Nov 172016
 

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Today is the birthday (1790) of August Ferdinand Möbius a Saxon mathematician and theoretical astronomer whose name is pretty well universally associated with the Möbius strip. Möbius was born in Schulpforta, Saxony-Anhalt, and was descended on his mother’s side from Martin Luther. He was home-schooled until he was 13 when he attended the College in Schulpforta in 1803 and studied there graduating in 1809. He then enrolled at the University of Leipzig, where he studied astronomy under the mathematician and astronomer, Karl Mollweide. In 1813 he began to study astronomy under the  renowned Carl Friedrich Gauss at the University of Göttingen while Gauss was the director of the Göttingen Observatory. From there he went to study with Carl Gauss’s instructor, Johann Pfaff at the University of Halle, where he completed his doctoral thesis, The Occultation of Fixed Stars in 1815. In 1816 he was appointed as Extraordinary Professor of astronomy and higher mechanics at the University of Leipzig. Möbius died in Leipzig in 1868 at the age of 77. His son Theodor was a noted philologist.

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He is best known for his discovery of the Möbius strip, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space. It was independently discovered by Johann Benedict Listing around the same time. The Möbius configuration, formed by two mutually inscribed tetrahedra, is also named after him. Möbius was the first to introduce homogeneous coordinates into projective geometry.

Many mathematical concepts are named after him, including the Möbius plane, the Möbius transformations which are important in projective geometry, and the Möbius transform of number theory. His interest in number theory led to the Möbius function μ(n) and the Möbius inversion formula. In Euclidean geometry, he systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.

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OK, I promise not to get much more mathematical. I know how easily eyes glaze over. Just be assured that Möbius was a smart guy and his ideas have many practical applications as well as being important to pure mathematics. The Möbius strip or Möbius band can be created by taking a paper strip and giving it a half-twist, and then joining the ends of the strip to form a loop. However, the Möbius strip is not a surface of only one exact size and shape, such as the half-twisted paper strip shown in the illustration. Rather, mathematicians refer to the closed Möbius band as any surface that is homeomorphic (topologically identical) to this strip. Its boundary is a simple closed curve, i.e., homeomorphic to a circle.

A half-twist of the band clockwise gives an embedding of the Möbius strip different from that of a half-twist counterclockwise – that is, a Möbius strip can be right- or left-handed, although the underlying topological spaces within the Möbius strip are homeomorphic in each case. There are an infinite number of topologically different Möbius strips since they can be formed by twisting the strip an odd number of times greater than one, or by knotting and twisting the strip, before joining its ends.

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The Möbius strip has several curious properties. A line drawn starting from the seam down the middle meets back at the seam but at the other side. If continued the line meets the starting point, and is double the length of the original strip. This single continuous curve demonstrates that the Möbius strip has only one boundary. Cutting a Möbius strip along the center line with a pair of scissors yields one long strip with two full twists in it, rather than two separate strips; the result is not a Möbius strip. This happens because the original strip only has one edge that is twice as long as the original strip. Cutting creates a second independent edge, half of which was on each side of the scissors. Cutting this new, longer, strip down the middle creates two strips wound around each other, each with two full twists.

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If the strip is cut along about a third of the way in from the edge, it creates two strips: one is a thinner Möbius strip – it is the center third of the original strip, comprising 1/3 of the width and the same length as the original strip. The other is a longer but thin strip with two full twists in it – this is a neighborhood of the edge of the original strip, and it comprises 1/3 of the width and twice the length of the original strip.

Other analogous strips can be obtained by similarly joining strips with two or more half-twists in them instead of one. For example, a strip with three half-twists, when divided lengthwise, becomes a strip tied in a trefoil knot. (If this knot is unraveled, the strip is made with eight half-twists in addition to an overhand knot.) A strip with N half-twists, when bisected, becomes a strip with N + 1 full twists.

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There have been several technical applications for the Möbius strip and they can be found naturally occurring at the micro- and macro-level. Giant Möbius strips have been used as conveyor belts that last longer because the entire surface area of the belt gets the same amount of wear, and as continuous-loop recording tapes (to double the playing time). Möbius strips are common in the manufacture of fabric computer printer and typewriter ribbons, as they let the ribbon be twice as wide as the print head while using both halves evenly.

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A Möbius resistor is an electronic circuit element that cancels its own inductive reactance. Nikola Tesla patented similar technology in 1894: “Coil for Electro Magnets” explored a possible system of global transmission of electricity without wires.

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Charged particles caught in the magnetic field of the earth that can move on a Möbius band.

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The cyclotide (cyclic protein) kalata B1, active substance of the plant Oldenlandia affinis, contains Möbius topology for the peptide backbone.

Möbius strips of bacon are the most obvious and completely pointless uses of Möbius geometry. Still, they’re fun. This site http://www.instructables.com/id/M%C3%B6bius-Bacon/ gives you all the instructions you need including a description of meat glue – yup, meat glue !! Even though Möbius bacon has a never-ending surface, it doesn’t last long on the table.

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When you are done fiddling with strips of bacon you can turn your attention to Gose beer. Today, by happy coincidence is International Gose Beer day, and Gose beer is native to Möbius’ homeland of Saxony and Leipzig. Gose beer is unusual, and has never been tremendously popular because it is a sour wheat beer (as opposed to bitter), flavored with coriander and salt – a most definitely acquired taste.

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Gose was first brewed in the early 16th century in the town of Goslar, from which its name derives. It became so popular in Leipzig that local breweries copied the style. By the end of the 19th century it was considered to be local to Leipzig and there were numerous Gosenschänken (gose taverns) in the city.

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Originally, gose was spontaneously fermented. A description in 1740 stated “Die Gose stellt sich selber ohne Zutuung Hefe oder Gest” (“Gose ferments itself without the addition of yeast”). Some time in the 1880s, brewers were achieving the same effect by using a combination of top-fermenting yeast and lactic acid bacteria.

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By the outbreak of World War II, the Rittergutsbrauerei Döllnitz, between Merseburg and Halle, was the last brewery producing gose. When it was nationalized and closed in 1945, gose disappeared temporarily. In 1949, the tiny Friedrich Wurzler Brauerei opened in Leipzig; Friedrich Wurzler had worked at the Döllnitz brewery and had known the techniques for brewing gose. Before his death in the late 1950s, Wurzler passed the recipe to his stepson, Guido Pfnister. Brewing of gose continued in the small private brewery, though there appears to have been little demand. By the 1960s there were no more than a couple of pubs in Leipzig and possibly one in Halle that were still selling it. When Pfnister died in 1966 the brewery closed and gose production again ceased. Since then it’s been pretty much on again, off again – currently on. Who knows?

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I’m not a fan of sour or wheat beers, and, besides, I don’t drink any more. But I do still cook with various alcoholic beverages including beer. I know that cooking with beer seems like a waste to drinkers. Live with it. You can cook beef with beer very successfully and still drink it. Last I heard, beer is not in short supply in the world. Gose is an excellent beer to cook with because of its tart notes along with the coriander and salt.

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Use your imagination. You don’t need me to give you an exact recipe. My most usual way to cook beef in beer is to sauté a mix of chopped leeks and onions in olive oil until lightly browned, reserve, and then brown chunks of stewing steak – all in a large skillet. Add back the leeks and onions plus a half-and-half mix of beef stock and beer to cover. Usually I add additional flavorings but with gose there is no need, and they will mask the notes of the beer. Simmer covered for about 2 hours, or until the beef is in shreds. Uncover and reduce the remaining stock. Serve over noodles or with boiled new potatoes.

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