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Posts Tagged ‘Euler

“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

 

Written by LW

August 6, 2018 at 1:01 am

“I’m gonna put a curse on you and all your kids will be born completely naked”*…

 

More at “Rejected Titles for Kids Say the Darnedest Things.” (Younger readers click here for explanatory background.)

* Jimi Hendrix

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As we remark that an acorn never falls far from the tree, we might spare a thought for Christian Goldbach; he died on this date in 1764.  A mathematician, lawyer, and historian who studied infinite sums, the theory of curves and the theory of equations, he is best remembered for his correspondence with Leibniz, Euler, and Bernoulli, especially his 1742 letter to Euler containing what is now known as “Goldbach’s conjecture.”

In that letter he outlined his famous proposition:

Every even natural number greater than 2 is equal to the sum of two prime numbers.

It has been checked by computer for vast numbers– up to at least 4 x 1014– but remains unproved.

(Goldbach made another conjecture that every odd number is the sum of three primes; it has been checked by computer for vast numbers, but remains unproved.)

Goldbach’s letter to Euler (source, and larger view)

Written by LW

November 20, 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.

 source

Written by LW

December 4, 2014 at 1:01 am

“Nothing is more memorable than truth beautifully told”*…

 

If physicists and mathematicians can’t be rock stars, they can at least have rock star logos.  Dr. Prateek Lala, a physician and amateur calligrapher from Toronto has obliged with 50 nifty “scientific typographics” of important cosmologists and scientists through the ages.

 

Inspired by the “type biographies” of Indian graphic designer Kapil Bhagat, Lala designed his logos to make the lives and discoveries of various scientists more engaging and more immediately relatable to students.

Dr. Lala’s work was for a poster that was published in the latest issue of Inside The Perimeter, the official magazine of Canada’s Perimeter Institute for Theoretical Physics.  One can subscribe to the magazine by email for free here.

Meantime, one can read the backstory, and see many more of Dr. L’s lyrical logos at CoDesign.

* Rick Julian

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As we ponder personal branding, we might send dynamic birthday greetings to Daniel Bernoulli; he was born on this date in 1700.  One of the several prominent mathematicians and physicists in the Swiss Bernoulli family, Daniel is best remembered for or his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics.  His name is commemorated in the Bernoulli principle, a particular example of the conservation of energy, which describes the mathematics of the mechanism underlying the operation of two important technologies of the 20th century: the carburetor and the airplane wing.

A contemporary and close friend of Leonhard Euler (see above), Bernoulli was the son of Johann Bernoulli (one of the early developers of calculus), nephew of Jakob Bernoulli (who was the first to discover the theory of probability), and the brother of Johann II (an expert on magnetism and the propagation of light).  Daniel is said to have had a bad relationship with his father: when they tied for first place in a scientific contest at the University of Paris, Johann, unable to bear the “shame” of being compared as Daniel’s equal, banned Daniel from his house.  Johann Bernoulli then plagiarized some key ideas from Daniel’s book Hydrodynamica in his own book Hydraulica, which he backdated to before Hydrodynamica.  Despite Daniel’s attempts at reconciliation, his father carried the grudge until his death.

 source

 

 

Written by LW

February 8, 2014 at 1:01 am

No reservation? No problem!…

 

Jeff Dekofsky explains Hilbert’s paradox of the Grand Hotel, a thought experiment proposed in the 1920s by German mathematician David Hilbert to illustrate some surprising properties of infinite sets, in this TED-Ed animated lecture

email readers click here for video

As a special bonus, another amusing video (via Kottke)– an explanation of why it is that the sum of all positive integers (1 + 2 + 3 + 4 + 5 + …) = -1/12…  Euler actually proved this result in 1735, but the result was only made rigorous later; and now physicists have been seeing this result actually show up in nature.  (Spoiler alert: the answer turns on what one means by “sum” mathematically…)

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As we pray for more fingers and toes, we might spare a thought for Harald August Bohr; he died on this date in 1951.  While materially less well-known than his brother Niels, Harald was a formidable mathematician (founder of the field of almost periodic functions), a gifted athlete (an accomplished footballer who won a silver medal at the 1908 Summer Olympics as a member of Denmark’s team), an inspirational teacher (the annual award for outstanding teaching at the University of Copenhagen is called “the Harald” in his honor), and an out-spoken critic of the anti-Semitic policies that took root in the German mathematical establishment in the 1930s.

 source

Spiraling into control…

 The Fibonacci spiral (source)

As we remark that math really is beautiful, we might send elegantly parsimonious birthday greetings to one of Fibonacci’s spiritual descendants, a father of Pure Mathematics, Leonhard Euler; he was born on this date in 1707.  While crafting “the most remarkable formula in mathematics,” Euler made foundational contributions to number theory, graph theory, mathematical logic, and applied math; he originated many commonly-used figures of mathematical notation, and invented the concept of the “mathematical function.”  And he was no slouch in physics either, making renowned contributions in work in mechanics, fluid dynamics, optics, and astronomy.

 source

Written by LW

April 15, 2012 at 1:01 am

It’s all about the ink…

Euler’s Identity (source)

Called “the most remarkable formula in mathematics” by Richard Feynman, Leonhard Euler‘s Identity, as the equation in the tattoo is known, was named in a reader poll conducted by Mathematical Intelligencer as the most beautiful theorem in mathematics. Another reader poll conducted by Physics World named it the “greatest equation ever.” *

One can find other mathematical and scientific tattoos here…  and if one wishes to design one’s own, well…

… just click here.

As we steel ourselves for the needle, we might recall that on this date in 1947, George C. Marshall, a former general serving as Secretary of State, gave the speech at Harvard that laid the foundation for what became known as The Marshall Plan– the program under which the U.S. provided around $12 Billion (a fraction of the sum that the Federal government is “investing” in G.M, but real money in those days… ) to help finance the economic recovery of Europe in the wake of World War II.

George C. Marshall

Oh, and lest we forget, June is Accordion Appreciation Month.

* Why is Euler’s Identity considered beautiful?  Three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:

-The number 0.
-The number 1.
-The number π, which is ubiquitous in trigonometry, geometry of Euclidean space, and mathematical analysis (π ≈ 3.14159).
-The number e, the base of natural logarithms, which also occurs widely in mathematical analysis (e ≈ 2.71828).
-The number i, imaginary unit of the complex numbers, which contain the roots of all nonconstant polynomials and lead to deeper insight into many operators, such as integration.

And the equation is “balanced,” with zero on one side.

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