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Posts Tagged ‘Mathematics

“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…

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* Steven Wright

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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.

Banjo Clock

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Simon Willard

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Written by LW

August 30, 2014 at 1:01 am

“It isn’t that they cannot see the solution. It is that they cannot see the problem.”*…

 

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From Zogg from Betelgeuse , “Mathematics: Measuring x Laziness²,” the latest entry in the Earthlings 101 series– a beginner’s guide for alien visitors.

* G.K. Chesterton

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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.

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Written by LW

July 22, 2014 at 1:01 am

“Perfect numbers like perfect men are very rare”*…

 

From Spiked Math, Dirty Math: a N-altogether-SFW collection of mathematical concepts…

* Rene Descartes

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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.

Hargrave (seated in the apparatus, left) and his assistant James Swain demonstrate the “manlift kite”

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Written by LW

July 6, 2014 at 1:01 am

“Geometry is not true, it is advantageous”*…

 

4dcube

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

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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.

kleinBottle_anim

A Klein bottle

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225px-Felix_Klein source

 

Written by LW

June 22, 2014 at 1:01 am

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

 

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

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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).

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Written by LW

June 18, 2014 at 1:01 am

“Arithmetic! Algebra! Geometry! Grandiose trinity! Luminous triangle! Whoever has not known you is without sense!”*…

 

In 1915, Polish mathematician Wacław Sierpiński described what’s now known as “the Sierpinski triangle” in 1915.  He was explicating the properties of a pattern that had appeared in the 13th-century Cosmati mosaics in the cathedral of Anagni,Italy, and other places of central Italy, in carpets in many places (e.g., in the nave of the Roman Basilica of Santa Maria in Cosmedin), and in isolated triangles positioned in the rotae of several churches and Basiliche.

Sierpiński noted that, like our old friend Gabriel’s Horn (AKA Torricelli’s Trumpet), the pattern generates a paradoxical outcome:

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See how this fabulous fractal maps onto the rather better-known Pascal’s Triangle.

Comte de Lautréamont

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As we ponder patterns, we might send efficient birthday greetings to Lillian Evelyn Moller Gilbreth; she was born on this date in 1878.   One of the first working female engineers holding a Ph.D., she was arguably the first true industrial/organizational psychologist.  With her husband Frank Gilbreth. she was one of the first “efficiency experts” helping establish the fields of motion study and human factors.  She is perhaps best remembered as the subject of Cheaper by the Dozen and Belles on Their Toes (charming books written by their children Ernestine and Frank Jr.) recounting the couple’s family life with their twelve children, and their application of time and motion study to the organization and daily routines of such a large family.

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“How wonderful that we have met with a paradox. Now we have some hope of making progress”*…

 

Consider the simple function Y=1/X:

Take one half and rotate it around X.

It creates the shape you see at the top of this post, known as “Torricelli’s Trumpet” for its discoverer, the 17th century mathematician Evangelista Torricelli.  It’s noteworthy for its peculiar topographical qualities:  while both the volume and the surface area can be calculated, and the volume is a finite number, the surface area is Infinite.  That’s to say that, while one can fill that three dimensional shape with a calculable quantity of paint, one cannot coat the exterior surface, as it would require an infinite amount of paint…  (Supporting math, here.)

(The figure is also known as “Gabriel’s Horn,” a reference to the Archangel Gabriel, who blows his horn to announce Judgment Day– an association of the divine, or infinite, with the finite.)

This contribution (from Pablo Ramos) is just one of the fascinating answers to the question of Quora: “What are the weirdest science paradoxes that are mathematically true but counter-intuitive?

* Niels Bohr

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As we rotate our minds around x, we might send post-industrial birthday greetings to Daniel Bell; he was born on this date in 1919.  Bell spent the first twenty years of his adult life as a journalist, exploring sociological issues; in 1960, on the strength of a book he’d written– The End of Ideology: On the Exhaustion of Political Ideas in the Fifties– he was awarded a PhD by Columbia University, where he taught briefly before moving for the rest of his career to Harvard.  One of the leading intellectuals of the Post-War era, Bell is best known for his contributions to the study of “post-industrialism,” and for his acute unpacking of the interactions among science, technology and politics.

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