A Sense of Balance

 A Sense of Balance

John D Barrow Gresham Professor of Geometry

This first talk from my new series of lectures on maths from everyday life is going to be about some matters of balance and equilibrium.  I am going to start by showing you some examples from the world of sport, which is going to motivate us to thinking a little more clearly about what it is that determines balance and stability.  We will move on from there to look at rather more esoteric applications, in outer space, and some even in your living room.

Balance is a precarious thing.  You might ask yourself what it is that determines whether things can remain in balance when they are static like that, but there is a secondary question which is much more interesting, and that is, when things are moving, or perhaps wobbling, what is going to determine whether they are stable or not?

Our first motivating sporting question considers the high-jump.  If you had been a high-jumper before 1968, you would not be high-jumping like Stefan Holm, for instance, who won the Olympic Gold Medal eight years ago in Athens with a height of about 2.35m.  What is extraordinary is he is only 1.80m tall.  The world record is 2.45m, which is rather astonishing, but the person who set that record was considerably taller (almost 2m tall) than Holm. 

You can see the technique that is employed by high-jumpers in all competitions: you go over the bar backwards.  It did not always happen like that; if you had, you would have ended up in the hospital, because it was only in the mid-1960s that air-beds became commonplace for high-jump landing areas at major athletics meetings.  It is no coincidence that it was around 1967/1968 that the American high-jumper Dick Fosbury invented the technique of high-jumping that bears his name, the so-called Fosbury Flop, which every high-jumper now uses.

But what is happening here that is different perhaps from other techniques?  When you were at school and perhaps eight or nine years old, the technique that you would have been introduced to for high-jumping would have been called the scissors technique, and really, it is just a glorified step over the bar, so you effectively just hop over.  You can see that somehow it is much less efficient because you are taking the whole of your body over the bar.   The technique of the so-called straddle, was commonplace until 1968, so it would have been used by the world high-jump record holder until 1968 or so, where you sort of rolled over the bar, with your chest passing over the bar as closely as you possibly could.  So you would run, you would raise your outside leg, and then you would roll over the bar.  This was a very difficult technique to learn.  The best high-jumpers tended to be quite old, simply because it took a long time to perfect that technique. 

One of the advantages of Fosbury's technique is actually very simple, despite appearances.  An athletic child introduced to this technique with a good landing area will be able to do it very quickly.  You do not have to worry about where your arms are flying around because it is your back that is close to the bar.  So what passes closest to the bar is not a part of the body that has got lots of bits sticking out which you have to worry about managing.

So, what is the advantage then of this technique?  Why do all high-jumpers in the world use it?  The answer is a mathematical one, because what you have to think about when you are jumping is that you have a certain amount of energy of motion that you use to launch yourself in the air, and that energy of motion will determine how high off the ground you can get, fighting against the force of gravity trying to keep you down on the ground.  But what matters is the height to which your centre of gravity can be raised, and your centre of gravity is about two-thirds of your height, on an average person.  It is your kinetic energy - which is determined by how much oomph you have, as it were, how much power you have got in your muscles, how much elasticity to launch off the runway - raises your weight, your mass times the acceleration due to gravity, by a height h, and that h is the height that your centre of gravity is raised:

            Launch Kinetic Energy = M x g x h             h = height of centre of gravity is raised

Because the old-fashioned scissors technique keeps the torso vertical, you can understand that using that technique will mean that your centre of gravity is going way over the top of the bar.  Your centre of gravity is probably going one foot or 18 inches above the bar, so you are not going to be able to clear a very high bar because you are being really rather profligate with your kinetic energy, since you're losing 18 inches or so on what you have earned from your kinetic energy.

The trick of the Fosbury Flop, is that the centre of gravity is actually going underneath the bar.  So, if you are particularly bendy, like a banana, the centre of gravity of a banana is not on the surface of the banana itself; it is somewhere below the banana bend.  And so it is with a modern high-jumper: the centre of gravity is somewhere below the body as it arches over the bar.  Therefore, by arching yourself in that way and using that technique, you can send your centre of gravity under the bar, while your body slides over the bar in a curved path.  So, in this way, you use that launch kinetic energy most effectively since the centre of gravity is going underneath.  This is very much in contrast to the old method, where the centre of gravity was going over the top of the bar.

What sort of difference does this technique make?  A little bit of mathematics will tell us.  We can start by working out what the kinetic energy that you would have when you launched with is, which would be a half times your mass times the launch speed squared:

        Vertical kinetic energy K = ½mv2

I said just now, if your height is L, your centre of gravity will be about two-thirds of your height off the ground.

        Centre of mass h = 2L/3 for jumper of height L

If we put these two together, we can see that the highest point when you jump, Y, will be the height that you have raised your centre of gravity by is that maximum height minus where your centre of gravity started out at.  So it is Y minus h, which is V squared over 2g, or launch speed squared over twice the acceleration due to gravity, which is a standard formula for throwing something up, under gravity, or dropping it:

        Y - h = v2/2g is height CoM raised

We can put a few numbers in now.  Let us suppose that our high-jumper is 1.98m tall, which is a typically very tall high-jumper, and so the centre of gravity will be about 1.32m off the ground, and you want to clear a bar that is at 2.13m.  If we put all these numbers in the formulae, what you find is that the amount of kinetic energy that you would have needed to launch with is 0.81 times your weight, and there are some units of metres, so you can think of it as 81% of your weight in the appropriate units:

        L = 1.98m h= 1.32m to clear Y=2.13m:         Y - h = v2/2g         2.13 - 1.32 = v2/2g         KS = 0.81´mg

But that is what happens if you use the straddle and you lay absolutely flat as you pass over the bar.   But if you are using the flop, then you are going to bend your body so that you send the centre of gravity underneath the bar.  So, in this case, with these numbers, the centre of gravity of the high-jumper's body goes about 10cm underneath the bar rather than over the top.  So now, you just need to be able to achieve a rise of the centre of gravity up to 2.03cm in order to get your body over a height of 2.13m, and that means a real saving on what you have to do at the launch.  So instead of 0.81 times your weight as the launch energy, 0.71 is sufficient, so this is a big saving of your energy resources:

        C of M goes 10cm below bar, hips 7cm above.         Now Y = 2.03 to clear 2.13m         KF = (2.03-1.32)´mg = 0.71 xmg         A 12% saving on take-off impulse using flop

So it is pretty obvious why this Fosbury technique is so popular: it is easy to learn, and energetically very advantageous.

That problem was about the centre of gravity, and lots of people think therefore that centre of gravity really is the key factor in determining all aspects of balance and stability and optimisation, but let us look at another type of problem.

What I am interested in is: why tightrope walkers carry long poles?

If you stop the average engineer or even mathematician in the street and pose this question, a lot of people will tell you straightaway, 'Oh, it is to lower their centre of gravity,' that the lower your centre of gravity, the more stable you will be.  It is true that you are more stable if your centre of gravity is lower.  That is why Formula 1 cars are so flat on the ground and have such a wide wheel base, since it makes their centre of gravity is very low.  They will not easily turn over going round a corner.  But if you think about it twice, holding the pole actually seems to raise your centre of gravity, so it does the opposite to what people tend to think initially.  So the pole is not actually about centre of gravity.

What it is about is something that mathematicians and engineers call inertia, or moment of inertia.  If you want to move something, what matters is not just its weight or where its centre of gravity is located, but how its mass is distributed.

For example, you might have a solid sphere and a hollow sphere, like a shell, which have the same mass and the same size, but they are made out of different materials - perhaps one might be made out of lead and the other out of aluminium.  How would you tell which was the hollow one and which was the solid one?

There is a simple way to tell and that is to just roll them downhill or try to move them.  What you will find is that the hollow one is much harder to move and rolls more slowly than the solid one.  So what this shows is that there is a quality of objects which we call inertia, or moment of inertia, which reflects how the mass is distributed.  In the case of the hollow sphere, the mass is far from the centre - it is a long way away; and the object has a high inertia - which means that it resists being moved, more than does an object where the mass is distributed evenly throughout.  Objects with high inertia are slow to move, but objects with low inertia are quick to move.

This is the source of the little tricks as to how you tell a hard-boiled egg from an uncooked one.  So if you spin them around, the uncooked egg will push its interior contents, which are rather liquidy, out towards the boundaries, and will become more as if it where hollow, whereas the hard-boiled egg is solid and will stay with a low inertia, so they will behave differently when you spin them.

The mathematical formula for inertia is as follows (where Mass is 'M', Average radius is 'R' and mass spread is 'C'):

        Inertia = C x Mass x (Radius)2

So the inertia of an object is its mass times the square of its radius or its diameter, times some factor which tells you how the mass is concentrated. So, for a solid sphere, that concentration factor is two-fifths, and for the hollow shell, it is two-thirds, so the inertia for the hollow shell is larger.  So the message here is that, if you can move mass away from the centre of an object, you will increase its inertia and make it harder to move, or it will move more slowly.  In fact, the period of wobble oscillation is proportional to the square root of the inertia:

        Period of wobble oscillation ∝ √(inertia)

This of course is what is going on with our tightrope walker who holds the long pole.  By holding the long pole, the tightrope walker is increasing their moment of inertia by moving mass a long way away from their centre, where the wire is.  Sometimes you see exotic tightrope walkers trying to walk across bits of the Niagara Falls and they go even further: they carry heavy buckets on each end, so that is moving even more mass away from the centre.  So what this does is that it means when you wobble on the wire, you wobble more slowly, and you have more time to correct and restore your balance, making the tight-rope walk easier.

If you have a six-inch ruler or a one-foot ruler or a one-metre ruler, and you try to balance them on your finger, you will find it rather harder to balance the shorter ruler on your finger because you won't be able to react quickly enough to restore the balance.  But if you put a metre ruler stick on your finger, you will find it much easier to respond and restore the balance.  This is an example of exactly the same principle.

If you look around the world of sport and movement, you find this type of feature reflected quite often.  For instance, you will note that speed cyclists use different types of bikes to you or I might in going down to the shops.  The important difference is that speed cyclists have discs for their wheels instead of having spokes like the rest of us do.  Indeed, I guarantee that none of you will be riding around the streets with bikes with discs for wheels because you would fall off pretty quickly.  If you remember what we have just seen: if you have a solid object and you spread the mass evenly through it, it has a different and lower moment of inertia than if you have all the mass at the outside, with just a few light spokes.  So, a disc wheel will respond differently because it has got a different inertia to the typical bicycle wheel.  So the idea is that, when you apply some pressure by peddling, it will respond and move more quickly.  Of course, this is only going to work if you are on an indoor velodrome, because if you try to cycle down the road with a wheel like this, as soon as you turn it slightly away from directly ahead, the air resistance will make it flip round and you will fall off.  So to make this work, the cyclist has to cycle very carefully, making sure that the wheel is always at 90 degrees to the track surface and always pointed exactly along a tangential direction.  As a result, if you see a serious road cyclist, you might find that they have that type of low inertia real wheel, so it responds quickly and moves quickly when you press on the pedals, but for safety and stability reasons, it has a hollow type of front wheel, so it has a slightly larger inertia.

These are simple examples in many ways because they are just considering motion about one axis.  But, many objects in nature are three-dimensional, like ourselves, and there are three different directions about which they could rotate, and the distribution of mass within that object will determine how it will rotate about different axes. 

For example, a squash racquet has three axis about which one might spin it.  So, if you were holding it vertically in the air by the handle, you could spin in around the axis of the body of the racket, spinning it in your hands as you see squash players do between points; or you could lay it flat on the floor and spin it around on that axis; or you could throw it in the air and throw it along the axis at 90 degrees to that one.

Three-dimensional objects then have these three axes of rotation, and there is a rather remarkable, sometimes alarming, property of these three axes.  The inertia is different about each of them.  So you can see, on a squash racquet, there is a different distribution of mass away from the centre in a horizontal axis direction, compared to a vertical axis.  So the stability properties will be different for these three rotations, and a key property of a three-dimensional object is that the rotation about the in-between axis of inertia, so where the inertia is neither the biggest nor the smallest, is always unstable.  So if I rotate my racquet about the axis for which I think the inertia is the smallest, it is stable; I can sort of spin it round.  If I were to put it flat on the ground here and spin it around, it would also be stable.  But if I throw it upwards so that it spins in the air handle-over-head so that the face of the racquet is opposite to what it began as, so there has to be a twist, this is the manifestation of the instability.  This instability of rotation about the intermediate axis is something one sees this in all sorts of situations.

If one watches the gymnasts carefully, particularly female gymnasts, at the Olympic Games who do exercises on the beam, you will often see them perhaps moving along the beam and then doing a somersault in which there will be a twist, so they will do a somersault or a cartwheel and they will end up facing in the other direction on the beam.  You seem to get extra credit for doing that, but you really should get extra credit for not going it, because it is like the instability of the rotating squash racquet: you cannot avoid it.

Of course, divers, trampolinists and gymnasts, exploit this very carefully, because you can change your body shape to alter what is the intermediate axis of inertia, so that you will either twist or not twist according as to how your organise your mass distribution.  We will come back to that in a moment.

There was, some years ago, a rather remarkable example of this problem that was nearly an enormous space disaster.   It was in 1977.  It was not really publicised at the time, but it came to light later on through space agencies.  This was to do with the Mir Space Station, which, in 1977, was a great celebrity thing in space.  The Americans and Russians were collaborating, both sending astronauts to stay there, and in fact Britain had an astronaut, Michael Foale, who was on the space station (I think he had to take American citizenship in order to take part in this programme because Britain was not a collaborator technically in the project).

There was a disaster with one of the bits that are typical of space ships in that they always appear to be precariously stuck onto the side.  But this type of structure is possible because there is no gravity, so you do not have to worry about things falling down.  For instance, there were great sails which collected solar energy.  What happened was the Russian docking device was supposed to come in with more supplies, so a smaller spacecraft was supposed to fly in and dock on one of the ends and transfer supplies. 

Foale told me that Russian steering technology at the time was really very primitive, so, in order to slow down and meet this docking section of the station, they were having to make key manoeuvres hundreds of miles away.  It was as though you are driving to Edinburgh and if you want to change gear on the outskirts of Edinburgh, you have got to start moving the clutch somewhere around the Watford Gap service station.  So, perhaps inevitably, there was a crash, and the docking vehicle crashed into Mir.  It was sent spinning as a result of the impact, and the astronauts had to retreat to a little emergency capsule.  Had they done that, they would have in effect been abandoning the ship and it would undoubtedly eventually have broken up, leaving a quarter of a million tons of space debris in orbit around the Earth and putting an end to a very large amount of any future space programme work above the Earth's atmosphere - so it was a potentially huge disaster. 

But, even staying put, there was a big problem, because it is rotating, spinning, recessing, doing all sorts of complicated things.  You have got to bring this thing back under control, and you have some rocket fuel effect which you can fire in any direction you choose to try to cancel out the rotation.  But, very quickly, people back at Mission Control and Foale himself, who was trained as an astrophysicist at Imperial College and then at Cambridge, realised that the disaster could be compounded if you did not understand which were these inertial axes of rotation of the space station, because if it was rotating about the intermediate axis, giving a little push could make it flip over, just like my squash racquet, and then it would break in pieces and everyone would be killed.  What you wanted to do was to make sure that it got pushed into one of the stable rotationary motions and then slow it down by just the reverse rotation.

Nobody knew what the principal moments of inertia of the space station were.  Nobody had ever even thought of such a question before.  So, Foale, with his laptop up in the spinning space station, had to sit down, getting information from Mission Control about the geometry and the weight of everything in the space station, work it out, and then solve the equations of motion for the rotation of the spacecraft about these different axes, and calculate precisely what the direction in which you should fire the rockets to cancel out the rotation effect would be, without passing through that unstable axis of rotation.  So, fortunately, Foale's background in mathematical physics enabled him to do this and to do it right.  So he calculated what was the right direction in which to fire that rocket, so that you cancelled out the rotation and stabilised the motion of the space station.  So we see that this business of instability about the intermediate axis has all sorts of application, some really some which are of gigantic importance to us.

There are many examples of this feature appearing in sport, as we mentioned earlier.  I indicated that you often see gymnasts, divers, particularly people in trampoline, utilising the findings that we have arrived at through our mathematics.  They have lots of time to do lots of things on the way up and on the way down, and what they will be doing is changing the position of their body, a tuck position or a spread position, so that they alter which axis is the stable one and which is the unstable one.  So if you just want to do a somersault, you want to be in a particular configuration, so that you spin and rotate in a stable way.  If you want to do a twist as well, you want to spread out a bit so that your somersault is about the unstable axis now and you will do a twist as well.

In winter sports, we see another specific application of this idea.  So you remember inertia looks like mass times the square of the distance over which you are spreading it.  When you see skaters pirouetting and spinning very rapidly, what they do is start with their arms out and then they bring their arms in.  What they are doing is reducing their inertia, so they spin more rapidly as they bring their arms in.  In fact, the rate of spin is inversely proportional to the square of the extent of their arm, so as they bring their arms in, they will spin up very dramatically. 

On a more mundane level, I realise that those people who have cars and enjoy paying lots of money to garages for things like tuning and repairs, something like a small metal clasp goes on your wheel when your car disappears into that mysterious place that reduces your bank balance quite dramatically.  This is to do with wheel alignment.

What is going on when your wheels are balanced is that the axis of symmetry on which the wheel is fitted is being brought into balance with the moment of inertia axis, the principal moment of inertia about which it rotates and wobbles.  So if those two axes are not parallel, are not aligned, then you are going to get wobbling, in a small way, of the wheels.  So how the alignment is done is to add these little weights to create a change in one direction until things are balanced.  So the whole balancing of your car and the avoidance of this type of mismatch between the axis of rotation and the axis of symmetry on which the wheels are mounted is another problem of moments of inertia and balance.

We mentioned rotation.  I want to look at a much more elaborate example now.  This is the unusual problem of a cat.  If you drop your cat from a height, it will not injure itself (but perhaps it might be best not to try it).  You might know what happens: cats go through an unusual sequence of movements to make sure that they always land on their feet.  This seems to be innate in cats.  Within just a week or so, a small kitten will be able to do this, to some extent, and they have a skeletal flexibility which helps them in doing this.

What is odd about this, for someone in engineering or mathematics, when you think about it, is that there is a quality of rotational motion, which we call angular momentum, which is conserved.  So if you start something rotating, then there is a measure of that rotation which cannot be changed.  So once a cat starts to fall, it cannot change the magnitude of the momentum of angular motion, but what it can do is redistribute it in different ways, just like the diver can do movements in one direction and then in another direction, cannot create rotation from nothing, but can redistribute it.  So what is happening here with a falling cat; what does the cat do?

The flexibility of the cat's spine is such that it can orientate itself so that its front legs are rotating in one direction, while its back legs are rotating in another direction, so it is like a jointed thing - it is not like a single object; it has a joint.  What it does, first of all, is that it bends in the middle, and makes sure then that the back legs and the front legs are free to rotate in different ways, in different directions, about different axes, and so forth.  The aim of the cat is to make sure it ends up landing nicely on its feet.  So, if it has enough time, what it wants to do first of all, it wants to get those front legs round, so it tucks in the front legs, so it reduces the inertia, whereby there is less mass far away, so the front legs will rotate quicker, and the back legs are stretched out so that they have a larger inertia and they rotate more slowly.  So the first step in the game is to get the front legs rotated round quickly, but not have the back legs moved as well, so the back legs move more slowly, while the front half really rotates a lot.  You do not want the back legs rotating at the same time because there is a conserved quantity of rotation, so in order to rotate the front a lot, you have got to not rotate the back in the same direction.

The next step is to forget about the front legs, tuck the back legs in, extend the front legs, so you have got lots of inertia for the front movement, and the back legs can now move quickly.  So what then happens is that the back legs then rotate a lot, and the front legs hardly rotate at all, so you bring the back legs round, and then you are in the correct position.  If you were to drop the cat from a very great height it might try and go through this process more than once in order to make sure it is in the nicest possible landing position.

There was a strange story I saw a few years ago, which was some veterinary surgeons reporting on injuries to cats in New York, and it seems that there are quite a number of cases where cats fall out of the windows of skyscrapers. So they perhaps have wandered around on window ledges high up, and it becomes much windier than they ever imagined, and they fall very great distances.  The curiosity was that it was the cats that fell the greatest distance that tended to have least injuries compared with ones that fell the smaller distances.  So what was happening, presumably, is the ones that fell a great distance had more time to get themselves in the best position for a soft landing, with the back also arched so as to absorb all the stress; whereas, if they were falling just two floors rather than ten floors, they might not have enough time to effect all these cunning manoeuvres and land in this rather nice way.

The last topic I want to talk about is about a very unusual object, whose stability has been something of a shock to mathematicians and engineers.  You might have heard about it.  It has been on the television occasionally, and it has a Hungarian name, it is called a Gombóc, which means a sort of sphere or scoop.

This object was created by two Hungarian mathematicians, Gabor Domokos, a friend of mine, who was in Budapest, and Peter Varkonyi, based in Princeton.  The manufacture and theory behind making this object was something that went on for many years, so Gabor told me they really started thinking about this about twenty years ago, and finally, they managed to make this rather unusual type of object that nobody thought was possible.

This is a solid object.  It has uniform density, so it has not got a big heavy lump at the bottom, and it is convex, so everywhere on its surface it points outwards, what that means, if you were to draw a line from any point on the surface to any other point on the surface, that line would have to go through the interior of the object.  So there are no cavities cut the object.  This object has the property that it has one stable equilibrium point and one unstable equilibrium point.

Such a situation would be impossible if this were in effect a two-dimensional object.  So two-dimensional objects have more than one stable equilibrium point.  You cannot have a two-dimensional object that has only one stable equilibrium point and one unstable equilibrium point.  And for a long time, I think probably every mathematician in the world, except one maybe, thought that it would be impossible to make a three-dimensional object that had one stable and one unstable equilibrium point.

A stable equilibrium point is like the bottom of a valley, so if you move away from it, you end up falling back to that point.  An unstable equilibrium point is like standing a pencil on its point - if you move it at all, it will move away.

If an object has just one stable and one unstable point, like the Gombóc, then however you put it down, it will always end up in the same configuration: at the stable equilibrium point.  It may vary in how much in needs to wobble and rock in order to reach the stable point, but it will always reach it eventually.

It was predicted by Vladimir Arnold, a very famous Russian mathematician, that it would be possible to find an object of this sort, and after many years, Domokos and his colleague were able to establish that such an object should exist.  To make this object is rather extraordinary.  You need an engineering precision of one part in 100,000 in order to make an object that has precisely the properties you need, so this is more accurate than for the finest Swiss watch.  So you cannot make a Gombóc any smaller than about the size of a large thumb-nail, because you cannot make this any smaller due to the engineering precision required to do so would be too great for current technology to handle.

These objects are therefore expensive.  Domokos and his company, which set about making these - I think it was a German engineering company - made a set of 2,000 numbered ones some years ago, and you could buy the numbered ones.  The first one was presented to Arnold.  If you want to buy one of the first ten, I think there are one or two still available (there is a website), it would cost you around €50,000.  If you buy an unnumbered one, which is what I have, as a gift from Domokos, it is something like £400.  So it is a reflection of the engineering that is required, not some novelty value, but it really does require very expensive engineering.  It is like buying an expensive watch where the price reflects the engineering that went into it.

The first unusual aspect of a Gombóc is its uniform density and it is second important aspect is that it is convex and the final aspect is that it is smooth.  Then, of course, there is the Gombóc's distinctive shape, which I won't go into.  So, if you have an object that has a big hole cut in it, you can just sit it down and it will find an equilibrium point in all sorts of places, so it behaves quite differently.

If you have a two-dimensional oval shape, say a sort of slice through an egg, then there is a stable place to sit it down, if you turn it the other way up, there will be another stable place, but if you stand it on the other end, like on the point of the egg, it is going to fall over.  So there are two stable and two unstable places if you have a flat oval, like a cut-off egg.  The point about the Gombóc is that you have just got one stable and one unstable point.  You cannot do that with a flat object.  What you need to do that is the absence of another type of equilibrium point. 

If you think about a saddle for a moment, the sort of saddle you put on a horse, this has the property that, if you were to move in the saddle in line with the direction of the horse, it is like a stable valley, so if you were to wobble in this direction, you will drop down to the bottom of the saddle and stay on the horse.  But if you roll off in the direction perpendicular to the direction in which the horse is going, it is unstable. So a saddle point is like a combination of a stable valley and an unstable hill, it matters in which direction you move if you want to be stable or unstable. 

With three-dimensional objects, there is a rather pretty mathematical theorem that has been known for a long time, that if you have got some number of stable equilibrium points, of valleys, call them i, and some number of unstable ones we will call j, then the number of saddle points that there has to be is equal to the sum of those minus two:

        If the object has i stable and j unstable equilibria, then?         It has i+j-2 saddle points

Let us look at a very simple example.  Suppose that you have a cube. There are six equilibrium points, so we can put it flat on any of the six faces, so there are six equilibrium points.  The disequilibria, or unstable equilibrium points are like the corners. So for a cube there are eight disequilibrium points.  So the number of stable plus unstable points is six plus eight, which is fourteen, take away two is twelve, so there are going to be twelve saddle points on a cube.  Where are they?  They are the sides, the edges, because if you moved along the edge, things would be stable, they remain on there, but if you moved perpendicular to that, you would fall over.  How many edges are there? 

So what does this mean for the Gombóc?  Well, if there is one stable equilibrium point and one unstable, then the number of saddle points is one plus one minus two, which is zero.  So the novelty of this is that there are none of these saddle points.  So the challenge for Domokos and co was to construct the geometry of a solid object that had those properties.

Another curious point about this is that, in some rather precise sense, it is an object that is really neither flat nor thin.  You are familiar with thin things, like pencils, and they have usually two unstable equilibrium points: you can try and balance them on one end or the other end.  Things like Frisbees or that squash racquet will have two stable equilibrium points, which will be either side.  What one can do here is to measure what you mean, or define rather precisely what you mean by thinness and what you mean by flatness, and you find that the Gomboc is defined by the fact that it has the smallest possible combination of these features of flatness and thinness, which means that instead of having two stable or two unstable points, it has just the one of each.

What has this got to do with the bigger outside world?  Well, Domokos and his colleagues talked to zoologists for a while, and they were always very interested in finding places in the natural world where you might discover manifestations of this object.  First of all, they started sifting through hundreds and thousands of pebbles on beaches to see if natural erosion processes had made pebbles that had this type of structure, but unfortunately, they never found a single one.  So the process of erosion, if you started off with your beach covered with pebbles that looked like Gombócs, the unusual shapes and plates and corners would soon get eroded and they would become much more rounded, so natural processes tended to remove this type of configuration. 

But then they discovered that there are some unusual types of tortoise in the natural world - there is the one called the Indian Star, and there is another type of tortoises that are called radiated tortoises - and these tortoises have a very unusual shell structure on their back, which is highly raised.  When you look at the geometry of the individual plates on that shell, they reflect a number of the features of the convex arrangement on the Gombóc.  Of course the tortoise is not homogeneous, it is not uniform in density, but the purpose here is that if these tortoises find themselves on their backs, they can, by a sequence of manoeuvres get themselves back on their feet, so they use the curved plates to roll in one direction, and then in another, and then flip themselves round onto their feet.  So this appears to be a process by which natural selection has selected for tortoises that have these nice convex shields on the outer area of their plate to enable them to right themselves more efficiently and more effectively than your average turtle.

So, I hope that I have given you some perspective on some unusual aspects of balance.  Remember what we learnt first about centre of gravity, and then about inertia.  There is a real difference between weight and inertia.  So inertia is about how your weight is distributed, so if too much of your weight is distributed far from your centre, you will find it harder to get up out of your seat than if it is mostly near your centre.  Then we looked at how this was exploited in all sorts of different sports, and in particular with rotational motion, and how the falling cat exploits an ability to change its inertia by bringing in its front legs and its back legs at different times, in order to guarantee that, however you drop it, or throw it in the air, it always lands on its four feet, with its back arched to absorb the shock.  And then finally, this rather curious object, recently discovered, thought to be impossible, that will right itself, find its equilibrium state, no matter how you start it.  Hopefully these will give you a nice overview of the mathematics of this area of the real world all around us.

© John Barrow, 6th October 2009