Posts Tagged ‘algebra’
“‘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.
“Those who wish to know the art of calculating, its subtleties and ingenuities, must know computing with hand figures”*…
The House of Wisdom sounds a bit like make believe: no trace remains of this ancient library, destroyed in the 13th Century, so we cannot be sure exactly where it was located or what it looked like.
But this prestigious academy was in fact a major intellectual powerhouse in Baghdad during the Islamic Golden Age, and the birthplace of mathematical concepts as transformative as the common zero and our modern-day “Arabic” numerals.
Founded as a private collection for caliph Harun Al-Rashid in the late 8th Century then converted to a public academy some 30 years later, the House of Wisdom appears to have pulled scientists from all over the world towards Baghdad, drawn as they were by the city’s vibrant intellectual curiosity and freedom of expression (Muslim, Jewish and Christian scholars were all allowed to study there).
An archive as formidable in size as the present-day British Library in London or the Bibliothèque Nationale of Paris, the House of Wisdom eventually became an unrivalled centre for the study of humanities and sciences, including mathematics, astronomy, medicine, chemistry, geography, philosophy, literature and the arts – as well as some more dubious subjects such as alchemy and astrology.
To conjure this great monument thus requires a leap of imagination (think the Citadel in Westeros, or the library at Hogwarts), but one thing is certain: the academy ushered in a cultural Renaissance that would entirely alter the course of mathematics.
The House of Wisdom was destroyed in the Mongol Siege of Baghdad in 1258 (according to legend, so many manuscripts were tossed into the River Tigris that its waters turned black from ink), but the discoveries made there introduced a powerful, abstract mathematical language that would later be adopted by the Islamic empire, Europe, and ultimately, the entire world.
Tracing the House of Wisdom’s mathematical legacy involves a bit of time travel back to the future, as it were. For hundreds of years until the ebb of the Italian Renaissance, one name was synonymous with mathematics in Europe: Leonardo da Pisa, known posthumously as Fibonacci. Born in Pisa in 1170, the Italian mathematician received his primary instruction in Bugia, a trading enclave located on the Barbary coast of Africa (coastal North Africa). In his early 20s, Fibonacci traveled to the Middle East, captivated by ideas that had come west from India through Persia. When he returned to Italy, Fibonacci published Liber Abbaci, one of the first Western works to describe the Hindu-Arabic numeric system.
When Liber Abbaci first appeared in 1202, Hindu-Arabic numerals were known to only a few intellectuals; European tradesmen and scholars were still clinging to Roman numerals, which made multiplication and division extremely cumbersome (try multiplying MXCI by LVII!). Fibonacci’s book demonstrated numerals’ use in arithmetic operations – techniques which could be applied to practical problems like profit margin, money changing, weight conversion, barter and interest…
Fibonacci’s great genius was not just his creativity as a mathematician, however, but his keen understanding of the advantages known to Muslim scientists for centuries: their calculating formulas, their decimal place system, their algebra. In fact, Liber Abbaci relied almost exclusively on the algorithms of 9th-Century mathematician Al-Khwarizmi. His revolutionary treatise presented, for the first time, a systematic way of solving quadratic equations. Because of his discoveries in the field, Al-Khwarizmi is often referred to as the father of algebra – a word we owe to him, from the Arabic al-jabr, “the restoring of broken parts”—and in 821 he was appointed astronomer and head librarian of the House of Wisdom…
Centuries ago, a prestigious Islamic library (tragically burned in the the Siege of Baghdad) brought Arabic numerals to the world; its mathematical revolution changed our world: “How modern mathematics emerged from a lost Islamic library.”
For more on The House of Wisdom– and the sad stories of other libraries and archives that have been destroyed through the ages– see Richard Ovenden‘s remarkable new Burning the Books- a History of the Deliberate Destruction of Knowledge.
* Leonardo da Pisa, known posthumously as Fibonacci [see here]
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As we count our blessings, we might spare a thought for John Pell; he died on this date in 1685. An English mathematician, 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.)
“Oh, there you are Peter”*…
The missing links between galaxies have finally been found. This is the first detection of the roughly half of the normal matter in our universe – protons, neutrons and electrons – unaccounted for by previous observations of stars, galaxies and other bright objects in space.
You have probably heard about the hunt for dark matter, a mysterious substance thought to permeate the universe, the effects of which we can see through its gravitational pull. But our models of the universe also say there should be about twice as much ordinary matter out there, compared with what we have observed so far.
Two separate teams found the missing matter – made of particles called baryons rather than dark matter – linking galaxies together through filaments of hot, diffuse gas…
Get galactic at: “Half the universe’s missing matter has just been finally found.”
* meme
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As we heed E.M. Forster, 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.
“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]
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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.
Now you don’t…
Camouflage– cryptic coloration– is common in the majority of species that have lived on earth; but military camouflage is a relatively recent development. Through the early 19th Century, almost all soldiers tended to dress in bright colors chosen precisely to make them more identifiable on the battlefield; it was during World War I that camouflage found common use.
The earliest military camouflage drew on the work of zoologist and artist Abbott Thayer, applying lessons from the animal kingdom to secreting troops and tanks.
But World War I was as importantly a naval war. Norman Wilkinson, a marine painter [and your correspondent’s relatively distant ancestor] who was in the Royal Navy, is credited with being the first to develop camouflage for ships– “dazzle,” a kind of camouflage that is “disruptive” like zebra’s stripes. The Royal Navy allowed him to test his idea; and when the test went well, Wilkinson was told to proceed… but was given no office space. So he went to his alma mater, the Royal Academy, and was given a classroom. Wilkinson hired Vorticist Edward Wadsworth to be a port officer in Liverpool, England and to oversee the painting of dazzle ships. In 1918, Wilkinson came to United States to share his dazzle plans. 1,000 plans were developed through this partnership.
One of Wilkinson’s U.S. collaborators was Maurice L. Freedman, the district camoufleur for the 4th district of the U.S. Shipping Board, Emergency Fleet Corporation (a precursor to today’s Merchant Marine). Maurice’s job was to take the plans, adjust them if necessary, then hire painters (artists, house painters) to paint the ships accordingly.
Freedman, who attended the Rhode Island School of Design after the war, donated the plans and photos in his collection to the Fleet Library at RISD. Now (through the end of March) those plans are on display at the library– and online.
As we dress discretely, we might recall that it was on this date in 1258 that (a decidedly un-camouflaged) Hulagu Khan (a grandson of Genghis) and his Mongol force sacked Baghdad, and brought the Abbasid Caliphate (source of, among other marvels, algebra) to an end.
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