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Topology in chemistry: designing Möbius molecules.

In 1747, Johann Sebastian Bach wrote a series of canons and fugues on the occasion of his visit to King Frederick the Great of Prussia in Potsdam. His music of homage, the “Musikalisches Opfer” (Musical Offering, BWV1079), includes a canon that is probably one of the oldest man-made examples of an object with Mobius topology.1 The canon cancricans (crab canon) is written for two violins playing the same score in two different directions. If you cut the score, paste it in such way that it forms a band with the notes of the first violin on top and those of the second violin below, fold it lengthwise between the two notes, give one end a half twist, and join both ends, you end up with a endless Mobius-shaped score (Figure 1). After every turn the two violins change their parts. Karl Amadeus Hartmann, Nicolas Slonimsky,2 and others several centuries later wrote further examples of Mobius music. Note that there are two different ways to form the Mobius score, by twisting the strip either clockwise or anticlockwise. The two Mobius strips are enantiomers (the music, unfortunately, is not) and exhibit the highest symmetry a Mobius object can attain: C2. More than 100 years later, August Ferdinand Mobius and Johann Benedict Listing (Figure 2), both students of Carl Friedrich Gauss in Gottingen, published their seminal work on projective planes and one-sided surfaces. Nowadays, the credit goes almost exclusively to Mobius3 after whom the famous band is named, even though Listing published 4 years earlier.4 Both mathematicians mentioned the “Mobius” band in unpublished papers for the first time in 1858; however, again Listing’s unpublished note (July) predates the one of Mobius (September) by 2 months. It would probably be fair to name the one-sided band a “Listing band”. There are good reasons to assume that Listing was not adequately credited because of social rather than scientific reasons.5 Most of the objects we are dealing with in our everyday lives are two-sided. They have an inside and an outside surface like a sphere, a cube, or a torus. This is also true for most two-dimensional mathematical objects. According to the mathematical definition, a normal vector on the surface of a two-sided object cannot be shifted to any other point on the surface without crossing a border or interpreted in anthropomorphic terms: walking upright on the outside surface you cannot enter the inside and vice versa. The Mobius band is probably the simplest and best-known exception (Figure 3). Any closed band with an odd number of 180° twists is one-sided, and those with an even number of 180° twists are two-sided. Unfortunately, the sidedness is not an intrinsic property, because it requires that the surface is embedded in a † In memory of Professor Edgar Heilbronner. * To whom correspondence should be addressed. Telephone: ++49 431 88