About the image

Contemporary definitions of mathematics include:

• the study of pattern and structure,
• the quantitative language of the world.

The image attempts to capture these ideas and to involve a number of areas of modern mathematics. The surface shown is one of a family of complete minimal surfaces discovered by Costa, Hoffman and Meeks. The image was generated using 3D-Filmstrip, a computer package developed by Richard Palais.

Terminology: A minimal surface is one for which every small piece of the surface has less area than anything else with the same boundary as that piece. Such a surface is complete if it approaches infinity in various directions and has no boundary (of course the diagram can only show a part of the surface).

History: After the discovery of the catenoid and the helicoid at the end of the eighteenth century it remained an open problem for 200 years if there were any other examples of complete minimal surfaces (other than infinitely periodic ones). The image is one of a family of new examples discovered in the 1980's.

Computers: These surfaces were postulated to exist on the basis of computer experimentation. Their existence was proved a few years later using the patterns indicated by the computer as a starting point in the proof.

Applications: Minimal surfaces and their generalisations arise in many areas of science and technology, including eqi-potential surfaces, crystal growth, and architectural constructions.

Mathematical significance: The equations used to describe minimal surfaces are prototypes of nonlinear partial differential equations. Such equations arise in modelling almost any phenomena which varies in space or time (that doesn't leave much out!). The computational methods developed in these settings are widely applicable in engineering and other fields. Minimal surfaces and their generalisations play a major role in contemporary geometry and theoretical physics.