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Posts Tagged ‘Mathematics

“God made the integers; all the rest is the work of Man”*…

 

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From Alex Bellos: the results of his global online poll to find the world’s favorite number…

The winner?  Seven—  and it wasn’t even close…

Leopold Kronecker

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As we settle for anything but snake eyes, we might send symbolic birthday greetings to John Pell; he was born on this date in 1611.  An English mathematician of accomplishment, he is perhaps best remembered for having introduced the “division sign”– the “obelus”: a short line with dots above and below– into use in English.  It was first used in German by Johann Rahn in 1659 in Teutsche Algebra; Pell’s translation brought the symbol to English-speaking mathematicians.  But Pell was an important influence on Rahn, and edited his book– so may well have been, many scholars believe, the originator of the symbol for this use.  (In any case the symbol wasn’t new to them:  the obelus [derived from the word for “roasting spit” in Greek] had already been used to mark passages in writings that were considered dubious, corrupt or spurious…. a use that surely seems only too appropriate to legions of second and third grade math students.)

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Written by LW

March 1, 2015 at 1:01 am

“Prime numbers are what is left when you have taken all the patterns away”*…

 

Der Hl. Augustinus und der Teufel

 

1, followed by 13 zeros, then 666, and then another 13 zeros, and a final 1:  a palindromic prime number named for Belphegor (or Beelphegor), one of the seven princes of Hell.  Reputed to help people make discoveries, Belphegor is the demon of inventiveness.  He figures in Milton’s Paradise Lost as the namesake of one of the “Principalities of the Prime”… So it is only fitting that these devilish digits bear his name.

The symbol of Belphegor’s Prime resembles Pi, only upside down. It is derived from a bird glyph first seen embedded in the Voynich Manuscript.

 

More prime provocation at Cliff Pickover‘s “Belphegor’s Prime: 1000000000000066600000000000001.”

* Mark Haddon, The Curious Incident of the Dog in the Night-Time

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As we try to divine divisors, we might recall that it was on this date in 1968 that the first-ever 9-1-1 call was placed by Alabama Speaker of the House Rankin Fite, from Haleyville City Hall, to U.S. Rep. Tom Bevill, at the city’s police station.

Emergency numbers date back to 1937, when the British began to use 999.  But experience showed that three repeated digits led to many mistaken/false alarms.  The Southern California Telephone Co. experimented in 1946 in Los Angeles with 116 for emergencies.

But 911– using just the first and last digits available– yielded the best results, and went into widespread use in the 1980s when 911 was adopted as the standard emergency number across most of the country under the North American Numbering Plan.

And yes, “911” is a prime…

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Written by LW

February 16, 2015 at 1:01 am

“The day after tomorrow is the third day of the rest of your life”*…

 

French Republican Calendar of 1794, Philibert-Louis Debucourt

 

… The Earth’s orbit is almost — but not quite — a round number, and so we continually try to fit the natural world into a mathematical order that makes sense. Even though the Gregorian Calendar solved one major problem (a year now aligned with the time of the Earth’s orbit), in the eyes of many it’s still far from perfect, and two quirks of its construction have continued to nag those inclined towards a more rational calendar. First is the inconsistent number of days in each month, and second, the fact that 365 is not divisible by seven, so that each year calendar dates fall on different days of the week…

Colin Dickey explores some the modern attempts to “correct” these short-comings in “Tempo Shifts.”

… As the Sumerian God Gozer tells Bill Murray and friends at the climax of Ghostbusters, we choose the means of our destruction. The End we imagine, Kermode writes, “will reflect [our] irreducibly intermediary preoccupations,” which is why the Apocalypse is always assumed to be happening within years or decades, rather than centuries or millennia. The plain fact being that no matter how we try to organize and structure the calendar — be we French Revolutionaries, post-Soviet mathematicians, or American evangelicals — we design it so that we are the center of history. Time and tide may wait for no man, but the calendar always revolves around the calendar-makers.

The full– and fascinating– story here.

* George Carlin

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As we count the days, we might spare a thought for Immanuel Kant; he died on this date in 1804.  One of the central figures of modern philosophy, Kant is remembered primarily for his efforts to unite reason with experience (e.g., Critique of Pure Reason [Kritik der reinen Vernunft], 1781), and for his work on ethics (e.g., Metaphysics of Morals [Die Metaphysik der Sitten], 1797) and aesthetics (e.g., Critique of Judgment [Kritik der Urteilskraft], 1790).  But he made important contributions to mathematics as well:  Kant’s argument that mathematical truths are a form of synthetic a priori knowledge was cited by Einstein as an important early influence on his work.

There is … only a single categorical imperative and it is this: Act only on that maxim through which you can at the same time will that it should become a universal law.

– Chapter 11, Metaphysics of Morals

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Written by LW

February 12, 2015 at 1:01 am

“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

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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.

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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.

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Written by LW

February 4, 2015 at 1:01 am

“What is the difference between a taxidermist and a tax collector? The taxidermist takes only your skin”*…

 

A Ferris wheel of chipmunks, formerly part of the Dead Pals of Sam Sanfillippo

 

Last October, Atlas Obscura co-presented a Rogue Taxidermy Fair with its fellow Brooklyn-based (and former Roughly Daily subject) Morbid Anatomy in celebration of a new book on “rogue taxidermy.”

The Crucified Sheep

 

Read more about– and see more of– the Fair at “The Crucified Sheep, Tattooed Frogs, and Crocheted Skeletons of a Rogue Taxidermy Fair in Brooklyn,” and revisit (R)Ds earlier look at rogue taxidermy here.

* Mark Twain

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As we strike a pose, we might recall that it was on this date in 1697 that Isaac Newton received and solved Jean Bernoulli’s brachistochrone problem.  The Swiss mathematician Bernoulli had challenged his colleagues to solve it within six months.  Newton not only solved the problem before going to bed that same night, but in doing so, invented a new branch of mathematics called the calculus of variations.  He had resolved the issue of specifying the curve connecting two points displayed from each other laterally, along which a body, acted upon only by gravity, would fall in the shortest time.  Newton, age 55, sent the solution to be published, at his request, anonymously.  But the brilliant originality of the work betrayed his identity, for when Bernoulli saw the solution he commented, “We recognize the lion by his claw.”

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Written by LW

January 26, 2015 at 1:01 am

“I know numbers are beautiful. If they aren’t beautiful, nothing is”*…

 

Euler’s identity: Math geeks extol its beauty, even finding in it hints of a myste­rious connected­ness in the universe. It’s on tank tops and coffee mugs [and tattoos]. Aliens, apparently, carve it into crop circles (in 8-bit binary code). It’s appeared on The Simpsons. Twice.

What’s the deal with Euler’s identity? Basically, it’s an equation about numbers—specifically, those elusive constants π and e. Both are “transcendental” quanti­ties; in decimal form, their digits unspool into infinity. And both are ubiquitous in scientific laws. But they seem to come from different realms: π (3.14159 …) governs the perfect symmetry and closure of the circle; it’s in planetary orbits, the endless up and down of light waves. e (2.71828 …) is the foundation of exponential growth, that accelerating trajectory of escape inherent to compound interest, nuclear fission, Moore’s law. It’s used to model everything that grows…

Now, maybe you’ve never thought of math equations as “beautiful,” but look at that result: It combines the five most fundamental numbers in math—0, 1, e, i, and π—in a relation of irreducible simplicity. (Even more astonishing if you slog through the proof, which involves infinite sums, factorials, and fractions nested within fractions within fractions like matryoshka dolls.) And remember, e and π are infinitely long decimals with seemingly nothing in common; they’re the ultimate jigsaw puzzle pieces. Yet they fit together perfectly—not to a few places, or a hundred, or a million, but all the way to forever…

But the weirdest thing about Euler’s formula—given that it relies on imaginary numbers—is that it’s so immensely useful in the real world. By translating one type of motion into another, it lets engineers convert messy trig problems (you know, sines, secants, and so on) into more tractable algebra—like a wormhole between separate branches of math. It’s the secret sauce in Fourier transforms used to digi­tize music, and it tames all manner of wavy things in quantum mechanics, electron­ics, and signal processing; without it, computers might not exist…

More marvelous math at “The Baffling and Beautiful Wormhole Between Branches of Math.”

[TotH to @haarsager]

Paul Erdős

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As we wonder if Descartes wasn’t right when he wrote that “everything turns into mathematics,” 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.

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Written by LW

December 4, 2014 at 1:01 am

“I’d imagine the whole world was one big machine”*…

 

From Harvard’s Houghton Library (where your correspondent is currently ensconced), a pair of plates (click here for larger) from Jean Errard‘s Instruments mathematiques mechaniques, 1584.  Errard, who was a pioneering mathematician, engineer, and developer of military fortifications, is thought by some scholars to have based these drawings on thoughts from Archimedes.  In any case, they’re a treat.

* Hugo Cabret (in Brian Seltznick’s The Invention of Hugo Cabret)

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As we muse on mechanization, we might send well-suspended birthday greetings to John M. Mack; he was born on this date in 1864.  At the turn of the 20th century, mack and his brother Augustus developed a successful gasoline-powered sightseeing bus; then in 1905, they joined with three other brothers to form the Mack Brothers Motor Car Company.  They continued to build sightseeing buses, but shifted their focus increasingly to heavy-duty trucks; then, in 1909, they produced the first engine-driven fire truck in the United States.  With financing from J.P. Morgan, the company grew into what we now know as the Mack Truck Company.

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Written by LW

October 27, 2014 at 1:01 am

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