Professor Stewart's Hoard of Mathematical Treasures (41 page)

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 and ordered a beer.

‘Sorry’, said the barman. ‘We don’t serve strings.’

The string stomped out, muttering darkly about his funicular
⁴⁰
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.’

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?

 

The largest number of pieces with up to four cuts.

In 1647, the English mathematician William Oughtred wrote 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.

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.

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
L
₁
-
L
₅
.