Posts Tagged ‘geometry’
“Why, sometimes I’ve believed as many as six impossible things before breakfast”*…
Imaginary numbers were long dismissed as mathematical “bookkeeping.” But now, as Karmela Padavic-Callaghan explains, physicists are proving that they describe the hidden shape of nature…
Many science students may imagine a ball rolling down a hill or a car skidding because of friction as prototypical examples of the systems physicists care about. But much of modern physics consists of searching for objects and phenomena that are virtually invisible: the tiny electrons of quantum physics and the particles hidden within strange metals of materials science along with their highly energetic counterparts that only exist briefly within giant particle colliders.
In their quest to grasp these hidden building blocks of reality scientists have looked to mathematical theories and formalism. Ideally, an unexpected experimental observation leads a physicist to a new mathematical theory, and then mathematical work on said theory leads them to new experiments and new observations. Some part of this process inevitably happens in the physicist’s mind, where symbols and numbers help make invisible theoretical ideas visible in the tangible, measurable physical world.
Sometimes, however, as in the case of imaginary numbers – that is, numbers with negative square values – mathematics manages to stay ahead of experiments for a long time. Though imaginary numbers have been integral to quantum theory since its very beginnings in the 1920s, scientists have only recently been able to find their physical signatures in experiments and empirically prove their necessity…
Learn more at “Imaginary numbers are real,” from @Ironmely in @aeonmag.
* The Red Queen, in Lewis Carroll’s Through the Looking Glass
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As we get real, we might spare a thought for two great mathematicians…
Georg Friedrich Bernhard Riemann died on this date in 1866. A mathematician who made contributions to analysis, number theory, and differential geometry, he is remembered (among other things) for his 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, regarded as one of the most influential papers in analytic number theory.
Andrey (Andrei) Andreyevich Markov died on this date in 1922. A Russian mathematician, he helped to develop the theory of stochastic processes, especially those now called Markov chains: sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors. (For example, the probability of winning at the game of Monopoly can be determined using Markov chains.) His work on the study of the probability of mutually-dependent events has been developed and widely applied to the biological, physical, and social sciences, and is widely used in Monte Carlo simulations and Bayesian analyses.
“There’s no idea in economics more beautiful than Arrow’s impossibility theorem”*…
Tim Harford unpack’s Kenneth Arrow‘s Impossibility Theorem (which feels a bit like a socio-economic “Monty Hall Problem“) and considers it’s implications…
… if any group of voters gets to decide one thing, that group gets to decide everything, and we prove that any group of decisive voters can be pared down until there’s only one person in it. That person is the dictator. Our perfect constitution is in tatters.
That’s Arrow’s impossibility theorem. But what does it really tell us? One lesson is to abandon the search for a perfect voting system. Another is to question his requirements for a good constitution, and to look for alternatives. For example, we could have a system that allows people to register the strength of their feeling. What about the person who has a mild preference for profiteroles over ice cream but who loathes cheese? In Arrow’s constitution there’s no room for strong or weak desires, only for a ranking of outcomes. Maybe that’s the problem.
Arrow’s impossibility theorem is usually described as being about the flaws in voting systems. But there’s a deeper lesson under its surface. Voting systems are supposed to reveal what societies really want. But can a society really want anything coherent at all? Arrow’s theorem drives a stake through the heart of the very idea. People might have coherent preferences, but societies cannot…
On choice, law, and the paradox at the heart of voting: “Arrow’s Impossibility Theorem,” from @TimHarford in @WhyInteresting. Eminently worth reading in full.
* Tim Harford
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As we contemplate collective choice, we might send grateful birthday greetings to the man who “wrote the book” on perspective, Leon Battista Alberti; he was born on this date in 1404. The archetypical Renaissance humanist polymath, Alberti was an author, artist, architect, poet, priest, linguist, philosopher, cartographer, and cryptographer. He collaborated with Toscanelli on the maps used by Columbus on his first voyage, and he published the the first book on cryptography that contained a frequency table.
But he is surely best remembered as the author of the first general treatise– Della Pictura (1434)– on the the laws of perspective, which built on and extended Brunelleschi’s work to describe the approach and technique that established the science of projective geometry… and fueled the progress of painting, sculpture, and architecture from the Greek- and Arabic-influenced formalism of the High Middle Ages to the more naturalistic (and Latinate) styles of Renaissance.
“‘It’s magic,’ the chief cook concluded, in awe. ‘No, not magic,’ the ship’s doctor replied. ‘It’s much more. It’s mathematics.’*…
Michael Wendl (and here) dissects some variants of the magic separation, a self-working card trick…
Martin Gardner—one of history’s most prolific maths popularisers [see here]—frequently examined the connection between mathematics and magic, commonly looking at tricks using standard playing cards. He often discussed ‘self-working’ illusions that function in a strictly mechanical way, without any reliance on sleight of hand, card counting, pre-arrangement, marking, or key-carding of the deck. One of the more interesting specimens in this genre is a matching trick called the magic separation.
This trick can be performed with 20 cards. Ten of the cards are turned face-up, with the deck then shuffled thoroughly by both the performer and, importantly, the spectator. The performer then deals 10 cards to the spectator and keeps the remainder for herself. This can be done blindfolded to preclude tracking or counting. Not knowing the distribution of cards, our performer announces she will rearrange her own cards ‘magically’ so that the number of face-ups she holds matches the number of face-ups the spectator has. When cards are displayed, the counts do indeed match. She easily repeats the feat for hecklers who claim luck…
All is revealed: “An odd card trick,” from Chalkdust (@chalkdustmag).
* David Brin, Glory Season
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As we conjure, we might spare a thought for Persian polymath Omar Khayyam; the mathematician, philosopher, astronomer, epigrammatist, and poet died on this date in 1131. While he’s probably best known to English-speakers as a poet, via Edward FitzGerald’s famous translation of the quatrains that comprise the Rubaiyat of Omar Khayyam, Omar was one of the major mathematicians and astronomers of the medieval period. He is the author of one of the most important works on algebra written before modern times, the Treatise on Demonstration of Problems of Algebra (which includes a geometric method for solving cubic equations by intersecting a hyperbola with a circle). His astronomical observations contributed to the reform of the Persian calendar. And he made important contributions to mechanics, geography, mineralogy, music, climatology, and Islamic theology.
“All you really need to know for the moment is that the universe is a lot more complicated than you might think, even if you start from a position of thinking it’s pretty damn complicated in the first place”*…
When you gaze out at the night sky, space seems to extend forever in all directions. That’s our mental model for the universe, but it’s not necessarily correct. There was a time, after all, when everyone thought the Earth was flat, because our planet’s curvature was too subtle to detect and a spherical Earth was unfathomable.
Today, we know the Earth is shaped like a sphere. But most of us give little thought to the shape of the universe. Just as the sphere offered an alternative to a flat Earth, other three-dimensional shapes offer alternatives to “ordinary” infinite space.
We can ask two separate but interrelated questions about the shape of the universe. One is about its geometry: the fine-grained local measurements of things like angles and areas. The other is about its topology: how these local pieces are stitched together into an overarching shape.
Cosmological evidence suggests that the part of the universe we can see is smooth and homogeneous, at least approximately. The local fabric of space looks much the same at every point and in every direction. Only three geometries fit this description: flat, spherical and hyperbolic…
Alternatives to “ordinary” infinite space: “What Is the Geometry of the Universe?”
* Douglas Adams, The Hitchhiker’s Guide to the Galaxy
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As we tinker with topology, we might recall that it was on this date in 1811 that Percy Bysshe Shelley was expelled from the University of Oxford for publishing the pamphlet The Necessity of Atheism. Shelley, of course, went on to become a celebrated lyric poet and one of the leaders of the English Romantic movement… one who had a confident (if not to say exalted) sense of his role in society:
Poets are the hierophants of an unapprehended inspiration; the mirrors of the gigantic shadows which futurity casts upon the present; the words which express what they understand not; the trumpets which sing to battle, and feel not what they inspire; the influence which is moved not, but moves. Poets are the unacknowledged legislators of the world.
“The laws of nature are but the mathematical thoughts of God”*…
2,300 years ago, Euclid of Alexandria sat with a reed pen–a humble, sliced stalk of grass–and wrote down the foundational laws that we’ve come to call geometry. Now his beautiful work is available for the first time as an interactive website.
Euclid’s Elements was first published in 300 B.C. as a compilation of the foundational geometrical proofs established by the ancient Greek. It became the world’s oldest, continuously used mathematical textbook. Then in 1847, mathematician Oliver Byrne rereleased the text with a new, watershed use of graphics. While Euclid’s version had basic sketches, Byrne reimagined the proofs in a modernist, graphic language based upon the three primary colors to keep it all straight. Byrne’s use of color made his book expensive to reproduce and therefore scarce, but Byrne’s edition has been recognized as an important piece of data visualization history all the same…
Explore elemental beauty at “A masterpiece of ancient data viz, reinvented as a gorgeous website.”
* Euclid, Elements
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As we appreciate the angles, we might spare a thought for Kurt Friedrich Gödel; he died on this date in 1978. A logician, mathematician, and philosopher, he is considered (along with Aristotle, Alfred Tarski— whose birthday this also is– and Gottlob Frege) to be one of the most important logicians in history. Gödel had an immense impact upon scientific and philosophical thinking in the 20th century. He is, perhaps, best remembered for his Incompleteness Theorems, which led to (among other important results) Alan Turing’s insights into computational theory.
Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Gödel’s achievement. — John von Neumann
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