that symmetry.

We can just wrap the show up now if you want.

Well, I'd like to.

I'd like to.

We've talked about symmetries of objects, so spheres and cubes, triangles, and so on.

But we also talk about the laws of nature themselves being symmetric.

So we're talking about equations having symmetries here.

So what do we mean by the symmetries of an equation?

Well, that's really interesting.

If you take something like Dirac's discovery of antimatter, that's somehow a symmetry in the equations there.

If I ask you what's the square root of 4, well, most people say 2, but there's a mirror symmetry answer to that, which is minus 2 times minus 2 is also 4.

So in equations, you can get symmetries happening where solutions have sort of mirror symmetries or more complicated symmetries.

And those often give you some indication that there's probably something corresponding to that.

So Dirac eventually led to the discovery of antimatter.

Dirac thought it was protons, I think, rather than but actually, the sort of positive solution was the electron, and we discovered antimatter through being the negative solution.

So, very often, these ideas of symmetry and equations can lead you to the discovery of new particles, which is what you're doing at CERN, basically.

Yeah, this raises a very interesting question because we're talking about something that sounds very abstract and mathematical, but in fact, the idea that these symmetries also describe nature, describe the laws of nature, describe the universe, is interesting.

Yeah, and I think it comes down to the fact that symmetry often gives you economy.

Symmetry is very often there to create the solution that needs the least amount of energy.

So for example, why do bees choose hexagons to make their beehive?

The hexagon, we can prove, is the shape which uses the least amount of wax to contain that particular area of honey.

So any other shape is less economical.

Why is a bubble spherical?

A bubble, again, is a shape with the least surface area and so a minimal energy.

So I think that the reason we're finding symmetry all over the place, especially in fundamental fundamental physics, where somehow an extraordinary sort of symmetrical object which lives in very high dimensions seems to explain everything that's happening in the LHC, is about the fact that symmetry is the kind of the easiest way to make things.

And Adam, we've heard there about the symmetries in particle physics, we've heard about symmetries in beehives.

So symmetries in animals are symmetries in plants.

What is the origin of that symmetry?

So why are we bilaterally symmetric?

And why do flowers have the symmetries that they do?

Well, basically, because we evolved from tube worm-like things, and because they are tube worms in the sense that they are tubes, they have a head and a tail, but a tube is a symmetrical object, right?

And that laid out deep in our evolutionary past the basic body plan from which all animals have evolved.

And we still have basically the same body plan now that we did 600 million years ago, which is we have a head and an anus.

Much underrated point in the evolution of life on Earth was the evolution of the anus.

Don't go on about that too much, it goes out at four thirty this show, so it's just a certain point.

But the whole idea that we have an axis, we have a head and a tail, that applies to almost all animals.

Almost all, not all.

I I've already mentioned that there are a group of round animals that

which we don't really talk about that much.

But the the spherical ones.

The spherical ones, yeah.

But yeah, basically we we have exactly the same axis as almost all animals, which is that we have a head and a tail.

And the most efficient way to have a head and a tail and to have useful functions associated with having a head and a tail is to have it in a bilateral symmetrical way, a left hand and a right hand.

I think it's intriguing that we don't have symmetry inside our bodies, though.

You know, the heart is on one side of the body, it's the external side which is symmetrical.

I mean, is that something to do with the fact that

we're communicating information about our DNA?

It's hard to make symmetry, therefore, the more symmetrical you are, I think that very often people associate that with being more beautiful.

Is that a kind of DNA indicator?

Well, possibly, kind of.

So our internal organs are not selected for.

You generally don't choose a mate based on what their lungs look like.

I say generally, because biology is all about exception.

But the reason we have asymmetry on the inside is because it's a sort of packing problem more than anything.

We need much longer guts than we have the physical structure to have in

long ways.

What about?

Don't get the symmetries of the brain.

I mean, because you've got these two two symmetrical halves which control the opposite halves of the body, and presumably that must lead to some striking asymmetries.

I mean, like I used to have a theory that if your left eye is governed by your right brain and vice versa, then you should be able to sort of judge people by covering up half of their face in a photograph.

This is a strange theory.

I can't actually

saying it out loud, I realise that it doesn't sound very likely.

But

I would be looking at mainly at pictures of murderers.

You know how you do sometimes.

You know,

you have wicks like that, don't you?

But all of them have got wildly asymmetrical faces.

They've all got a strange left eye.

I think Tony Blair actually had the same condition.

I think that was his crazy eye, wasn't it?

And the only exception to that that I found was the murderer Jean-Alandrew, who was called Bluebeard.

And his face is absolutely symmetrical.

And it's not until you see a face that is where the two halves are like,

complete mirror images, that you realise how odd that is.

I don't want to take it back to asymmetry before you finish the symmetry part of the show, but

consolidation.

Infinite monkey cage, bringing back the Victorian science of physiognomy out of the wilderness and back on the air.

That's one of the things that you actually mentioned in your book, Finding Moonshine, as well, is that symmetry takes a certain amount of energy and effort to achieve.

You used as an example, for instance, the hen's egg.

You say, you know, there is, I believe, evidence that says a battery hen more often, because of the stress and the energy taken on stress, means that malformed eggs are more likely.

Whereas, so actually,

I'm interested in the idea of what is required to create symmetry in the middle.

Yeah, it's very interesting because actually, sort of, I think in the life sciences, it's quite hard to make symmetry.

Things can often slip, and so any sort of hardship in upbringing causes asymmetries, which again, I think, is why we tend to associate symmetry with beauty, because we're drawn to something with symmetry, because it's likely to be, you know, have good genes, good DNA, good upbringing, and make a good mate.

But on the physical sciences side, actually, symmetry is something that it seems to be a low-energy state that things naturally are drawn to.

Like the bubble, it's trying to make itself symmetrical.

So it's kind of a strange sort of tension there between biology, symmetry seems to be difficult, but in the physical sciences it seems to be the state that things want to assume.

Adam, there are notable exceptions in biology.

There are the crabs that are asymmetric,

fiddler crabs.

Owls, I think you mentioned to me earlier, have asymmetric ears.

So what are these notable exceptions and why?

So what Marcus is talking about is subtle deviations from perfect symmetry, which almost no one has.

The archetypes of beauty that are sometimes discussed do tend to be more symmetrical.

Like you say, there are some really significant examples of proper, really measurable asymmetry.

The fiddler crab has one enormous left claw and a normal, weedy right claw.

That is relatively easy to explain.

It's sexually selected, it is an ornament, it is a male ornament, as you mentioned.

But basically, it's no different from

a peacock's tail or the antlers on a deer.

It's something that has been selected because females prefer a bigger one.

The barn owl, I'm just, I offer that with no comment.

Only on the left.

Again, though, it appears to be like this is evolution being efficient.

It happened once, it began to evolve in that direction.

The claw began to get bigger over time, over generational time, and so it stayed on

the left-hand side.

The owl's a really interesting one because our ears are perfectly, well, they should be perfectly set as being symmetrical, which means that if they are perfectly symmetrical, if you close your eyes and listen to a noise which is absolutely in front of you, you can't tell whether it's absolutely in front of you or absolutely behind you.

Now, there are several owl species that use their slightly comically shaped face as a funnel for sound.

So, when they're listening for a little innocent doormouse that they're going to tear the head off,

it's better that they have asymmetric ears than perfectly symmetrical ones because they can then pinpoint the target more accurately.

But that's a perfect example of Darwinian natural selection.

You really are doing all the sex and violence in this, aren't you?

All to tear its head off and hangs on the left, bigger ones are.

Biology is.

Biology is sex and violence.

Yeah.

And Alan, I mean in terms of when you are using ideas of symmetry or asymmetry, when you're trying to create, for instance,

ideas or creatures that you wish to disconcert your audience with, how much when we hear of something like the fiddler crab, for instance, there, the idea of taking shape and pattern and then then malforming it?

This is often something that the artists will do.

Perhaps, I remember that there was a creature in Captain Britain that the artist had made deliberately asymmetrical, and it hasn't even got the components that you can make into a face, and humans can make nearly anything into a face.

So that's disturbing.

I suppose the thing about symmetry is that surely it depends upon the level upon which you're actually perceiving it.

Like we were talking about the messiness of the natural landscape.

Yeah, cloud, a sunset does not appear to be symmetrical.

Very beautiful, but not symmetrical.

And yet, the components of a sunset, and I mean this would probably be something for you, Marcus,

in that

we discovered that things like clouds

are obeying strict laws laws of mathematics.

They're just more complex.

Well, in some ways,

they're simple.

I mean, you're getting, in a sense, to the ideas of chaos theory.

That chaos theory can have formation of shapes like clouds or even in nature as well, can have very simple rules which can still create something which looks incredibly complicated.

Yes, it appears to have no rules at all.

So it's a matter of orders of complexity to a degree, and our perceptions.

There will be layers of the universe which will appear orderly and symmetrical to us

because our perceptual apparatus is geared for a certain level of complexity.

We can't see the order in clouds, but we can have it explained to us in the form of fractal mathematics.

Well, I think we're incredibly sensitive to symmetry because anything with symmetry really has a message kind of hidden inside it.

It's there for a reason.

And so I think we've all kind of evolved.

I mean, you mentioned earlier how we seem to be very sensitive to picking out things with symmetry.

If you think about just, you know, when we're in the jungle, the chaos of the jungle, all the leaves, and suddenly you see something with symmetry, well, that's likely to be an animal, and either it's going to eat you or you're going to eat it.

So you suddenly become those who can spot symmetries, survived in this world.

In symmetry, there is this idea of the monster.

And can you just tell us a little bit about what this is, the monster in the world of?

The monster.

Basically, we understood through Galois' work that we can create a periodic table of symmetry, the building blocks of symmetry, which include things like coins with a prime number of sides, for example.

But one of the strangest things that was discovered in the middle of the 1980s was this extraordinary symmetrical object which lives in 196,883-dimensional space.

You can't see the thing.

It's kind of like an incredible snowflake that lives there.

And it can't be broken down into smaller symmetries.

It fits into no patterns.

It's a kind of strange symmetrical object.

It has more symmetries than there are atoms in the Sun, I think.

Well, Brian's better at those kind of big numbers than I am.

How many atoms is it in the Sun?

How many symmetries does it have?

It has

billions and billions and billions and billions of.

Was it not like.

Sorry to ask us my

Manchester accent is.

Sorry, was it the thing that we were talking about?

Was it 10 to the 50?

It's 803 sextotillion or something, which is

8 times 10 to the 53 number of symmetries.

Yeah.

So that's quite a lot.

But how does someone imagine?

I mean, that's the thing I find fascinating.

That must take a very special kind of mind to even go into that realm.

Yeah,

I'm being careful.

Absolutely.

There's a group in Cambridge that, when I started my PhD, had just finished this project of classifying all the sort of building blocks of symmetry, this periodic table.

And they are an obsessive lot.

John Conway, who people might know because he's created the game of life, one of the things, and he's just somebody who just loves playing with stuff.

And he can actually imagine this shape in 800, you know, 180, well however many

dimensions he gets.

I'm just fascinated how you go about finding that.

I mean, did you check four dimensions, five, six?

Six,

how do you?

No, it's it's it's I mean it's very interesting because um it's a little bit works the same way as fundamental physics where often you'll you'll make a prediction for something like the Higgs boson and then you've got to go out and find it.

So we played around with this language of group theory, which helps us to understand symmetry, and realized that there were kind of restrictions on what could happen.

And we saw that there should be one in that dimension.

And then the thing was to go out and try and find and construct an object in this, you know, huge thousand-dimensional space or whatever.

I think the power of mathematics is that you can create something which you can never see, but using the language of mathematics, you can create an object which lives in so many dimensions beyond our three-dimensional world.

So, this is uh, we were talking about the great minds and minds that uh perhaps cannot really be imagined or understood.

And so, to help us with that, we asked the audience a question.

So,

the question we asked them because Stonehenge is an example of man's fascination with symmetry.

So, we wanted to know from our audience here, what do you think Stonehenge, the Stonehenge?

Stonehenge.

What do you think?

Stonehenge.

Stone Engel.

So, we asked the audience, what do you think Stonehenge was really built for?

And this is what they came up with.

Stonehenge, we got as a perfect backdrop for Brian to be filmed.

I think it says Brian to be filmed panting at the stars,

pointing, sorry, panting though.

Oh, look at that one.

Oh,

I've not seen that one before.

I said, man, oh.

That's from Emma T, thank you.

This one appears to be from a creationist because it says it was built as giant tables for dinosaurs so they could eat off them, which implies that humans and dinosaurs were on the earth at the same time.

Which will please a very small fraction of our audience.

They are gateways to parallel universes, which we'll be dealing with next week.

D-Ream's Comeback Gig, that's for you as well.

And a medieval hairdressing salon for goths.

So, there we are.

So, thank you very much.

There are just a few of the ideas of Stonehenge.

And thank you very much for all our guests for helping us with the ideas of symmetry and asymmetry.

They were Marcus DeSotoy, Adam Rutherford, and Alan Moore.

Next week, we'll be looking at parallel universes with John Lloyd, Professor Sir Martin Rees, and Dr.

Lucy Green.

So, finally, some people have been complaining about the time slot of this show and say that they've been missing it due to being either at work or asleep.

So, Brian will now explain how, possibly utilizing any wormholes you may well have lying around the house, you can manipulate the fabric of time to ensure that you don't miss it.

Brian

because space-time's hyperbolic, and the speed of light, which is, I suppose, the conversion factor if you like between distance and time, is finite and agreed upon by all observers, you can travel into the future arbitrarily far by travelling relative to the person that's stationary.

So you get to next week any time you choose if you can just calculate the velocity.

The gamma factor, by the way, is square root 1 minus v squared over c squared.

Alternatively, you can just download the podcast, which is available all the week.

So thank you very much for listening.

Goodbye.

If you've enjoyed this program, you might like to try other Radio 4 podcasts, including Start the Week, Lively Discussions chaired by Andrew Marr, and a weekly highlight from Radio 4's evening arts program, Front Row.

To find out more, visit bbc.co.uk/slash radio 4.

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