Communication networks, soap films and vectors

The problem of constructing the least-cost network of connections between arbitrarily placed points is one that is common and which can be very important financially. The network may consist of motorways between towns, a grid of electric power lines, buried gas or oil pipe lines or telephone cables. Soap films trapped between parallel planes with vertical pins between them provide a 'shortest path' network and Isenberg (1975) has suggested that soap films of this sort be used to model communication networks. However soap films are unable to simulate the different costs of laying, say, a three-lane motorway instead of a two-lane one or of using a larger pipeline to take the flow from two smaller ones. Soap films, however, have considerable intrinsic interest. In the article the emphasis is on the use of soap films and communication networks as a practical means of illustrating the importance of vector and matrix methods in geometry. The power of vector methods is illustrated by the fact that given any soap film network the total length of the film can be written down by inspection if the vector positions of the pins are known. It is also possible to predict the boundaries at which 'catastrophes' occur and to decide which network has the least total length. In the field of communication networks a method is given of designing the minimum cost network linking, say, a number of oilwells, which produce at different rates to an outlet terminal.