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  • feedwordpress 08:01:49 on 2018/10/16 Permalink
    Tags: Andreas Cellarius, , , Harmonia Macrocosmica, , , Mathematics, , Sir William Rowan Hamilton,   

    “The Bible shows the way to go to heaven, not the way the heavens go”*… 

     

    cellarius-seasons-banner-1024x689

     

    Harmonia Macrocosmica (1660), an atlas of the stars from the Dutch Golden Age of cartography, maps the structure of the heavens in twenty-nine extraordinary double-folio spreads. We are presented with the motions of the celestial bodies, the stellar constellations of the northern hemisphere, the old geocentric universe of Ptolemy, the newish heliocentric one of Copernicus [as above], and Tycho Brahe’s eccentric combination of the two — in which the Moon orbits the Earth, and the planets orbit the Sun, but the Sun still orbits the Earth. The marginal area of each brightly coloured map is a hive of activity: astronomers bent over charts debate their findings, eager youngsters direct their quadrants skywards, and cherubs fly about with pet birds in tow…

    northern stars

    The Northern Stellar Hemisphere of Antiquity

    More marvelous maps of the heavens at “The Celestial Atlas of Andreas Cellarius (1660)

    * Galileo (quoting a librarian at the Vatican)

    ###

    As we look to the stars, we might recall that it was on this date in 1843 that Sir William Rowan Hamilton conceived the theory of quaternions.  A physicist, astronomer, and mathematician who made important contributions to classical mechanics, optics, and algebra, he had been working since the late 1830s on the basic principles of algebra, resulting in a theory of conjugate functions, or algebraic couples, in which complex numbers are expressed as ordered pairs of real numbers.  But he hadn’t succeeded in developing a theory of triplets that could be applied to three-dimensional geometric problems.  Walking with his wife along the Royal Canal in Dublin, Hamilton realized that the theory should involve quadruplets, not triplets– at which point he stopped to carve carve the underlying equations in a nearby bridge lest he forget them.

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  • feedwordpress 08:01:41 on 2018/10/05 Permalink
    Tags: , , , , , Mathematics, , , ,   

    “It might help to think of the universe as a rubber sheet, or perhaps not”*… 

     

    mobius strip

    You have most likely encountered one-sided objects hundreds of times in your daily life – like the universal symbol for recycling, found printed on the backs of aluminum cans and plastic bottles.

    This mathematical object is called a Mobius strip. It has fascinated environmentalists, artists, engineers, mathematicians and many others ever since its discovery in 1858 by August Möbius, a German mathematician who died 150 years ago, on Sept. 26, 1868.

    Möbius discovered the one-sided strip in 1858 while serving as the chair of astronomy and higher mechanics at the University of Leipzig. (Another mathematician named Listing actually described it a few months earlier, but did not publish his work until 1861.)…

    The discovery of the Möbius strip in the mid-19th century launched a brand new field of mathematics: topology: “The Mathematical Madness of Möbius Strips and Other One-Sided Objects.”

    * Terry Pratchett, Hogfather

    ###

    As we return from whence we came, we might wish a Joyeux Anniversaire to Denis Diderot, contributor to and the chief editor of the Encyclopédie (“All things must be examined, debated, investigated without exception and without regard for anyone’s feelings.”)– and thus towering figure in the Enlightenment; he was born on this date in 1713.  Diderot was also a novelist (e.g., Jacques le fataliste et son maître [Jacques the Fatalist and his Master])…  and no mean epigramist:

    From fanaticism to barbarism is only one step.

    We swallow greedily any lie that flatters us, but we sip only little by little at a truth we find bitter.

    Man will never be free until the last king is strangled with the entrails of the last priest.

    A thing is not proved just because no one has ever questioned it.

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  • feedwordpress 08:01:55 on 2018/09/20 Permalink
    Tags: , , Jeremy Baumberg, Mathematics, Paul Erdos, , ,   

    “To change something, build a new model that makes the existing model obsolete”*… 

     

    science

    There is remarkably little reflection taking place about the state of science today, despite significant challenges, rooted in globalization, the digitization of knowledge, and the growing number of scientists.

    At first glance, all of these seem to be positive trends. Globalization connects scientists worldwide, enabling them to avoid duplication and facilitating the development of universal standards and best practices. The creation of digital databases allows for systematic mining of scientific output and offers a broader foundation for new investigations. And the rising number of scientists means that more science is being conducted, accelerating progress.

    But these trends are Janus-faced…

    Jeremy Baumberg argues that we live in an age of hyper-competitive, trend-driven, and herd-like approach to scientific research: “What Is Threatening Science?

    * Buckminster Fuller

    ###

    As we rethink research, we might spare a thought for Paul Erdős; he died on this date in 1996.  One of the most prolific mathematicians of the 20th century (he published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed), he is remembered both for his “social practice” of mathematics (he engaged more than 500 collaborators) and for his eccentric lifestyle (he spent his waking hours virtually entirely on math; he would typically show up at a colleague’s doorstep and announce “my brain is open”, staying long enough to collaborate on a few papers before moving on a few days later).

    Erdős’s prolific output with co-authors prompted the creation of the Erdős number, the number of steps in the shortest path between a mathematician and Erdős in terms of co-authorships.  Low numbers are a badge of pride– and a usual marker of accomplishment: as of 2016, all Fields Medalists have a finite Erdős number, with values that range between 2 and 6, and a median of 3.  Physics Nobelists Einstein and Sheldon Glashow have an Erdős number of 2.   Baseball Hall of Famer Hank Aaron can be considered to have an Erdős number of 1 because they both autographed the same baseball (for number theorist Carl Pomerance).  Natalie Portman’s undergraduate collaboration with a Harvard professor earned her an Erdős number of 5; Danica McKellar (“Winnie Cooper” in The Wonder Years) has an Erdős number of 4, for a mathematics paper coauthored while an undergraduate at UCLA.

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  • feedwordpress 08:01:51 on 2018/08/29 Permalink
    Tags: Hermann Hankel, , , history of mathematics, , Mathematics, , supertasks, , Zeno   

    “I am incapable of conceiving infinity, and yet I do not accept finity”*… 

     

    hilberts_hotel

    Suppose you’re working at a hotel with infinitely many rooms in it, numbered 1, 2, 3, 4, 5, … all the way up forever and ever. (This is known as a Hilbert Hotel.) One evening when every single room is occupied, a traveler arrives and requests to be accommodated too. You’re the manager. What do you do to help the traveler?

    Simple. You just ask each occupant to one room forward. 1 goes to 2, and 2 goes to 3, and so on. Every previous occupant gets a new room. And the first room is now open for the traveler.

    The procedure above is characterized by an infinite number of actions or tasks to be carried out in a finite amount of time. Procedures with this character are known as supertasks…

    More on the ins and outs of infinities at “Introducing Supertasks.” (More fun musings on infinity here and here; and more on Hilbert’s Hotel here.)

    * Simone de Beauvoir, La Vieillesse

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    As we muse on many, we might spare a thought for Hermann Hankel; he died on this date in 1873.  A mathematician who worked with Möbius, Riemann, Weierstrass,  and Kronecker (among others), he made important contributions to the understanding of complex numbers and quaternions… and to work begun by Bernard Bolzano on infinite series.

    220px-Hankel source

     

     
  • feedwordpress 08:01:27 on 2018/08/06 Permalink
    Tags: , , , , Johann Bernoulli, Mathematics, , ,   

    “Once is happenstance. Twice is coincidence. Three times, it’s enemy action.”*… 

     

    complexity

    A couple of weeks ago, we considered the human urge to find significance, meaning in everyday occurrences: “All mystical experience is coincidence; and vice versa, of course.” Today, we consider the same phenomena from a more mathematical point-of-view…

    Was it a chance encounter when you met that special someone or was there some deeper reason for it? What about that strange dream last night—was that just the random ramblings of the synapses of your brain or did it reveal something deep about your unconscious? Perhaps the dream was trying to tell you something about your future. Perhaps not. Did the fact that a close relative developed a virulent form of cancer have profound meaning or was it simply a consequence of a random mutation of his DNA?

    We live our lives thinking about the patterns of events that happen around us. We ask ourselves whether they are simply random, or if there is some reason for them that is uniquely true and deep. As a mathematician, I often turn to numbers and theorems to gain insight into questions like these. As it happens, I learned something about the search for meaning among patterns in life from one of the deepest theorems in mathematical logic. That theorem, simply put, shows that there is no way to know, even in principle, if an explanation for a pattern is the deepest or most interesting explanation there is. Just as in life, the search for meaning in mathematics knows no bounds…

    Noson Yanofsky on what math can teach us about finding order in our chaotic lives.

    * Ian Fleming

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    As we consider the odds, we might send carefully-calculated birthday greetings to Johann Bernoulli; he was born on this date in 1667.  A member of the mathematically-momentous Bernoulli family, Johann (also known as Jean or John) discovered the exponential calculus and (with Leibniz and Huygens) the equation of the catenary.  Still, he be best remembered as teacher and mentor of Leonhard Euler.

    220px-Johann_Bernoulli2 source

     

     
  • feedwordpress 08:01:18 on 2018/07/31 Permalink
    Tags: , , , , , , Mathematics, sentimental cartography, Seymour Papert,   

    “I have an existential map. It has ‘you are here’ written all over it.”*… 

     

    map

    A detail from illustrator James Turner‘s Map of Humanity.

     

    A long time ago, I made a map of the rationalist community.  This is in the same geographic-map-of-something-non-geographic tradition as the Greater Ribbonfarm Cultural Region or xkcd’s map of the Internet. There’s even some sort of therapy program that seems to involve making a map like this of your life, though I don’t know how seriously they take it.

    There’s no good name for this art and it’s really hard to Google. If you try “map of abstract concept” you just get a bunch of concept maps. It seems the old name, from back when this was a popular Renaissance amusement, is “sentimental cartography”, since it was usually applied to sentiments like love or sorrow. This isn’t great – the Internet’s not a sentiment – but it’s what we’ve got and I’ll do what I can to try to make it catch on…

    See the marvelous examples (like the one above) collected by Scott Alexander at “Sentimental Cartography.”

    * Steven Wright

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    As we find our place, we might spare a thought for Seymour Papert; he died on this date in 2016.  Trained as a mathematician, Papert was a pioneer of computer science, and in particular, artificial intelligence. He created the Epistemology and Learning Research Group at the MIT Architecture Machine Group (which later became the MIT Media Lab); he directed MIT’s Artificial Intelligence Laboratory; he authored the hugely-influential LOGO computer language; and he was a principal of the One Laptop Per Child Program.  Called by Marvin Minsky “the greatest living mathematics educator,” Papert won a Guggenheim fellowship (1980), a Marconi International fellowship (1981), the Software Publishers Association Lifetime Achievement Award (1994), and the Smithsonian Award (1997).

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  • feedwordpress 08:01:10 on 2018/07/12 Permalink
    Tags: , , , Fullerenes, , hole, Mathematics, nothing, , , zero   

    “I love to talk about nothing. It’s the only thing I know anything about.”*… 

     

    zero

    The computer you’re reading this article on right now runs on a binary — strings of zeros and ones. Without zero, modern electronics wouldn’t exist. Without zero, there’s no calculus, which means no modern engineering or automation. Without zero, much of our modern world literally falls apart.

    Humanity’s discovery of zero was “a total game changer … equivalent to us learning language,” says Andreas Nieder, a cognitive scientist at the University of Tübingen in Germany.

    But for the vast majority of our history, humans didn’t understand the number zero. It’s not innate in us. We had to invent it. And we have to keep teaching it to the next generation.

    Other animals, like monkeys, have evolved to understand the rudimentary concept of nothing. And scientists just reported that even tiny bee brains can compute zero. But it’s only humans that have seized zero and forged it into a tool.

    So let’s not take zero for granted. Nothing is fascinating. Here’s why…

    It is indeed fascinating, as you’ll see at “The mind-bendy weirdness of the number zero, explained.”

    Pair with: “Is a hole a real thing, or just a place where something isn’t?” and with The Ministry of Ideas’ podcast “Nothing Matters.”

    * Oscar Wilde

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    As we obsess about absence, we might box a dome-shaped birthday cake for inventor, educator, author, philosopher, engineer, and architect R(ichard) Buckminster Fuller; he was born on this date in 1895.  “Bucky” most famously developed the geodesic dome, the only large dome that can be set directly on the ground as a complete structure, and the only practical kind of building that has no limiting dimensions (i.e., beyond which the structural strength must be insufficient).  But while he never got around to frankfurters, he was sufficiently prolific to have scored over 2,000 patents.

    “Fullerenes” (molecules composed entirely of carbon, in the form of a hollow spheres, ellipsoids, or tubes), key components in many nanotechnology applications, were named for Fuller, as their structure mimes that of the geodesic dome.  Spherical fullerenes (resembling soccer balls) are also called “buckyballs”; cylindrical ones, carbon nanotubes or “buckytubes.”

    I have to say, I think that we are in some kind of final examination as to whether human beings now, with this capability to acquire information and to communicate, whether we’re really qualified to take on the responsibility we’re designed to be entrusted with. And this is not a matter of an examination of the types of governments, nothing to do with politics, nothing to do with economic systems. It has to do with the individual. Does the individual have the courageto really go along with the truth?

    God, to me, it seems
    is a verb,
    not a noun,
    proper or improper.

    For more, see “And that’s a lot.”

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  • feedwordpress 08:01:40 on 2018/07/01 Permalink
    Tags: , , , Mathematics, , practical joke, , Pythagoras, Pythagorean Cup,   

    “All practical jokes, friendly, harmless or malevolent, involve deception, but not all deceptions are practical jokes”*… 

     

    28134677457_5f485d8423_z

    When you think of the ancient Greeks, practical jokes might not be the first thing that comes to mind. But along with art, architecture, and philosophy, you can add trick cups to their list of accomplishments.

    The Pythagorean cup is so-named because it was allegedly invented by Pythagoras of Samos (yes, the same guy who gave us theories about right triangles). It’s a small cup with a column in its center. It doesn’t look like much, but when an unsuspecting drinker fills it past a designated level, the liquid mysteriously drains out. Legend has it that Pythagoras used it as a way to punish greedy drinkers who poured themselves too much wine…

    A timeless practical joke, brought to you by the ancient Greeks: more merriment at “Pythagorean Cup.”

    * W. H. Auden, The Dyer’s Hand

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    As we ponders pranks, we might send a “Alles Gute zum Geburtstag” to the polymathic Gottfried Wilhelm Leibniz, the philosopher, mathematician, and political adviser, who was important both as a metaphysician and as a logician, but who is probably best remembered for his independent invention of the calculus; he was born on this date in 1646.  Leibniz independently discovered and developed differential and integral calculus, which he published in 1684;  but he became involved in a bitter priority dispute with Isaac Newton, whose ideas on the calculus were developed earlier (1665), but published later (1687).

    As it happens, Leibnitz was no mean humorist.  Consider, e.g…

    If geometry conflicted with our passions and our present concerns as much as morality does, we would dispute it and transgress it almost as much–in spite of all Euclid’s and Archimedes’ demonstrations, which would be treated as fantasies and deemed to be full of fallacies. [Leibniz, New Essays, p. 95]

    28134677537_d79a889e6a_o source

     

     
  • feedwordpress 08:01:48 on 2018/06/03 Permalink
    Tags: , , , mathematical economics, Mathematics, merit, RGD Allen, , , wealth inequality   

    “Oh, I am fortune’s fool!”*… 

     

    The distribution of wealth follows a well-known pattern sometimes called an 80:20 rule: 80 percent of the wealth is owned by 20 percent of the people. Indeed, a report last year concluded that just eight men had a total wealth equivalent to that of the world’s poorest 3.8 billion people.

    This seems to occur in all societies at all scales. It is a well-studied pattern called a power law that crops up in a wide range of social phenomena. But the distribution of wealth is among the most controversial because of the issues it raises about fairness and merit. Why should so few people have so much wealth?

    The conventional answer is that we live in a meritocracy in which people are rewarded for their talent, intelligence, effort, and so on. Over time, many people think, this translates into the wealth distribution that we observe, although a healthy dose of luck can play a role.

    But there is a problem with this idea: while wealth distribution follows a power law, the distribution of human skills generally follows a normal distribution that is symmetric about an average value. For example, intelligence, as measured by IQ tests, follows this pattern. Average IQ is 100, but nobody has an IQ of 1,000 or 10,000.

    The same is true of effort, as measured by hours worked. Some people work more hours than average and some work less, but nobody works a billion times more hours than anybody else.

    And yet when it comes to the rewards for this work, some people do have billions of times more wealth than other people. What’s more, numerous studies have shown that the wealthiest people are generally not the most talented by other measures.

    What factors, then, determine how individuals become wealthy? Could it be that chance plays a bigger role than anybody expected? And how can these factors, whatever they are, be exploited to make the world a better and fairer place?…

    A new computer model of wealth creation confirms that the most successful people are not the most talented, just the luckiest. Learn more at: “If you’re so smart, why aren’t you rich? Turns out it’s just chance.

    * Shakespeare, Romeo and Juliet

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    As we muse on merit, we might send carefully-calculated birthday greetings to a forbearer of the researchers who did the work recounted above, Sir Roy George Douglas Allen; he was born on this date in 1906.  A mathematician and statistician turned economist, he was a leader in the field of mathematical economics, writing a number of influential texts including  Mathematical Analysis for EconomistsStatistics for Economists, and Mathematical Economics.

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  • feedwordpress 08:01:22 on 2018/03/14 Permalink
    Tags: , , Mathematics, , , ,   

    “All numbers are by their nature correct. Well, except for Pi, of course. I can’t be doing with Pi. Gives me a headache just thinking about it, going on and on and on and on and on…”*… 

     

    It’s Pi Day!

    In celebration, a few amusing– and illuminating– links:

    The history of pi

    Pi day magic revealed

    10 stunning images show the beauty hidden in pi

    The history of Pi Day

    How to Memorize Pi if You’re a Word Person (from whence, the image above)

    * Neil Gaiman, Anansi Boys

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    As we enumerate endlessly, we might pause for a piece of pi(e)…

     source

    … in celebration of Albert Einstein’s birthday; he was born on this date in 1879.

    source

    “Everything should be made as simple as possible, but not simpler.”

     

     
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