Problem Solvers
Archimedes ( 287 – 212 BCE ) might be the most famous problem solver of all time. Surely everyone who has heard the story of Archimedes and the king's crown has been tempted to shout “ Eureka !” when first finding the solution to a difficult problem. Yet the story cannot possibly be true. If water overflowed when Archimedes got into his bathtub, we have to imagine that this brilliant man had never taken a bath before and stupidly filled the tub to the brim. Perhaps the story is famous because it comically commemorates the Archimedes principle, which says that a submerged object displaces an equivalent volume of water, and hence a graduated container of water can be used to measure the volume of an irregular object, like the king's crown. In fact, Archimedes solved many problems, and we are lucky to have one of his books about mathematics and physics survive until today. Archimedes knew that the circumference of a circle has a constant ratio to its diameter, but what exactly was that ratio? If the diameter of the circle is given a value of 1, then the circumference ( ? d) will equal the unknown ratio. To find the circumference mathematically, Archimedes drew two polygons, one inscribed within the circle and the other circumscribed outside the circle. The length of the circumference would have to be somewhere between the perimeters of these two polygons. As the number of sides of the polygons increases, the difference between their perimeters becomes less and less and the closer their perimeters approximate the circumference of the circle. Archimedes first calculated the perimeters of two triangles and then increased the number of sides using a formula developed by Euclid . When he calculated the difference between the perimeters of two polygons with 96 sides, he had a close approximation of the circumference of the circle. Using this method, one can calculate the value of ? to 3.14185 . . . But Archimedes did not call his ratio “pi.” It was not called “pi” until 1706 when William Jones used the Greek letter ? in a book called A New Introduction to Mathematics .
http://www.mcs.drexel.edu/~crorres/Archimedes/contents.html
http://www.archimedespalimpsest.org/
Archimedes used a geometric model to calculate pi, which he expressed in terms of a lower and an upper limit : > 3 10 / 71 and < 3 1 / 7 . But modern mathematicians use a variety of algorithms suitable for the reiterative power of computers. In 1949, computers were a novelty. Although they were physically huge, they were electronically tiny compared to the supercomputers of today. In 1949, operators of one of these antique computers, the Electronic Numerical Integrator and Computer (ENIAC), at the United States Army Ballistic Research Laboratory, entered a program to calculate pi and waited to see what would happen. After 70 hours, the machine had calculated pi to 2,035 decimal places—in telling contrast to William Shanks (1812-1882) who took 15 years to calculate pi to 707 places using only pen and ink, and the last 180 digits of his solution were incorrect! More recently, Yasumasa Kanada (1948 - ) at the University of Tokyo , using a supercomputer, the HITACHI SR8000/MPP, has calculated pi to 1.2411 trillion decimal places.
http://www.jstor.org/view/0025570x/di020965/02p04056/0
http://www.super-computing.org/
Jules Henri Poincare (1854-1912) was both a physicist and a mathematician. He shared many ideas with Henrik Lorentz (1823-1928) and together they developed formulas that Poincare called the “Lorentz transformations,” which are used to convert measurements in space and time from the point of view of one observer to the point of view of another observer, either at rest or at speeds as high as the speed of light. The Lorentz transformations were an integral step in the formulation of Einstein's theory of relativity. Poincare left a mathematical problem, known as Poincare's conjecture, for later mathematicians to solve. Topologists describe the surface of a sphere as a two-dimensional space without holes. If a hypothetical rope were looped around a sphere, it could be tightened and gradually removed as it becomes smaller and smaller. Poincare's conjecture says that a three-dimensional space with the same characteristic would be the surface of a four-dimensional sphere.
Proving this conjecture became very important to mathematicians and physicists, because the proven theorem would contribute much to their understanding of the universe, which exists in the four dimensions of space/time. The problem was so important that the Clay Mathematical Institue offed a prize of $1,000,000 to the first peron to prove the conjecture. The problem was unsolved for almost one hundred years before Grigori Perlman (1966 - ) solved it.
http://mathworld.wolfram.com/PoincareConjecture.html
http://mathworld.wolfram.com/news/2003-04-15/poincare/
Grigori Perelman (1966 - ) studied and then worked at American universities before returning to Russia, where he was better able to work without distractions, he said, at the Seklov Institute of Mathematic in St. Petersburg. In 2002 and 2003, he posted three articles on the Internet in which he outlined the proof of a more general conjecture than Poincare's, but for which Poincare's is a special case. When teams of mathematicians had studied his proof and found it complete, Perelman was awarded the Fields Medal, the most prestigious, internationally recognized award for mathematics, and with it a prize of $1,000,000 offered by the Clay Mathematical Institute to the first person who could prove Poincare's conjecture. Incredibly to most academics, who covet the prestige and the cash needed to fund their research, Perelman refused the medal and the prize.
The Fields Medal is presented every four years at the International Congress of Mathematicians, together with a prize of 15,000 Canadian dollars. The first Fields Medal was awarded in 1936 at the World Congress in Oslo . The Fields Medal is made of gold, and shows the head of Archimedes (287-212 BC) together with a quotation attributed to him: Transire suum pectus mundoque potiri (Rise above oneself and grasp the world).
http://mathworld.wolfram.com/FieldsMedal.html
http://www.newyorker.com/archive/2006/08/28/060828fa_fact2?currentPage=1