Posts Tagged ‘Einstein’
“A child[’s]…first geometrical discoveries are topological…If you ask him to copy a square or a triangle, he draws a closed circle”*…
Topology is the Silly Putty of mathematics. Indeed, sometimes, topology is called “rubber-sheet geometry” because topologists study the properties of shapes that don’t change when an object is stretched or distorted. As Cliff Pickover explains, this leads to the creation of some pretty confounding shapes…
Mathematicians continue to invent strange objects to test their intuitions. Alexander’s horned sphere [above] is an example of a convoluted, intertwined surface for which it is difficult to define an inside and outside. Introduced by mathematician James Waddell Alexander (1888 – 1971), Alexander’s horned sphere is formed by successively growing pairs of horns that are almost interlocked and whose end points approach each other. The initial steps of the construction can be visualized with your fingers. Move the thumb and forefinger of each of your hands close to one another, then grow a smaller thumb and forefinger on each of these, and continue this budding without limit!
Although this may be hard to visualize, Alexander’s horned sphere is homeomorphic to a ball. In this case, this means that it can be stretched into a ball without puncturing or breaking it. Perhaps it is easier to visualize the reverse: stretching the ball into the horned sphere without ripping it. The boundary is, therefore, homeomorphic to a sphere…
Read more at “The Official Alexander Sphere Appreciation Page.”
* Jean Piaget
As we twist and turn, we might send artfully-folded birthday greetings to Sir Erik Christopher Zeeman; he was born on this date in 1925. While he is probably most-widely known as a popularizer of Catastrophe Theory, his primary contributions to math have been in topology, more particularly in geometric topology (e.g., in knot theory) and in dynamical systems. The Christopher Zeeman Medal for Communication of Mathematics of the London Mathematical Society and the Institute of Mathematics and its Applications is named in his honor.
We might also spare a thought for Satyendra Nath Bose; he died on his date in 1974. A physicist and mathematician, he collaborated with Albert Einstein to develop a theory of statistical quantum mechanics, now called Bose-Einstein statistics. Paul Dirac named the class of particles that obey Bose–Einstein statistics, bosons, after Bose.
For over two decades, The Simpsons has been one of the best written and most entertaining programs on television. Simon Singh believes that he’s discovered the series’ secret sauce: it’s written by math geeks who unreservedly lard the show with math gags…
The first proper episode of the series in 1989 contained numerous mathematical references (including a joke about calculus), while the infamous “Treehouse of Horror VI” episode presents the most intense five minutes of mathematics ever broadcast to a mass audience. Moreover, The Simpsons has even offered viewers an obscure joke about Fermat’s last theorem, the most notorious equation in the history of mathematics.
These examples are just the tip of the iceberg, because the show’s writing team includes several mathematical heavyweights. Al Jean, who worked on the first series and is now executive producer, went to Harvard University to study mathematics at the age of just 16. Others have similarly impressive degrees in maths, a few can even boast PhDs, and Jeff Westbrook resigned from a senior research post at Yale University to write scripts for Homer, Marge and the other residents of Springfield…
More on the numerical nuttiness here.
And readers can test themselves against The Simpsons writing room in this multiple choice test (wherein one will find, among other amusements, the answer to the riddle in the title above).
As we wonder how cartoon characters count with only four fingers, we might pause to remember Sir Arthur Stanley Eddington, OM, FRS; he died in this date in 1944. An astrophysicist, mathematician, and philosopher of science known for his work on the motion, distribution, evolution and structure of stars, Eddington is probably best remembered for his relationship to Einstein: he was, via a series of widely-published articles, the primary “explainer” of Einstein’s Theory of General Relativity to the English-speaking world; and he was, in 1919, the leader of the experimental team that used observations of a solar eclipse to confirm the theory.
Dictionary of Numbers is an award-winning Google Chrome extension that tries to make sense of numbers encountered on the web by providing descriptions of those numbers in human terms. Just as a dictionary describes words one doesn’t know in terms one does, so Dictionary of Numbers puts unfamiliar quantities in understandable, recognizable terms… “Because ‘8 million people’ means nothing, but ‘population of New York City’ means everything.”
As we graduate from our fingers and toes, we might spare a thought for Jules Henri Poincaré; he died on this date in 1912. A mathematician, theoretical physicist, engineer, and a philosopher of science, Poincaré is considered the “last Universalist” in math– the last mathematician to excel in all fields of the discipline as it existed during his lifetime.
Poincaré was a co-discoverer (with Einstein and Lorentz) of the special theory of relativity; he laid the foundations for the fields of topology and chaos theory; and he had a huge impact on cosmogony. His famous “Conjecture” held that if any loop in a given three-dimensional space can be shrunk to a point, the space is equivalent to a sphere; it remained unsolved until Grigori Perelman completed a proof in 2003.
[TotH to Richard Kadrey’s Damn Tumbler]
As we purchase our peek behind the curtain, we might note that it was on this date in 1923 that Albert Eistein demonstrated that time is relative: he delivered his Nobel Prize lecture… two years late.
In 2010, Japanese engineer Shigeru Kondo set a record, calculating the value of pi to 5 trillion digits… then last October, he smashed his own mark, identifying the first 10 trillion decimal places. (He used a home-made computer that ran so hot that the temperature in his apartment was over 100 degrees…)
The quest will no doubt continue– pi is an irrational number that exerts an irrational fascination. Meantime, readers can take a peek at this work-perpetually-in-progress. Web design firm firm Two-N has created this nifty visualization and search tool, allowing one to find any one of the first 4,000,000 digits of pi:
Bonus: “50 Interesting Facts About Pi”
As we ruminate on randomness, we might send carefully-calculated birthday greetings to Hermann Minkowski; he was born on this date in 1864. Minkowski developed the geometry of numbers and used geometrical methods to solve difficult problems in number theory and mathematical physics; he is probaly best remembered for realizing that his former student Albert Einstein’s special theory of relativity (1905), presented algebraically by Einstein, could also be understood geometrically as a theory of four-dimensional space-time. Einstein embraced the geometric approach in the development of his theory of general relativity– and the four-dimensional space (the three physical dimensions plus time) involved has since been known as “Minkowski spacetime.”
Minkowski’s best friend was “mathematical hotelier” David Hilbert.
Could Quantum Mechanics be wrong?
The philosophical status of the wavefunction — the entity that determines the probability of different outcomes of measurements on quantum-mechanical particles — would seem to be an unlikely subject for emotional debate. Yet online discussion of a paper claiming to show mathematically that the wavefunction is real has ranged from ardently star-struck to downright vitriolic since the article was first released as a preprint in November 2011.
The paper, thought by some to be one of the most important in quantum foundations in decades, was finally published last week in Nature Physics (M. F. Pusey, J. Barrett & T. Rudolph Nature Phys. http://dx.doi.org/10.1038/nphys2309; 2012), enabling the authors, who had been concerned about violating the journal’s embargo, to speak about it publicly for the first time. They say that the mathematics leaves no doubt that the wavefunction is not just a statistical tool, but rather, a real, objective state of a quantum system…
The authors have some heavyweights in their corner: their view was once shared by Austrian physicist and quantum-mechanics pioneer Erwin Schrödinger, who proposed in his famous thought experiment that a quantum-mechanical cat could be dead and alive at the same time. But other physicists have favoured an opposing view, one held by Albert Einstein: that the wavefunction reflects the partial knowledge an experimenter has about a system. In this interpretation, the cat is either dead or alive, but the experimenter does not know which. This ‘epistemic’ interpretation, many physicists and philosophers argue, better explains the phenomenon of wavefunction collapse, in which a quantum state is fundamentally changed by measuring it…
Read the full story in Nature.
As we listen for the tell-tale purr, we might spare a thought for Gerbert d’Aurillac (who became Pope Sylvester II); he died on this date in 1003. Gerbert/Sylvester was never canonized; indeed, in his day, he dogged with rumors that he was a sorcerer in league with the devil… which appear to have been the work of reactionary forces resisting both Sylvester’s attempts to rid the Church of corruption (especially simony, the sale of sacraments and indulgences) and his attempts to popularize mathematics, astronomy and mechanics for lay audiences. Inspired by translations of Arabic texts, he built clocks, invented the hydraulic organ, crafted astronomical instruments, and renewed interest in the abacus for use in mathematical calculations (in the process of which, he seems to have introduced Arabic numerals [except zero]). It’s not a stretch to suggest that Gerbert/Sylvester began Europe’s long march out of the Dark Ages.