Apr 292018

Today is the birthday (1854) of Jules Henri Poincaré, a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as “The Last Universalist,” since he excelled in all fields of the discipline as it existed during his lifetime. Outside of mathematics and science Poincaré is scarcely a household name, yet a reasonable argument can be made that both Einstein and Picasso were profoundly influenced by his work – yes, BOTH. I’ll try not to make your eyes glaze over with technicalities too much, and, instead, focus on generalities that just about anyone can grasp, including Poincaré’s ideas concerning original thinking, as well as his work habits.

As a mathematician and physicist, Poincaré, made a number of original and fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, which was one of the most famous unsolved problems in mathematics until it was solved in 2002–2003. In presenting the conjecture he helped found the field of topology. In his research on the three-body problem (calculating the motion of three interacting bodies – such as planets – using the laws of motion and gravity posited by Newton), Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. Poincaré was also one of the first to propose the existence of gravitational waves emanating from a body and propagating at the speed of light as a solution to problems in celestial mechanics. His proposal has proven correct experimentally.

Poincaré’s work itself was of great importance in numerous fields, but for the moment I would like to focus on that way in which he worked and how he construed the intellectual process. To begin, let me remind you that scientific discovery is almost never a step-by-step process. It requires imaginative leaps and “what-ifs” that are anything but logical. Poincaré tried to probe the mechanics of genius, and, though he explained his own process quite well superficially, he did not exactly provide much useful penetrating detail.

Poincaré’s work habits have been compared to a bee flying from flower to flower. One observer said:

Accustomed to neglecting details and to looking only at mountain tops, he went from one peak to another with surprising rapidity, and the facts he discovered, clustering around their center, were instantly and automatically pigeonholed in his memory.

The mathematician Darboux claimed he was un intuitif (intuitive), saying that this is demonstrated by the fact that he usually worked by visual representation. Poincaré himself wrote that he believed that logic was not a way to discover ideas, but a way to structure and manage ideas once imagination and intuition have uncovered them. Poincaré’s mental processes were not only interesting to Poincaré himself but also to Édouard Toulouse, a psychologist at the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote Henri Poincaré (published in 1910). In it, he discussed Poincaré’s personality and work habits:

He worked during the same hours each day for short periods of time. He did mathematical research for four hours a day, between 10 a.m. and noon then again from 5 p.m. to 7 p.m. He read articles in journals later in the evening.

His normal work habit was to solve a problem completely in his head, then commit the completed problem to paper.

His ability to visualize what he heard proved particularly useful when he attended lectures, since he was severely nearsighted and could not see what the lecturer wrote on the blackboard.

He was always in a rush and disliked going back for changes or corrections.

He never spent a long time on a problem since he believed that his unconscious would continue working on the problem while he consciously worked on another problem.

While most mathematicians worked from principles already established, Poincaré started from basic principles each time.

Not exactly helpful. What you get from this list is that Poincaré’s mind was a cauldron of stuff that he poked around in until he came up with something useful. I am, fortunately or unfortunately, familiar with the process, although my cauldron as not filled with as much technical stuff as his. Furthermore, filling your cauldron with stuff is not enough. You have to have ways of wading around in the stuff productively. Here Poincaré is not much help. He talks about intuition and imagination, but what are they and how do you get them? Poincaré studied his habits and gave a talk about his observations in 1908 at the Institute of General Psychology in Paris. He linked his way of thinking to how he made several discoveries. Some quotes are pertinent.

The chief aim of mathematics teaching is to develop certain faculties of the mind, and among these intuition is by no means the least valuable.

 It is by logic that we prove, but by intuition that we discover.

 Logic teaches us that on such and such a road we are sure of not meeting an obstacle; it does not tell us which is the road that leads to the desired end. For this, it is necessary to see the end from afar, and the faculty which teaches us to see is intuition. Without it, the geometrician would be like a writer well up in grammar but destitute of ideas.

All good up to a point. A good vocabulary and excellent command of grammar will not make you a poet; skilled brushwork and an array of paints will not make you an artist. No argument. What does make a Keats or a Picasso? Poincaré has no answer, and neither do I.

An array of pots and pans and a larder full of good ingredients will not make you a good cook either. Nor will training by the best chefs. I can give you recipes, however. They are the building blocks. Poincaré was born in Nancy, former capital of the duchy of Lorraine. Lorraine is, of course, the birthplace of quiche Lorraine, which you can find in qualities from wretched to divine the world over. It has become a rather mundane staple in many places. This recipe is serviceable, but you ought to go to Lorraine for a proper quiche. Even there you may be disappointed. You are best served by seeking the advice of a knowledgeable local. Eggs and cream in a pastry shell is not especially French, but the word “quiche” comes from Lorraine dialect (maybe from German, “kuchen”). The bacon was at one time lardons, and the cheese is a late addition also. Therefore, finding “authentic” quiche Lorraine is a lost cause.

Quiche Lorraine


For the crust

1 ¼ cups all-purpose flour
½ tsp salt
½ cup cold butter, cubed small
3 tbsp ice water

For the quiche

8 slices bacon
1½ cups shredded gruyere
1 shallot, minced
6 large eggs
1½ cups heavy cream
salt and black pepper


For the crust, sieve the flour and salt into a mixing bowl or food processor. Work the flour and butter together with your hands, or by pulsing in the food processor, until it resembles coarse sand.

Add the ice water one tablespoon at a time and work the mixture into a dough. Form into a disc, wrap in plastic wrap, and refrigerate until firm, (at least 30 minutes).

On a lightly floured surface, roll out crust until ¼” thick. Loosely drape it over a 9” pie plate (or quiche pan) and crimp the edges. Refrigerate until ready to use.

Preheat the oven to 350˚F/175˚C.

In a large, dry skillet over medium heat, cook the bacon until crispy. Drain and cool on wire racks and then break into bite-sized pieces.

Scatter the bacon pieces evenly on the pie crust and then spread over 1 cup of grated gruyere and the shallot.

In a large bowl, whisk together the eggs, cream, a pinch of cayenne and nutmeg, and season with salt and pepper to taste. Pour mixture over bacon and cheese. Sprinkle with the remaining cheese.

Bake for around 45 minutes until the crust is golden and the eggs are cooked through. Test by inserting a knife into the eggs near the center. It should come out clean when the eggs are cooked. Cool the quiche on a wire rack in the tin for 10 minutes before slicing into wedges and serving.

Nov 172016


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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?


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.


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.