Professor Stewart's Hoard of Mathematical Treasures (36 page)

Wrapping round without a twist creates a torus.

However, if you imagine a similar procedure for a Klein bottle, then the two ends of the cylinder don’t join up that way: one of them has to be given the opposite orientation. In 3D, one way to do this is to make it thinner, poke it through the side of the cylinder, poke it out of the open end, and then roll it back on itself like the neck of a sweater and finally join it to the other end of the cylinder. This leads to the standard ‘bottle’ shape, with a self-intersection where you poked it through. As Klein wrote: the shape ‘can be visualized by inverting a piece of a rubber tube and letting it pass through itself so that outside and inside meet’.

Joining the edges of a cylinder to make a Klein bottle.

With an extra dimension to play with, you can push the end of the cylinder off into the fourth dimension before poking it
through where the cylinder would have been; then pull it back into normal 3D space once it’s inside, and carry on as normal. That way, there’s no self-intersection.

The Klein bottle has a remarkable property, which has been celebrated in a limerick, whose author - perhaps mercifully - remains unknown:

Can you see how to achieve this?

 

 

For some brilliant visualisations, go to:

Another cute factoid: any map on the Klein bottle can be coloured with at most 6 colours, so that adjacent regions have different colours. This compares with 4 colours for the sphere or plane (Cabinet, page 10) and 7 for the torus. See:

In the Great Celestial Number Factory, where all numbers are made, the accountants keep tabs on how many times each digit 0-9 is used, to make sure that there are adequate stocks in the warehouse. They record these counts on a standard form, like this:

Typical inventory form.

So, for example, as the digit 4 occurs 3 times, Nugent writes ‘3’ in the lower row of boxes, underneath the printed 4. Numbers are written so that they end in the right-hand box, like the example, and leading zeros may or may not occur. (None of that matters for this puzzle, but people do worry... )

One day Nugent was filling in the form, as usual, when he suddenly noticed something remarkable: the two numbers (that is, sequences of digits) recorded in the two rows of boxes were identical.

What was the number concerned?

 

We all know how to measure a length when our ruler or tape measure is too short. We measure as far as we can, mark the end point, then continue measuring from there, and add the distances together. This puts into practice a basic principle of Euclid’s geometry: if you place two lines end to end - pointing in the same direction - then their lengths add.

This means that you can make an adding machine from two sticks. Just make marks along the edge distances 1, 2, 3, 4, and so on; then position the sticks to perform the addition sum.

The number on the top stick is 3 more than the corresponding one on the bottom stick.

Big deal, I hear you thinking, and it’s true that this gadget isn’t terribly practical. But a close relative is - or, to be honest, was. To get it, we change the markings, replacing each number by the corresponding power of 2.

Now the numbers on the top stick are the corresponding numbers on the bottom stick, multiplied by 8. Our adding-sticks have become multiplying-sticks. This trick works because of the well-known formula
Well, that’s fantastic. Now we can multiply powers of 2.

Back in the days when computers and calculators were undreamt of, and would have been seen as magic, multiplying two numbers was really hard work. But astronomers need to do a lot of multiplications to keep track of the stars and planets. So, around 1594, James Craig, court doctor to King James VI of Scotland, told John Napier, Baron of Murchiston, about something called prosthapheiresis. It sounds painful, and in a way it was: the Danish mathematicians had discovered how to multiply numbers using a formula discovered by François Viète:
Using tables of sines and cosines, you could use this formula to turn a multiplication problem into a short series of addition
problems. It was a bit complicated, but it was still quicker than conventional multiplication methods.

For years Napier had been thinking about efficient methods for doing sums, and it dawned on him that there was a better way. The formula for multiplying powers of 2 works for powers of any fixed number. That is,
for any number n. If you set n to something close to 1, such as 1.001, then the successive powers will be very closely spaced, so any number that interests you will be close to some power of n. Now you can use the formula to convert multiplication to addition. For instance, suppose I want to multiply 3.52 by 7.85. Well, to a good approximation

Therefore,
The exact answer is 27.632. Not bad!

Pages from Napier’s logarithm tables.

For more accuracy, you should replace 1.001 by something more like 1.0000001. Then you just draw up a table of the first million or so powers of that number, and you’ve got a quick way to multiply numbers to about 9-digit accuracy, just by adding the corresponding powers. Perversely, Napier chose to use powers of 0.9999999, which is less than 1, so the numbers got smaller as the powers got larger.

Fortunately, Henry Briggs, an Oxford professor, took an interest and sorted out a better way. The upshot of all this was the concept of a logarithm, which turns the calculations back to front. For example, since (1.001)
¹²⁵⁹
= 3.52, the logarithm of 3.52 to base 1.001 is 1,259. In general, log x (to base n) is whichever number a satisfies
Now the formula for
n
^a
+
b
can be reinterpreted as
whichever base you use. For practical purposes, base 10 is best, because we use decimals. Mathematicians prefer base e, which is roughly 2.71828, because it is better behaved with respect to the operations of calculus.