To infinity and beyond
QUESTION What is so special about the mathematical formula known as the Mandelbrot Set?
MATHEMATICIAN Benoit Mandelbrot had a remarkable facility for visualising geometric objects. He developed a simple yet elegant equation that mirrors the complexity of the natural world.
Born in Warsaw in 1924, his family emigrated to France when he was 12. He was influenced by the work of French mathematician Gaston Julia, who recognised the world is not naturally smooth edged and regularly shaped like the cones, circles, spheres and straight lines of Euclid’s geometry.
As Mandelbrot put it: ‘Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does lightning travel in a straight line.’
In 1975, he coined the term ‘fractal’ from the Latin word fractus, which means fragmented or broken, to describe the rough, but structured, forms he saw in the natural world.
Mandelbrot worked at the IBM Thomas J. Watson Research Center in New York modelling variations of stock prices.
The Mandelbrot Set stems from a famous equation that has parallels with Einstein’s E=MC in its simplicity and profundity: Zn+1 = Zn2 + C where c and z are complex numbers and n is zero or a positive integer.
Using the power of IBM’s computers, he was able to plot his set with incredible mathematical and artistic results.
To generate the Mandelbrot Set, a computer program calculates the equation over and over, a process called iteration. The points that produce a cycle fall in the set while those that diverge (give ever growing values) lie outside it.
When plotted on a computer screen using different colours to express various rates of divergence, the points outside the set shoot into infinity to produce pictures of great beauty.
The boundary of the Mandelbrot Set is a fractal curve of infinite complexity, any portion of which can be blown up to reveal extraordinary detail. Fractals have applications in mapping, music, painting, architecture and stock market analysis.
D. L. Dent, Reading, Berks.
QUESTION Was chess Grandmaster Paul Keres coerced into losing against Mikhail Botvinnik in the 1948 World Championship?
ESTONIAN Paul Keres was considered the world’s most talented chess player in the 1940s. The poor quality of some of his games in the post-war era suggest he was coerced into losing for political reasons.
The 1946 death of Alexander Alekhine left the chess world without a world champion. Previously, the title was gained by defeating the former champion.
Chess governing body FIDE chose to hold a tournament in The Hague and Moscow in 1948 to find a new champion.
Max Euwe from the Netherlands; Mikhail Botvinnik, Paul Keres and Salo Flohr from the Soviet Union; and Reuben Fine and Samuel Reshevsky from the U.S. were selected to play. Keres and Fine were considered the best players, but Fine dropped out.
Keres was favourite to win yet he lost 4-1 to the eventual winner Mikhail Botvinnik, playing some inexplicably poor games. Many are convinced Keres was ordered to throw the game.
During World War II, Estonia had fallen under Soviet and Nazi control. Keres played in a Soviet championship in Moscow in 1940 and later participated in Nazi-organised tournaments.
When the Soviets reoccupied Estonia in 1944, he unsuccessfully attempted to escape to western Europe.
In 1942, Latvian chess player Vladimir Petrovs disappeared into the Soviet Gulags never to return. It’s surmised that in return for escaping this fate, Keres agreed a Russian player would become world chess champion. He came second in a number of tournaments. Larry
Evans’s 1996 article Chess Life: The Tragedy Of Paul Keres concludes he ‘was forced to take a dive’.
A key piece of evidence was an interview with Botvinnik in which he stated: ‘At a very high level, it was proposed that the other Soviet players would lose against me on purpose in order to make sure there was going to be a Soviet world champion. It was Stalin himself who proposed this.’
Alex Spellman, Hale, Cheshire.
QUESTION Why was chewing food to a pulp called Fletcherism?
IN THE late 1890s, scientists around the U.S. were surprised to receive packages containing a sample of ‘economic ash’. The sender was Horace Fletcher and the samples his own excrement.
He wanted to show these learned men that ‘proper digestion’, or Fletcherism, produced stools ‘with no stench, no evidence of putrid bacterial decomposition, only with the odour of warm earth or a hot biscuit’.
Known as The Great Masticator, Fletcher promoted his theories on the lecture circuit. His acolytes included writers Mark Twain and business magnate John D. Rockefeller.
Fletcher claimed that not chewing thoroughly resulted in indigestion and a failure to assimilate nutrients. But if we chewed our food until it liquefied in our mouths, we could avoid poor digestion and disease.
A great self-promoter, Fletcher once chewed 722 times to liquefy a shallot. He claimed that those who followed his method would require only 12 to 15 mouthfuls of food to be satiated.
People started to pay attention when he underwent tests at Yale University where he supposedly beat younger men in bike races and weight-lifting.
Fletcher was regarded as a food faddist, but may have been on to something.
Since saliva contains enzymes that begin the process of breaking down food, chewing does aid digestion. Properly digested food is more readily absorbed, so, in theory, we could get away with eating less.
Sufferers of inflammatory bowel disease might also limit digestive irritation by thoroughly chewing their food.
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