London

London is one of the four largest cities in the world, has a population of over 8 million, and is famous for many things: Buckingham Palace, the Houses of Parliament and Big Ben, ‘London Bridge’, and the Tower amongst others. It is a fascinating mix of old and new. Visitors love its historic buildings, its many churches, its beautiful parks, the Thames River crossed by many bridges, the pigeons in Trafalgar Square. For many, a visit to London is a return to one’s roots, a fleeting glimpse of the past, a renewed association with names and places vaguely remembered from school days, but never before appreciated. One can spend literally months walking the streets, each day finding some new treasure, making some new discovery.

Of course there is also the traffic, the crowds, the litter, the pollution, ... . But somehow these are soon overlooked and forgotten – accepted as an inevitable part of this thriving, sprawling metropolis.    

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The Underground

The easiest way to travel around London is to use the Underground. The first Underground line was opened in 1863, and many extensions were made over the following years. By the year 1900, no fewer than nine main line companies were operating. The early lines were built using a ‘cut and cover’ technique, where a trench would be dug along a street, the tube and line laid, and then the hole backfilled. Later, tunnelling techniques were used which caused less chaos and damage on the surface. Today, a number of the 275 stations are being decorated with intricate and beautiful mosaics, making them worth a visit in their own right.

It is less well known that the British Post Office has its own underground railway, used solely for the movement of letters and parcels. The rail gauge is a mere two feet (70 cm), and two tracks are laid side by side, requiring a tube of just nine feet (2.75 m) diameter. Even with this narrow gauge, the specially constructed little trains can manage a top speed of 35 mph (56 km/hr).

The familiar London Underground sign of red circle and horizontal bar, and the stylized map of the Underground quickly become part of the visitor’s navigational equipment, enabling quick and easy access from one place to another. But it is the map of the Underground which holds special interest for us here.   































The Underground map (I)

Even at this small scale, we see that the Underground map, based on an idea of Harry Beck, is not an accurate representation. The railway lines are certainly not straight as indicated, nor are the stations equally spaced. Your city may have a similar transport map. For a detailed map of the London Underground see
http://www.thetube.com/content/tubemap/





















The Underground map (II)

What is the use of a map which does not give a true representation of the actual system? Would it not be more helpful to have a true picture of the lines and stations? In fact, the Underground map gives all the information the traveller requires, and is much easier to read than a map giving much accurate but unnecessary detail.

Investigate Imagine you are a traveller on the Underground. Which of the following pieces of information do you think would be helpful? necessary?

(a) the exact distance between two stations;
(b) whether or not there is a direct rail link between two stations;
(c) the exact direction of one station from another;
(d) whether the railway line between two stations is straight or curved;
(e) whether or not there is any rail link between two stations;
(f) whether a given station lies between two other stations on a line.

The map of the London Underground is an example of a network. One branch of mathematics concerned with networks is optimization, part of which deals with ‘shortest path’ problems. Another branch is topology.    





















Topology

Topology is a branch of mathematics which is sometimes called ‘rubber sheet geometry’, or ‘the mathematics of distortion’. Imagine that you have drawn some geometric figure on a flat sheet of rubber. The rubber sheet is then stretched without tearing in some arbitrary way. What properties of the geometric figure would remain unchanged? These are the properties which are studied in topology. For example, each of these two figures (above) can be obtained from the other by stretching. Clearly straightness, length and direction are all likely to change, so these will not be topological properties.

Two figures, each of which can be obtained from the other by stretching, are said to be topologically equivalent. In particular, two networks related in this way are said to be topologically equivalent.

Investigate Here is a set of letters in the Helvetica font:

(a) The letter I is equivalent to ten other letters in this set. Which are they?
(b) Which letter is equivalent to the letter O?
(c) Find three adjacent letters equivalent to the letter T. Are there any others?     
 



























Betweenness

We see that the simplification of the Underground map is a topological idea. The complicated twistings and turnings of the actual rail links are replaced by straight line segments. Distances and directions on the map are chosen, not according to actual distances and directions, but so that the map can be easily drawn, and so that it is easy to read. Now, what information does the map give us?

4. (a) The map shows the stations Tufnell Park, Kentish Town and Camden Park lying consecutively on the Northern Line. If travelling from Tufnell Park to Camden Park, would you expect the train to pass through Kentish Town? Why? What property is preserved here?

(b) A closed loop line called the Circle Line passes through the stations Victoria, Bayswater and Euston Square. Roughly sketch this. On the Circle Line, which of these stations lies between the other two? Does the question make sense?

A property preserved under stretching is betweenness: whether a point lies between two given points on a line. On a straight stretch of line (on the map) we can then obtain an ordering of the stations, and feel confident that when travelling by train the stations will appear outside our window in this order!

The same is true on a circular line, but as we have seen, for three points on a circle, in either direction, each point lies between the other two. This sort of behaviour cannot occur on a straight line.        

















Dimension

You will observe from the Underground map that three lines intersect at Oxford circus. This might bring to mind a vision of some form of Russian Roulette, with trains whizzing by, perhaps missing by a hair’s breadth. Of course this does not happen.

Investigate  Describe how you think the railway lines may be positioned at Oxford Circus. Why is this situation not truly shown by the Underground map?

The Underground lines are at various levels below ground. Because the map is a ‘flat’ picture, this fact is not apparent. This illustrates the importance of the mathematical concept of dimension. We know that we live in a ‘3-dimensional’ world. But what does this mean?

6. (a) What do you understand by the word ‘dimension’? How would you explain this to a pre-school child? Why do we live in a three dimensional world?
(b) We say that a flat surface has two dimensions. Can you explain why?
(c) How many dimensions has a straight line? a point?
(d) Would you be happy talking about four dimensions?

Further reading ...

Nock, O. S. (1973),Underground railways of the world, A. & C. Black, London.

Abbott, E. A. (1885, 1928), Flatland: a romance of many dimensions, Boston.