Professor Stewart's Hoard of Mathematical Treasures (46 page)

See Note on page 332

If you cut this doughnut with three straight slices, what is the largest number of pieces you can create? (You’re not allowed to move the pieces between cuts.)

 

How many pieces with three cuts?

If you try to surround a circular coin by coins of the same type, so that all the other coins touch the first one, you quickly discover that exactly six coins fit neatly round the first.

In 2 dimensions the kissing number is 6.

This isn’t exactly news to most of us, but it leads to a concept that turns out to be important in the theory of digital codes, as well as having mathematical interest in its own right. A coin is a circle, which is a 2-dimensional shape, so we’ve just seen that the kissing number in 2-dimensional space is 6. In n-dimensional space, the kissing number is similarly defined to be the largest number of non-overlapping unit (n - 1)-spheres that can touch (‘kiss’) a unit (n - 1)-sphere. Here an (n - 1)-sphere is the natural analogue of a circle (1-sphere) or sphere (2-sphere). The number drops from
n
to
n
- 1 because although a sphere, say, lives in 3-dimensional space, its surface has only 2 dimensions. And a circle is a curve (hence 1D) in a 2D space, the plane. The unit (n - 1)-sphere, in fact, comprises all points in n-dimensional space that are distance 1 from some fixed point, the centre of the (n - 1)-sphere.

The exact value of the kissing number is known for very few dimensions: 1, 2, 3, 4, 8 and 24, in fact. In 1D space, which is a line, a 0-sphere is a pair of points spaced 2 units apart (the diameter of a unit n-sphere is 2). So the kissing number in 1D is 2: one on the left, one on the right. We’ve just seen that in 2D space the kissing number is 6. What about higher dimensions?

In 3D space, it is easy to get 12 spheres to kiss a single sphere: you can do it with ping-pong balls and glue dots. But the arrangement is ‘sloppy’, with room to move the spheres around, and with quite a bit of space left between them. Can you fit in a 13th sphere? In 1694, David Gregory, a Scottish mathematician, thought it could be done; no lesser a luminary than Isaac Newton disagreed. The issue was sufficiently delicate that it was
not resolved until 1874; it then turned out that Newton was right. So the kissing number in 3D space is 12.

In 3 dimensions the kissing number is 12.

A similar story holds in 4D space, where it’s relatively easy to find an arrangement of 24 kissing 3-spheres, but there’s enough room left so that maybe a 25th might fit in. This gap was eventually sorted out by Oleg Musin in 2003: the answer is 24.

In most other dimensions, mathematicians know that some particular number of kissing spheres is possible, because they can find such an arrangement, and that some generally much larger number is impossible, for various indirect reasons. These numbers are called the lower bound and upper bound for the kissing number, and it must lie between them.

In just two cases beyond 4D, the known lower and upper bounds coincide, and their common value is therefore the kissing number. These dimensions are 8 and 24, where the kissing numbers are, respectively, 240 and 196,650. In these dimensions there exist two highly symmetric lattices, higher-dimensional analogues of grids of squares or more generally grids of parallelograms. These special lattices are known as E
₈
(or the Coxeter-Todd lattice) and the Leech lattice, and spheres can be placed at suitable lattice points. By an almost miraculous coincidence, the provable upper bounds for the kissing number in these dimensions are the same as the lower bounds provided by these special lattices.

The current state of play can be summed up in a table, where
I’ve used boldface for those dimensions where an exact answer is known:

The best known lower bounds, for all dimensions up to 40 and a few larger ones, can be found at:

The kissing number for regular arrangements, in which the centres of the spheres all lie on a lattice, is known exactly for 1-9 dimensions, as well as 24. In 1, 2, 3, 4, 8 and 24 dimensions, it is the value shown in the table. For 5, 6, 7 and 9 dimensions it is, respectively, 40, 72, 126 and 272. (The table entry 306 in 9 dimensions does not refer to a regular arrangement.)

The two positions of a tippe top.

The toy known as a tippe (or alternatively ‘tippy’) top is made by slicing a bit off a sphere and adding a cylindrical ‘stalk’. When
you spin it - fast enough - it turns upside down. Most of us have played with a tippe top at some point, but here’s a question we possibly haven’t thought about. Suppose that, when you first spin the top, while it is still the usual way up, you spin it clockwise, looking down from above. This is the natural direction for right-handers.

When it turns over, in which direction does it spin?

 

Topologists study things like knots, and they try to work out whether two knots are ‘topologically equivalent’, that is, can be deformed into each other. Or not. To do that, they invent cunning ‘invariants’, which are equal for equivalent knots, but may or may not be equal for two knots that are not equivalent. So knots with different invariants are definitely topologically different, but knots with the same invariants may or may not be topologically different.

It’s a knotty issue. Most of the useful invariants aren’t perfect: they’re a bit like using ‘odd/even’ to distinguish people’s ages. If Eva’s age is even and Ollie’s age is odd, then we know their ages must be different, even if we don’t know what their ages are. But if Evangeline’s age is even and Everett’s age is even, then their ages might be the same (for instance, 24 and 24) or maybe not (24 and 52). So in this case we can’t tell.

Sometimes topologists get lucky, and the invariant is good enough to tell them when a knot is not actually knotted, even if it can’t reliably distinguish all different knots. A case in point is the so-called ‘knot group’, one of the first knot invariants discovered. I mention all this not because of the topology, which is highly technical, but because in 1972, in the mathematical fanzine Manifold, it gave rise to a poem that summed up what was good and bad about the knot group. It bore the title Knode:

The early symbol for ‘factorial n’, which is
was
but this was tricky to print. So in 1808, the French mathematician Christian Kramp decided to change it to
which was easy to typeset. The old-fashioned version quickly went out of vogue, one of a number of examples where the practicalities of printing have affected mathematical symbolism.