Posts Tagged ‘Mathematics’
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 @]
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.
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)
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.
“I went to a restaurant that serves ‘breakfast at any time,’ so I ordered French toast during the Renaissance”*…
“When you wake up in the morning, Pooh,” said Piglet at last, “what’s the first thing you say to yourself?”
“What’s for breakfast?” said Pooh. “What do you say, Piglet?”
“I say, I wonder what’s going to happen exciting today?” said Piglet.
Pooh nodded thoughtfully. “It’s the same thing,” he said.
― A.A. Milne
How to prepare an essential– and exciting– part of any mathematically-correct breakfast…
* Steven Wright
As we tangle tastefully with topography, we might spare a thought for Simon Willard; he died on this date in 1848. A master clockmaker who created grandfather clocks and lobby/gallery clocks, Willard is best remembered for his creation of the timepiece that came to be known as the banjo clock, a wall clock that Willard patented in 1802. Only 4,000 authentic “Simon Willard banjo clocks” were made; and while he had many imitators turning out replicas, these originals are highly-prized collectibles.
* G.K. Chesterton
As we dazzle ‘em with differentials, we might spare a thought for Sir Sandford Fleming; he died on this date in 1915. A Scottish engineer who emigrated to Canada, Fleming designed much of the Intercolonial Railway and the Canadian Pacific Railway; was a founding member of the Royal Society of Canada; founded the Royal Canadian Institute; and designed the first Canadian postage stamp (the Threepenny Beaver, issued in 1851), But he is best remembered as the man who divided the world into time zones– the inventor of Worldwide Standard Time.
From Spiked Math, Dirty Math: a N-altogether-SFW collection of mathematical concepts…
* Rene Descartes
As we rethink using our fingers and toes, we might spare a thought for Lawrence Hargrave; he died on this date in 1915. An Australian engineer, explorer, astronomer, inventor, and aeronautical pioneer, he is probably best remembered as the inventor of the box kite. In November of 1894, Hargrave flew in one of his creations: he attached himself to a huge four kite construction tethered to the ground by piano wire. His demonstration of the ability of the box kite to carry heavy payloads and hold steady, high-altitude flight led to many industrial and military uses. For example, box kites were used until the 1930s to carry meteorological equipment for high altitude weather studies, and by the Royal Air Force to provide radio aerials for sea rescue.
The tesseract is a four dimensional cube. It has 16 edge points v=(a,b,c,d), with a,b,c,d either equal to +1 or -1. Two points are connected, if their distance is 2. Given a projection P(x,y,z,w)=(x,y,z) from four dimensional space to three dimensional space, we can visualize the cube as an object in familar space. The effect of a linear transformation like a rotation
| 1 0 0 0 | R(t) = | 0 1 0 0 | | 0 0 cos(t) sin(t) | | 0 0 -sin(t) cos(t) |
in 4d space can be visualized in 3D by viewing the points v(t) = P R(t) v in R3.
* Henri Poincare
As we follow the bouncing ball, we might spare a thought for Felix Klein; he died on this date in 1925. A mathematician of broad gauge, he is best remembered for his work in non-Euclidean geometry, perhaps especially for his work on synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Program, which profoundly influenced mathematics. He created the Klein bottle, a one-sided closed surface–a non-orientable surface with no inside and no outside– that cannot be constructed in Euclidean space.
A few years back, 12 million of us clicked over to watch the “Pachelbel Rant” on YouTube. You might remember it. Strumming repetitive chords on his guitar, comedian Rob Paravonian confessed that when he was a cellist, he couldn’t stand the Pachelbel Canon in D. “It’s eight quarter notes that we repeated over and over again. They are as follows: D-A-B-F♯-G-D-G-A.” Pachelbel made the poor cellos play this sequence 54 times, but that wasn’t the real problem. Before the end of his rant, Paravonian showed how this same basic sequence has been used everywhere from pop (Vitamin C: “Graduation”) to punk (Green Day: “Basket Case”) to rock (The Beatles: “Let It Be”).
This rant emphasized what music geeks already knew—that musical structures are constantly reused, often to produce startlingly different effects. The same is true of mathematical structures in physical theories, which are used and reused to tell wildly dissimilar stories about the physical world. Scientists construct theories for one phenomena, then bend pitches and stretch beats to reveal a music whose progressions are synced, underneath it all, in the heart of the mathematical deep.
Eugene Wigner suggested a half-century ago that this “unreasonable effectiveness” of mathematics in the natural sciences was “something bordering on the mysterious,” but I’d like to suggest that reality may be more mundane. Physicists use whatever math tools they’re able to find to work on whatever problems they’re able to solve. When a new song comes on, there’s bound to be some overlap in the transcription. These overlaps help to bridge mutations of theory as we work our way toward a lead sheet for that universal hum…
Read the harmonious whole at “How Physics is Like Three-Chord Rock.”
* Henri Poincare
As we hum the tune eternal, we might send astronomical birthday greetings to Allan Rex Sandage; he was born on this date in 1926. An astronomer, he spent his career first at the Palomar Observatory, then at the Carnegie Observatory in Pasadena, where at the outset, he was a research assistant to Edwin Hubble, whose work Sandage continued after Hubble’s death. Sandage was hugely influential on his field; he is probably best remembered for determining the first reasonably accurate value for the Hubble constant (there ate those chords again) and the age of the universe– and for discovering the first quasar (again, those chords).