ScienceDaily (Mar. 4, 2008) — A problem which has defeated mathematicians for almost 140 years has been solved by a researcher at Imperial College London. Professor Darren Crowdy, Chair in Applied Mathematics, has made the breakthrough in an area of mathematics known as conformal mapping, a key theoretical tool used by mathematicians, engineers and scientists to translate information from a complicated shape to a simpler circular shape so that it is easier to analyse.

This theoretical tool has a long history and has uses in a large number of fields including modelling airflow patterns over intricate wing shapes in aeronautics. It is also currently being used in neuroscience to visualise the complicated structure of the grey matter in the human brain.

A formula, now known as the Schwarz-Christoffel formula, was developed by two mathematicians in the mid-19th century to enable them to carry out this kind of mapping. However, for 140 years there has been a deficiency in this formula: it only worked for shapes that did not contain any holes or irregularities.

Now Professor Crowdy has made additions to the famous Schwarz-Christoffel formula which mean it can be used for these more complicated shapes. He explains the significance of his work, saying:

"This formula is an essential piece of mathematical kit which is used the world over. Now, with my additions to it, it can be used in far more complex scenarios than before. In industry, for example, this mapping tool was previously inadequate if a piece of metal or other material was not uniform all over - for instance, if it contained parts of a different material, or had holes."

Professor Crowdy's work has overcome these obstacles and he says he hopes it will open up many new opportunities for this kind of conformal mapping to be used in diverse applications.

"With my extensions to this formula, you can take account of these differences and map them onto a simple disk shape for analysis in the same way as you can with less complex shapes without any of the holes," he added.

Professor Crowdy's improvements to the Schwarz-Christoffel formula were published in the March-June 2007 issue of Mathematical Proceedings of the Cambridge Philosophical Society.

Journal reference: 'Schwarz-Christoffel mappings to unbounded multiply connected polygonal regions,' Math. Proc. Camb. Phil. Soc. (2007), 142, 319.

Related Stories

Knotty Four-decade-old Math Problem Solved By Team Including UGA Professor (Dec. 2, 2004) — It's not as famous as Fermat's Last Theorem. In fact, the math problem, which has not had a correct solution since it was proposed in the 1960s, doesn't even have a name. But a new, ...  > read more

The Mathematics Of A Lampshade (June 7, 2001) — Try to solve the following maths problem: does x^3+y^2+1 produce the same form as x^3+3y^2+xy^2? For cubic equations, it’s possible to solve this problem, but mathematicians found things more ...  > read more

Purdue Mathematician Claims Proof For Riemann Hypothesis (June 9, 2004) — A Purdue University mathematician claims to have proven the Riemann hypothesis, often dubbed the greatest unsolved problem in ...  > read more

Mathematicians Maximize Knowledge Of Minimal Surfaces (Aug. 16, 2006) — Mathematicians make breakthrough in understanding complex "minimal surfaces," revealing that pieces of planes, catenoids and helicoids are the building blocks of all minimal surfaces, and not merely ...  > read more

Search ScienceDaily

Why Icicles Are Long And Thin