Home >> United States & Canada >> Foreign Policy & Military

     Email   Print 

Infinite Space And Our Difficulties Getting There

Angelique van Engelen - 12/22/2007

Pioneers of mathematics often encountered significant resistance to their work during their lifetime. Donald Coxeter, a man described as the King of Infinity knew about this first hand. As did Rene Descartes. Both made significant strides into infinity.

When Donald Coxeter died in 2003, he was generally considered one of the 20th Century´s great minds in mathematics. Hyper-modern maths benefits from his work, even though he stuck to a remarkably old method; Euclidean geometry. Had he not done this in the late 1930s, we might not be where we are in our understanding of multi dimensional space.

Coxeter's biography was recently published in the UK and has been out in the US since a few months. The book's writer, Siobhan Roberts, entitled her work King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry

She describes Coxeter's memories of the defining moments of his life; those he experienced when being very ill at a young age. During high fevers, Coxeter, then 14, he began to hallucinate about three-dimensional shapes that his maths teacher had shown in a class he´d attended shortly before.The classical Greek shapes, included the cube, the dodecahedron and others. Coxeter said he recalled them going round in his head and that due to his high fevor he literally saw another dimension.

In later life, Coxeter made important advances in explaining multi-dimensional space. Coxeter's pioneering of new, but Euclidean-based, mathematical concepts coincided with that of a group of leading French mathematians who collaborated together and invented a character they called Nicolas Bourbaki who they promoted as a massive mathematical genius. The group lobbied for the replacement of geometry by something that is known as set theory, i.e. pure numbers-based maths. Donald Coxeter took them on and his biographer believes that if it hadn't been for that, they might have succeeded at eliminating all geometry from mathematics.

Marcus du Sautoy, the Guardian's reviewer of the biography, believes it's been crucial too. "Coxeter peddled a very particular and ancient view on geometry, not one that mathematicians of the 20th century considered mainstream. Nevertheless, despite his rather ancient perspective, the symmetries of these objects, things now named Coxeter groups, are of central importance to modern mathematics, and many Bourbaki members incorporated them into their work", he writes.

Incidentally, the works of Maurits Escher are in part inspired by Coxeter. The two met, and Coxeter inspired among others the Circle Limit series based on hyperbolic tessellations.

Whether it is true that we would never have landed ourselves where we are now in terms of our understanding of multi dimensional space is of course open to debate.

Descartes, who can be credited with creating methods for mathematic thinking and who also ventured out in to creating laws for the fourth dimension, himself was long ahead of his time even though nobody really found out until this century.

His theory about infinite space was only discovered by scientists living around 100 years later. A book about the secret theory appeared two years ago by bestseller writer Amir D. Aczel.

Aczel's book about Descartes entitled Descartes' Secret Notebook, describes why it took the world until 1987 to find Descartes' original theory.

Descartes wrote his notebook in heavily encrypted code because he knew he would land in a lot of trouble if the Catholic Church, which was already very iffy on anything that remotely smelt like Copernicus' ideas, found out about his brain gymnastics.

Descartes was beginning to discover the possibility to notulate the navigational points of the fourth (perhaps also the fifth) dimension. If the church would have learnt about his work, they would most certainly have sent him to the Inquisition. Descartes' notulations were the very basis of modern Global Positioning Systems.

Descartes' full notebook was lost, but Aczel's story revolves around the surviving 1.5 pages and also draws on Descartes' many published works. He also portrays the writer as a go getting soldier as well as a member of the Rosicrucians, who promoted the spread of forbidden knowledge.

That fact came to light when Descartes died. Twenty years after his death, the German mathematician Gottfried Wilhelm Leibniz copied the 1.5 pages. He also kept the information secret. His copy survived the ages however, but it wasn't before 1987 that someone -the French mathematician priest Piere Costabel- cracked the encryption.

The 'surreal' into which Descartes had already created the passageway of course had been entered long before 1987. Not only was Coxeter a 'frequent flyer', but before him, Bernhard Riemann in the 19th century also gained deep insights, by going about it the differential geometry way. Later on Einstein followed this direction further.