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Posts Tagged ‘Mathematics

“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.

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Written by LW

October 16, 2017 at 1:01 am

“Exploring pi is like exploring the universe”*…

 

pi

 

Pi is an infinite string of seemingly random numbers, but if you break down the first 1000 digits of Pi according to how many times each number from 0 to 9 appears, they’re all just about equal — with 1 being the outlier at 12% (although we wonder if they’d all average to ~10% given enough digits of Pi)…

More at “Visualizing The Breakdown Of The Numbers In The First 1000 Digits Of Pi Is Fascinating.”

* David Chudnovsky

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As we watch it even out in the end, we might spare a thought for Hannah Wilkinson Slater; she died on this date in 1812. The daughter and the wife of mill owners, Ms. Slater was the first woman to be issued a patent in the United States (1793)– for a process using spinning wheels to twist fine Surinam cotton yarn, that created a No. 20 two-ply thread that was an improvement on the linen thread previously in use for sewing cloth.

A waxen Hannah, at the Slaters’ Mill Museum in Pawtucket, RI

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Written by LW

October 2, 2017 at 1:01 am

“Mystery has its own mysteries”*…

 

Finally, an answer to a question that puzzled Cantor and Hilbert (proprietor of The Infinite Hotel) and challenged Cohen and Gödel…

In a breakthrough that disproves decades of conventional wisdom [and confounds common sense], two mathematicians have shown that two different variants of infinity are actually the same size. The advance touches on one of the most famous and intractable problems in mathematics: whether there exist infinities between the infinite size of the natural numbers and the larger infinite size of the real numbers…

Connecting the sizes of infinities and the complexity of mathematical theories:                        “Mathematicians Measure Infinities and Find They’re Equal.”

* “Mystery has its own mysteries, and there are gods above gods. We have ours, they have theirs. That is what’s known as infinity.”  – Jean Cocteau

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As we go big, 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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Written by LW

September 20, 2017 at 1:01 am

“Mathematics is the art of giving the same name to different things”*…

 

A 3,700-year-old clay tablet has proven that the Babylonians developed trigonometry 1,500 years before the Greeks and were using a sophisticated method of mathematics which could change how we calculate today.

The tablet, known as Plimpton 332, was discovered in the early 1900s in Southern Iraq by the American archaeologist and diplomat Edgar Banks, who was the inspiration for Indiana Jones.

The true meaning of the tablet has eluded experts until now but new research by the University of New South Wales, Australia, has shown it is the world’s oldest and most accurate trigonometric table, which was probably used by ancient architects to construct temples, palaces and canals…

More of the remarkable story at “3,700-year-old Babylonian tablet rewrites the history of maths – and shows the Greeks did not develop trigonometry.”

* Henri Poincaré

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As we struggle to remember the difference between a sine and a cosine, we might recall that it was on this date in 1842 that the United States Naval Observatory was authorized by an act of Congress. One of the oldest scientific agencies in the U.S., its primary task was to care for the Navy’s charts, navigational instruments, and chronometers, which were calibrated by timing the transit of stars across the meridian.  It’s now probably best known as the home of the “Master Clock“, which provides precise time to the GPS satellite constellation run by the United States Air Force… and for its non-scientific mission: a house located within the Naval Observatory complex serves as the official residence of the Vice President of the United States.

Initially located at Foggy Bottom in the District of Columbia (near the current location of the State Department), the observatory moved in 1893 to its present near Embassy Row.

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Written by LW

August 31, 2017 at 1:01 am

“Chaos is merely order waiting to be deciphered”*…

 

Let us say we were interested in describing all phenomena in our universe. What type of mathematics would we need? How many axioms would be needed for mathematical structure to describe all the phenomena? Of course, it is hard to predict, but it is even harder not to speculate. One possible conclusion would be that if we look at the universe in totality and not bracket any subset of phenomena, the mathematics we would need would have no axioms at all. That is, the universe in totality is devoid of structure and needs no axioms to describe it. Total lawlessness! The mathematics are just plain sets without structure. This would finally eliminate all metaphysics when dealing with the laws of nature and mathematical structure. It is only the way we look at the universe that gives us the illusion of structure…

Science predicts only the predictable, ignoring most of our universe.  What if neither Platonism nor the multiverse are the accurate approaches to understanding the reality we inhabit?  “Chaos Makes the Multiverse Unnecessary.”

[image above: source]

* José SaramagoThe Double

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As we impose order, we might spare a thought for Philipp Frank; he died on this date in 1966. A physicist, mathematician, and philosopher of science, he was Einstein’s successor as professor of theoretical physics at the German University of Prague– a job he got on Einstein’s recommendation– until 1938, when he fled the rise of Nazism and relocated to Harvard.  Frank’s theoretical work covered variational calculus, Hamiltonian geometrical optics, Schrödinger wave mechanics, and relativity; his philosophical work strove to reconcile science and philosophy and “bring about the closest rapprochement between” them.

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Written by LW

July 22, 2017 at 1:01 am

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty”*…

 

Maryam Mirzakhani did not enjoy mathematics to begin with. She dreamed of being an author or politician, but as a top student at her all-girls school in Tehran she was still disappointed when her first-year maths exam went poorly. Her teacher believed her – wrongly – to have no particular affinity with the subject.

Soon that would all change. “My first memory of mathematics is probably the time [my brother] told me about the problem of adding numbers from 1 to 100,” she recalled later. This was the story of Carl Gauss, the 18th-century genius whose schoolteacher set him this problem as a timewasting exercise – only for his precocious pupil to calculate the answer in a matter of seconds.

The obvious solution is simple but slow: 1+2+3+4. Gauss’s solution is quicker to execute, and far more cunning. It goes like this: divide the numbers into two groups: from 1 to 50, and from 51 to 100. Then, add them together in pairs, starting with the lowest (1) and the highest (100), and working inwards (2+99, 3+98, and so on). There are 50 pairs; the sum of each pair is 101; the answer is 5050. “That was the first time I enjoyed a beautiful solution,” Mirzakhani told the Clay Mathematics Institute in 2008.

Since then, her appreciation for beautiful solutions has taken her a long way from Farzanegan middle school. At 17 she won her first gold medal at the International Mathematics Olympiad. At 27 she earned a doctorate from Harvard University. The Blumenthal Award and Satter Prize followed, and in 2014 she became the first woman to be awarded the Fields Medal, the highest honour a mathematician can obtain.

Before this particular brand of wonder became perceptible to Mirzakhani, she experienced feelings many of us can relate to: to the indifferent, her subject can seem “cold”, even “pointless”. Yet those who persist will be rewarded with glimpses of conceptual glory, as if gifted upon them by a capricious god: “The beauty of mathematics,” she warned, “only shows itself to more patient followers.”

This concept of “beauty” found in maths has been referred to over centuries by many others; though, like beauty itself, it is notoriously difficult to define…

For an experienced mathematician, the greatest equations are beautiful as well as useful. Can the rest of us see what they see?  “What makes maths beautiful?

[From The New Humanist, via the ever-illuminating 3 Quarks Daily]

Maryam Mirzakhani died last Friday, a victim of breast cancer; she was 40.  As Peter Sarnak (a mathematician at Princeton University and the Institute for Advanced Study) said, her passing is “a big loss and shock to the mathematical community worldwide.”  See also here.

* Bertrand Russell, A History of Western Philosophy

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As we accede to awe, we might spare a thought for Andrey (Andrei) Andreyevich Markov; he died on this date in 1922.  A Russian mathematician, he helped to develop the theory of stochastic processes, especially those now called Markov chains: sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors.  (For example, the probability of winning at the game of Monopoly can be determined using Markov chains.)  His work on the study of the probability of mutually-dependent events has been developed and widely applied to the biological and social sciences.

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Written by LW

July 20, 2017 at 1:01 am

“Do not imagine that mathematics is hard and crabbed, and repulsive to common sense. It is merely the etherealization of common sense”*…

 

Indeed, mathematics can be pretty amazing.  Consider, for example, that a pizza (which is essentially a very short cylinder) that has radius “z” and height “a” has volume Pi × z × z × a.

More marvelous math here.

* William Thomson, 1st Baron Kelvin

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As we do the sums, we might send relaxing birthday greetings to  Edwin J. Shoemaker; he was born on this date in 1907.  In 1928, he and his cousin Edward M. Knabusch prototyped a porch chair out of some wooden slats taken from orange crates; it would automatically recline as a sitter leaned back.  Since it was a seasonal item, his sales improved when he added plush upholstery for year-round indoor use.  Still, his chairs were for the most part locally/regionally sold.  So he designed a manufacturing facility which utilized the mass-production methods of Detroit’s automotive industry– and in November of 1941 went national with the La-Z-Boy recliner.

Edwin (left) and Edward with their original creation

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Written by LW

June 2, 2017 at 1:01 am

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