And that's it, really. I don't really know how to pad this answer. Perhaps if I gave you a potted bio of the man who invented it. It would be a mistake to think that all there was to the invention of the Mobius strip was folding a strip of paper in an unusual way and gluing it together. It looks simple now but it took years of research to perfect. The following biography is stolen verbatim from this site, apart from the bits in bold which are my insertions and may not be true in a narrow factual sense but which I feel probably contain the essence of truth.
In 1809 Möbius graduated from his College and he became a student at the University of Leipzig. His family had wanted him study law and indeed he started to study this topic. However he soon discovered that it was not a subject that gave him satisfaction and in the middle of his first year of study he decided to follow him own preferences rather than those of his family. He therefore took up the study of folding strips of paper in amusing ways.
The teacher who influenced Möbius most during his time at Leipzig was his astronomy teacher Karl Mollweide who showed him how if you glue a strip of paper together in a certain way you can make a pretty bracelet.
In 1815 Möbius wrote his doctoral thesis on How to win friends and influence people by folding strips of paper in unusual ways.
However Möbius did not receive quick promotion to full professor. It would appear that he was not a particularly good lecturer and this made his life difficult since he did not attract fee paying students to his lectures. Because no-one would pay good money to study origami.
He was offered a post as an astronomer in Greifswald in 1816 and then a post as a mathematician at Dorpat in 1819. He refused both, partly through his belief in the high quality of Leipzig University, partly through his loyalty to the idea that he was on the verge of a major breakthrough in the folding bits of paper line.
From the time of his first appointment at Leipzig Möbius had also held the post of Observer at the Observatory at Leipzig. He was involved in the rebuilding of the Observatory and, from 1818 until 1821, he supervised the project. This is why the Observatory building has an outside but no inside.
In 1820 he married and he was to have one daughter and two sons who all died of internal injuries due to his unusual way of folding their nappies.
In 1844 Grassmann visited Möbius. He asked Möbius to review his major work Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (1844) which contained many results similar to Möbius's work. However Möbius did not understand what the fuck Ausdehnungslehre meant and did not review it.
Möbius's mathematical publications, although not always original, were effective and clear presentations. However it was hard to read them as they were twisted into weird shapes.
Möbius's 1827 work Der barycentrische Calkul, on analytical geometry, became a classic and was turned into a popular operetta. In it he introduced homogeneous coordinates, who had previously been too shy to meet each other, and also discussed geometric transformations, which meant the book had to be sold in a plain brown wrapper.
In a memoir, presented to the Académie des Sciences and only discovered after his death, he discussed the properties of one-sided surfaces including the Möbius strip which he had discovered in 1858. This discovery was made as Möbius worked on a question on the geometric theory of polyhedra posed by the Paris Academy and started folding paper out of boredom.
Although we know this as a Möbius strip today it was not Möbius who first described this object, rather by any criterion, either publication date or date of first discovery, precedence goes to Listing.
[Whaaat?? Oh, the tragedy! Oh, that's typical! There were not one but two great minds working on this invention and the wrong one got the credit.]
In 1825 Listing entered a Gymnasium where he studied for five years. He had a really buff body by the end of it. He mastered English, French, Italian and Latin and well as increasing his knowledge of folding bits of paper.
Listing entered Göttingen University in 1830 and attended a remarkably broad range of courses, much broader than the mathematics and architecture specified by his scholarship. In addition to these two topics he also took courses on astronomy, anatomy, physiology, botany, mineralogy, geology and how to glue two ends of a strip of paper together without oozing everywhere.
Soon Listing was attending mathematics courses given by Gauss and he was quickly spotted by Gauss as being both a very able and a very hard working student. Gauss invited him to join his circle of friends which later developed into a four-dimensional polyhedron of friends.
Listing was appointed to a professorship in physics 1839 despite never having published a paper. Although he had folded many.
In September 1846 Listing married Pauline Elvers. Almost immediately the couple were in financial problems as Listing would fold all their money into the shape of a swan.
Listing continued to think about topological ideas, however, and he wrote the book Vorstudien zur Topologie in 1847. It was the first published use of the word topology which had previously been considered unmentionable in polite society.
In 1848 the revolution which swept Europe had its consequences. The most marked effect was that Weber was reappointed to the chair at Göttingen which he had lost ten years before. Of course Listing had been appointed to fill Weber's chair so a compromise had to be reached. This was that Listing sat on the chair while Weber sat on his lap.
Neither Listing nor his wife seemed capable of managing the family finances. Listing borrowed frequently and heavily, sometimes from usurers; Pauline habitually suggested he got a proper job instead of folding bits of paper all the time.
Among the honours which Listing did receive were election to the Göttingen Academy and the Royal Society of Edinburgh. He was awarded an honorary doctorate by Tübingen. He died tragically of paper-cuts.
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Anyway, like the Klein Bottle and that bollocks about Schrodinger's Cat, the main point of the Mobius strip is to give science geeks orgasms and exalt the human ego at the expense of the Almighty. God will strike them down in the midst of their revels for creating something so unnatural.*
* Due to the embarrassing non-appearance of WMD in Iraq I have abandoned all pretensions to political punditry. However, as I still require an outlet for my self-righteousness and hatred of everything I have decided to embark upon a path I have been considering for some time and become an arch-reactionary and religious maniac.
17th Jun 03