Friday, September 26, 2008

Diary twist to Möbius strip

August Möbius, an important German mathematician, died 140 years ago today. Most students who have studied maths will recognise his name, largely because, at some stage in their education, they will have come across a Möbius strip - a twisted ring of paper that can be cut, as if by magic, into another ring twice as large. Historians know that Möbius discovered this ‘strip’ as early as 1858, 150 years ago, only because of an entry in his diary.

Möbius’s mother was descended from Martin Luther and his father, who died when Möbius was only three, was a dancing teacher. He studied mathematics, astronomy and physics at Leipzig University, then more astronomy at Göttingen with Johann Friedrich Gauss, and more maths in Halle under Johann Pfaff. By 1816, he was back in Leipzig having been appointed chair of astronomy and higher mechanics. He didn’t achieve a full professorship until 1844, but at the same time was also involved in building and running the Leipzig Observatory. Thanks to the MacTutor History of Mathematics archive, hosted by the University of St Andrews in Scotland, for providing this biographical information.

Although Möbius published works on astronomy, his most important contributions came in the field of mathematics. His publications, not always original, were thought to be effective and clear, MacTutor says. His biographer Richard Baltzer wrote about him as follows: ‘The inspirations for his research he found mostly in the rich well of his own original mind. His intuition, the problems he set himself, and the solutions that he found, all exhibit something extraordinarily ingenious, something original in an uncontrived way. He worked without hurrying, quietly on his own. His work remained almost locked away until everything had been put into its proper place. Without rushing, without pomposity and without arrogance, he waited until the fruits of his mind matured. Only after such a wait did he publish his perfected works . . .’

One of the areas studied by Möbius was the polyhedron, and how to define it. Even today, 150 years later, this remains an area of study. Wikipedia says a polyhedron (plural polyhedra or polyhedrons) is often defined as a geometric object with flat faces and straight edges, but that this definition is ‘not very precise, and to a modern mathematician is quite unsatisfactory’. In a book dedicated entirely to polyhedra, and imaginatively called Polyhedra, its author Peter R Cromwell discusses Möbius’s contribution to the subject. In 1865, Cromwell says, Möbius answered the question ‘What is a polyhedron?’ in a paper, ‘the same one in which he described his famous one-side strip.’

In a footnote, however, Cromwell adds the following: ‘From entries in his diary, we know that he had discovered the ‘Mobius strip’ as early as 1858. J B Listing also discovered it independently around the same time.’ (Listing was another German mathematician who wrote an important treatise on topology.) For much more on the Möbius strip see Wikipedia’s extensive article.

But, for simple instructions on how to make one, see this support site for school teachers hosted by Queen’s University, Kingston, Ontario. It provides a very short introduction: ‘August Ferdinand Möbius was born in Germany at the end of the 1700s (300 years ago). He worked in mathematics and astronomy at several German universities. In 1858 he wrote in his diary about a strip of paper with a ‘twist’. It is a circle of paper with a half-turn in it, and it can do things a regular circle strip of paper cannot.’ (Möbius would have been proud of the maths - the end of the 1700s being 200 not 300 years ago!)

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