Professor Stewart's Hoard of Mathematical Treasures (49 page)

A hosepipe seems to be 1D, but closer up we see it has two more dimensions. We can draw this schematically as a line with circles attached to each point.

Extra dimensions of space (here the 2D plane) shown schematically as spheres. In string theory, the spheres have more dimensions than we can draw. The spheres support quantum vibrations, which endow particles with properties like spin and charge.

All this talk of ‘hidden dimensions’ may be needlessly mystical. Physics presented us with very similar things long ago, but no one started babbling about increasing the dimension of space. An electromagnetic field - which we use to send radio, TV and phone calls - has six extra coordinates for each point in space: three for the magnetic field’s strength and direction, and three more for the electric field’s strength and direction. Maxwell’s equations for electromagnetism are naturally defined on a 9D space.

So the extra 7 dimensions required for string theory need not actually be spatial in any meaningful sense. They might be - in fact, are - new physical quantities, like colour or temperature, that enter into the string theory equations. So talking of them as hidden dimensions of space makes string theory seem more mysterious than it really is.

In the 1880s, the American medium Henry Slade used to convince people that he had access to the Fourth Dimension - the Spirit World - using a strip of leather with two cuts along it. He would get someone to make a mark on the leather, to prevent substitution. Then he would hold it under the table for a few moments, and produce it again - braided!

Start here . . .

. . . end here.

In 4D space, strips can be passed over each other and woven together by pushing one temporarily into the Fourth Dimension, moving it into the right position, and then pulling it back into ordinary 3D space. That is what Slade pointed out, and what he claimed proved he had the ability to access the Fourth Dimension.

How did he do it?

 

Place each of the digits 1-8 in the eight circles, so that neighbouring digits (that is, those that differ by 1) do not lie in neighbouring circles (connected directly by a line).

 

Keep the neighbours apart.

A mathematician who had spent his entire research career in pure mathematics - starting with topological algebra, then a bit of algebraic geometry, then some geometric topology, thinking of moving into algebraic topology or maybe geometric algebra - began to wonder if perhaps it was time he did something more obviously practical. He knew that those subjects did have applications, but he had never worked on such things, preferring the intellectual challenges of abstract thought.

He had never been against applied mathematics, you understand - just hadn’t done any himself.

Maybe, he thought, it’s time for a change.

Weeks went by, and still he had not translated his thoughts into deeds. The prospect of engaging with the real world made him very nervous. He’d never done it before. But he found the
idea appealing, nonetheless. The problem was to pluck up enough courage to take the plunge.

One day, walking along the corridor of the Mathematics Department, he saw a sign on a door. ‘Seminar on gears - today at 2.00.’ He looked at his watch: 1.56. Dare he? Could he actually ... go in? It was a big step. In an agony of indecision, he stood outside the door, shifting from one foot to the other, listening to the sounds of the lecturer preparing to start the talk. Finally, at 1.59, he plucked up his courage, opened the door, and slid into a vacant seat. Now he would begin his career move to practical applications of mathematics!

The speaker picked up his notes, cleared his throat, and began. ‘The theory of gears with an integer number of teeth is well known—’

A wheel of radius 1 metre rolls along a flat horizontal road at a constant speed of 10 metres per second, without slipping and without bouncing off the road. At a fixed instant of time, is any point on the wheel stationary? If so, which?

Assume that the wheel is a circular disc, the road is a straight line, and the wheel lies in a vertical plane. ‘Stationary’ means that the instantaneous velocity is zero.

 

You have a line of unit length, whose two endpoints at 0 and 1 are missing, and an unbounded supply of points - as one does. You are required to place the points successively on the line, so that: