In the end, the minister had to keep both hospitals open, unable to justify a decision either way if it were to be contested in court.
How to Turn a Sphere Inside Out
In 1958, the distinguished American mathematician Stephen Smale, then a postgraduate student, solved an important problem in topology. But his theorem was so surprising that at first his thesis adviser Arnold Shapiro didn’t believe it, pointing out that there was an obvious counterexample. That is, an example that proves the theorem false. One consequence of Smale’s claimed result was that you can turn a sphere inside out using only continuous, indeed smooth, deformations. That is, you can’t tear it, or cut holes in it, and you can’t even make a sharp crease in it.
Intuitively, this seemed absurd. But intuition was wrong, and Smale was right.
Now, we all know that no matter how we twist and turn a balloon, the outside stays on the outside and the inside stays on the inside. Smale’s work does not contradict this, because it permits one type of deformation that you can’t do with a balloon. Namely, the surface is allowed to move through itself. However, it must do so in a smooth way, without creating a sharp crease. If creases are allowed, ‘eversion’ of the sphere, as it is called, is easy. Just push opposite hemispheres through each other, leaving a tube round the equator, and keep pushing so that the tube shrinks and disappears. However, this method creates an ever-sharper crease round the equator, and the technical definitions in Smale’s theorem rule this out.
This is allowed . . .
. . . but this isn’t.
So Smale was right, and the proof of his theorem could in principle be followed step by step to find an explicit method for everting a sphere. However, in practice this was too complicated, and for several years no specific method was known. The first method was devised by Shapiro and Anthony Phillips, and it was the first of what are now called halfway models.
Topologists have long known that some surfaces are ‘onesided’. The best known example is the Möbius band (Cabinet, page 111), and another is the Klein bottle (page 181). A sphere is two-sided: you can paint the inside surface red and the outside blue, say. But if you try to do that with a Möbius band or a Klein bottle, the red paint eventually runs into the blue paint: the apparent ‘inside’ and ‘outside’ surfaces in any small region connect together further round the band.
Now, there is another one-sided surface, the projective plane, which is closely related to a sphere. In fact, you can construct it mathematically by taking a sphere and pretending that diametrically opposite points are the same - in effect ‘gluing’ them together. The resulting surface can’t be represented in three-dimensional space without passing through itself. But it can be ‘immersed’ in three-dimensional space, meaning that parts of it are permitted to pass smoothly through other parts.
Because the projective plane is a sphere with opposite points glued together, it can be pulled apart into a sphere by ungluing the pairs of points, which creates two separate layers, very close together. One of these is in effect the inside of the sphere, the other the outside. However, because the projective plane doesn’t have an inside and outside, it can be pulled apart in two different ways. If we call the layers ‘red’ and ‘blue’, then the colours match up as the layers are pulled apart in the two different ways, but the red layer is inside for one way and outside for the other, while the blue layer is outside for one way and inside for the other.
How red and blue layers interchange positions at the halfway stage.
The idea for a specific eversion, then, starts in the middle with an immersed projective plane. Pull it apart one way to create a sphere, with red on the outside and blue on the inside. Then deform that sphere, smoothly, until it looks like a normal round sphere, with only its red surface showing. This may not be easy, and it is not even obvious that it can be done, until you try. However, it works.
Now go back to the halfway stage, and pull the projective plane apart the other way, to create a sphere with blue on the outside and red on the inside. Then deform that sphere, smoothly, until it looks like a normal round sphere, with only its blue surface showing.
Fit these two deformations together by running the first one backwards. Now a sphere that is red on the outside and blue on the inside gets scrunged around, smoothly, until opposite pairs of points coincide at the midway projective plane. Pass the layers through each other, and pull them apart according to the second deformation. The result is a sphere that is blue on the outside and red on the inside.
Pull the projective plane apart two different ways . . .
... then reverse the first deformation and combine the two.
Many different immersions of the projective plane are known. A famous one is Boy’s surface. In 1901, the great German mathematician David Hilbert set his student Werner Boy a problem: to prove that the projective plane can’t be immersed in three-dimensional space. Boy, like Smale, disagreed with his adviser. Like Smale, he was right. Boy had a surface named after him for his discovery.
Boy’s surface.
An advanced stage in the Shapiro-Phillips method.
A completely different method for turning a sphere inside out emerged from some general observations made by William Thurston, one of the world’s greatest living geometers. Thurston devised a method in which the sphere is first corrugated, looking a bit like an exaggerated tangerine, with lots of segments poking out. This can be done by a smooth deformation. Then the north and south poles of the tangerine are pushed through each other, creating a series of handles round the equator. All the handles are simultaneously twisted through 180°. Then the north and south poles are pulled apart, creating another tangerine shape, but now the inside and outside of the original sphere have been swapped. It remains only to smooth away the corrugations.
Thurston’s corrugation method.
All these methods for turning a sphere inside out are seriously complicated and difficult to follow, even with a lot of extra pictures and explanation. If you want to understand this topic fully, there is a wonderful video on: www.youtube.com/watch?v=xaVJR60t4Zg which you can download and watch to your heart’s content. It was made by mathematicians at the Geometry Center at the University of Minnesota (unfortunately now closed), and it explains exactly how various sphere eversion methods work, with superb computer graphics. More information can also be found at: www.geom.uiuc.edu/docs/outreach/oi/
Interestingly, you can’t turn a circle inside out without creating creases - part of the intuition that made people think it was impossible for a sphere, too. This particular trick needs three dimensions to allow room to manoeuvre.
A Piece of String Walked into a Bar . . .
A piece of string walked into a bar and ordered a beer.
‘Sorry’, said the barman. ‘We don’t serve strings.’
The string stomped out, muttering darkly about his funicular40 rights. A little way up the street, he passed a stranger.
‘You look like you could do with a beer,’ said the stranger. ‘It sure is hot.’
‘I tried that, but the barman refused to serve me because I’m a string.’
‘I can fix that,’ said the stranger. He tied the string in a granny knot and frayed his ends. ‘Try again.’ So the string went back to the bar and asked again for a beer.
‘Aren’t you the piece of string that I just sent packing?’ the barman asked suspiciously. ‘You look just like him.’
‘No,’ the string replied. ‘I’m a frayed knot.’
Slicing the Cake
If you cut a circular cake with 1, 2, 3 or 4 straight slices, then the largest number of pieces you can get is 2, 4, 7 and 11, respectively. (You’re not allowed to move the pieces between cuts.)
What is the largest number of pieces you can create with five cuts?
Answer on page 328
The largest number of pieces with up to four cuts.
The Origin of the Symbol for Pi
In 1647, the English mathematician William Oughtred wr
ote δ/π for the ratio of the diameter of a circle to its circumference. Here δ (Greek ‘delta’) is the initial letter of ‘diameter’, and π (Greek ‘pi’, of course) is the initial letter of ‘perimeter’ and ‘periphery’. Isaac Barrow, another English mathematician, used the same symbols in 1664. The Scottish mathematician David Gregory (nephew of the famous James Gregory) similarly wrote π/ρ for the ratio of the circumference of a circle to its radius (ρ is the Greek ‘rho’, the initial letter of ‘radius’). But to all these mathematicians, the symbols referred to different lengths, depending on the size of the circle.
In 1706, the Welsh mathematician William Jones used π to denote the ratio of the circumference of a circle to its diameter, in a work that gave the result of John Machin’s calculation of π to 100 decimal places.
In the early 1730s, Euler used the symbols p and c, and history might have been different, but, in 1736, he changed his mind and started to use the symbol π in its modern sense. It came into general use after 1748, when he published his Introduction to the Analysis of the Infinite.
Hall of Mirrors
If someone lights a match in a hall of mirrors, can it be seen (reflected as many times as necessary) from any other location?
Let me make the question precise. We restrict attention to two dimensions of space - the plane. Recall that when a light ray hits a flat mirror, it bounces off again at the same angle. Suppose you have a room - a polygonal region - in the plane, whose boundary consists of flat mirrors. A point source of light is placed somewhere in the interior of the room. Can this source always be seen, perhaps after multiple reflections, from any other interior point? Light that hits any corner of the polygon is absorbed and stops.
Victor Klee published this question in 1969, but it goes back to Ernst Straus in the 1950s, if not earlier. In 1958, Lionel and Roger Penrose found a room with a curved edge for which the answer is ‘no’, but the question for polygons remained open until George Tokarsky solved it in 1995. Again, the answer is ‘no’. He found many rooms with that property: the picture shows one of them. It has 26 sides and every corner lies on a square grid.
Tokarsky’s hall of mirrors.
Greek and Trojan Asteroids
Two unusual clumps of asteroids occupy much the same orbit as Jupiter. Unlike the ‘clumps’ in the asteroid belt (page 120), these clumps really are clumps - the asteroids stay together in a cluster. Though they are still separated by huge distances: space is big. One clump, the Greeks, is spread out around a position 60° ahead of Jupiter; the other clump, the Trojans, lags 60° behind it. The individual asteroids are (mostly) named after characters in Homer’s Iliad, a story of the siege of Troy by the Greeks, belonging to the appropriate sides.
The discovery of the Trojans in the 1900s confirmed a prediction that the Italian-born mathematician Joseph Louis Lagrange made in 1772. He worked out the combined effects of gravity and centrifugal force in a miniature solar system containing a sun and one planet, in a circular orbit. The same goes for any two-body gravitational system with a circular orbit, such as the Earth and the Moon - to a good approximation, at least. His calculations showed that there are exactly five points, relative to these two bodies, at which gravity and centrifugal force cancel out exactly, so that a small mass located at such a point will stay in equilibrium. These are the Lagrangian points L1-L5.
Lagrangian points, and associated energy contours.
• L1 lies between the Sun and the planet.
• L2 lies on the far side of the planet, along a line joining the Sun and the planet.
• L3 lies on the far side of the Sun, along a line joining the Sun and the planet.
• L4 lies in the planet’s orbit, 60° ahead of it.
• L5 lies in the planet’s orbit, 60° behind it.
More precisely, around 1750, Leonhard Euler proved that the points L1, L2 and L3 exist, and Lagrange discovered the other two. Lagrange did this calculation as part of an attack on a more general question, the motion of three bodies under gravity. Isaac Newton had shown that, for two bodies, the orbits are ellipses, and it was natural to ask what happens with three bodies. This turned out to be a very difficult problem, and we now know why: the typical motion is chaotic (Cabinet, page 117).
The L4 and L5 points are stable, provided the mass of the Sun is at least
times that of the planet. That is, a mass located at such a point will remain nearby even if it is disturbed a little. The other three points are unstable. No natural occurrences of bodies orbiting at these points were known until astronomers noticed that unusually many asteroids are located near the Sun-Jupiter L4 and L5 points. They are spread out along Jupiter’s orbit in the same ‘banana’ shape as the energy contours near those points. Since then, other instances have been found:
• The Sun-Earth L4 and L5 points contain interplanetary dust.
• The Earth-Moon L4 and L5 points may contain interplanetary dust in so-called Kordylewski clouds.
• The Sun-Neptune L4 and L5 points contain Kuiper belt objects, a class of smallish bodies now including Pluto, most of which orbit further out than Pluto.
• The Saturn-Tethys L4 and L5 points hold the small moons Telesto and Calypso.
• The Saturn-Dione L4 and L5 points hold the small moons Helene and Polydeuces.
Although the other three Lagrangian points are unstable, they are surrounded by stable orbits, called halo orbits, so a space probe or other artefact can be maintained near those points with very little expenditure of fuel. The James Webb Space Telescope, successor to the Hubble Telescope, will be positioned at the Sun- Earth L2 point when it is launched in or after 2013. This location keeps the Earth and Sun in the same direction, as seen from the telescope, so that a single fixed shield can stop radiation from those two bodies warming it up and disturbing the delicate instruments. The only Lagrangian point that has not yet featured in an actual or planned space mission is L3. All five of them have been exploited in numerous science fiction stories.
A wealth of further information can be found at:
en.wikipedia.org/wiki/Lagrangian_point
Sliding Coins
A good pub puzzle. Start with six coins, numbered 1-6 and arranged as in the left-hand picture. Slide them one at a time, without disturbing the others, to rearrange them into the right-hand picture in the number order shown.
How can you achieve this by moving as few coins as possible?
Answer on page 328
Start here . . .
. . . and end here.
Beat That!
. . . and then what?
Chapter 94 of Snorri Sturluson’s Heimskringla: History of the Kings of Norway - which I’m sure you’re familiar with - tells of a game of chance between King Olaf I of Norway41 and the King of Sweden,42 to decide which country owned the island of Hísing.
According to Thorstein the Learned, the two kings agreed to throw a pair of dice, and whoever got the highest score also got the island.
The King of Sweden, who had won the right to go first by drawing lots, threw the dice, and scored a double six. ‘There is no use in you throwing,’ he said. ‘I cannot lose.’
‘There remain two sixes on the dice, my Lord,’ replied Olaf, as he shook the dice in his hand, ‘and it is a trifling matter for God to make the dice land that way.’ Then he rolled the dice ...
What do you think happened next?
Answer on page 328
Euclid’s Puzzle
Legend has it that the great geometer Euclid composed a puzzle which went as follows.
A mule and a donkey were stumbling along the road, each carrying several identical heavy sacks. The donkey started complaining, making a horrible groaning noise, and eventually the mule got fed up.
‘What are you complaining for? If you gave me one sack, I’d have twice as many as you! And if I gave you one sack, we’d be carrying the same load.’
How many sacks were the donkey and the mule carrying?
Answer on page 329
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The Infinite Monkey Theorem
It is said that if a monkey sat at a typewriter and kept hitting keys at random, then eventually it would type the complete works of Shakespeare. This statement dramatises two things about random sequences: anything can turn up, and, therefore, the result need not appear random. The infinite monkey theorem goes further, and states that, if the monkey keeps typing for ever, then the probability that it will eventually type any given text is 1.
To test this proposition, all you need is two dice, of different colours or otherwise distinguishable, and a table of symbols. The one at bottom right is a space.
Simulated monkey.
Throw the two dice, choose the corresponding symbol, and write it down. For instance, if you throw 4 /1, then you get the letter D. Keep going, and see how long it takes to get a sensible word with, say, three or more letters. Your experience should be confirmed by two calculations:
• On average, how many throws would it take to get DEAR SIR, including the space between the words?
Ian Stewart Page 18