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Posts Tagged ‘Mathematics

“What is mathematics? It is only a systematic effort of solving puzzles posed by nature.”*…




In 1998, the Cloisters—the museum of medieval art in upper Manhattan—began a renovation of the room where the seven tapestries known as “The Hunt of the Unicorn” hang. The Unicorn tapestries are considered by many to be the most beautiful tapestries in existence. They are also among the great works of art of any kind. In the tapestries, richly dressed noblemen, accompanied by hunters and hounds, pursue a unicorn through forested landscapes. They find the animal, appear to kill it, and bring it back to a castle; in the last and most famous panel, “The Unicorn in Captivity,” the unicorn is shown bloody but alive, chained to a tree surrounded by a circular fence, in a field of flowers. The tapestries are twelve feet tall and up to fourteen feet wide (except for one, which is in fragments). They were woven from threads of dyed wool and silk, some of them gilded or wrapped in silver, around 1500, probably in Brussels or Liège, for an unknown person or persons, and for an unknown reason—possibly to honor a wedding. A monogram made from the letters “A” and “E” is woven into the scenery in many places; no one knows what it stands for. The tapestries’ meaning is mysterious: the unicorn was a symbol of many things in the Middle Ages, including Christianity, immortality, wisdom, lovers, marriage. For centuries, the tapestries were in the possession of the La Rochefoucauld family of France. In 1922, John D. Rockefeller, Jr., bought them for just over a million dollars, and in 1937 he gave them to the Cloisters. Their monetary value today is incalculable…

As the construction work got under way, the tapestries were rolled up and moved, in an unmarked vehicle and under conditions of high security, to the Metropolitan Museum of Art, which owns the Cloisters. They ended up in a windowless room in the museum’s textile department for cleaning and repair. The room has white walls and a white tiled floor with a drain running along one side. It is exceedingly clean, and looks like an operating room. It is known as the wet lab, and is situated on a basement level below the museum’s central staircase.

In the wet lab, a team of textile conservators led by a woman named Kathrin Colburn unpacked the tapestries and spread them out face down on a large table, one by one. At some point, the backs of the tapestries had been covered with linen. The backings, which protect the tapestries and help to support them when they hang on a wall, were turning brown and brittle, and had to be replaced. Using tweezers and magnifying lenses, Colburn and her team delicately removed the threads that held each backing in place. As the conservators lifted the backing away, inch by inch, they felt a growing sense of awe. The backs were almost perfect mirror images of the fronts, but the colors were different. Compared with the fronts, they were unfaded: incredibly bright, rich, and deep, more subtle and natural-looking. The backs of the tapestries had, after all, been exposed to very little sunlight in five hundred years. Nobody alive at the Met, it seems, had seen them this way…

Philippe de Montebello, the director of the museum, declared that the Unicorn tapestries must be photographed on both sides, to preserve a record of the colors and the mirror images. Colburn and her associates would soon put new backing material on them, made of cotton sateen. Once they were rehung at the Cloisters, it might be a century or more before the true colors of the tapestries would be seen again.

The manager of the photography studio at the Met is a pleasant, lively woman named Barbara Bridgers. Her goal is to make a high-resolution digital image of every work of art in the Met’s collections. The job will take at least twenty-five years; there are between two and two and a half million catalogued objects in the Met—nobody knows the exact number. (One difficulty is that there seems to be an endless quantity of scarab beetles from Egypt.) But, when it’s done and backup files are stored in an image repository somewhere else, then if an asteroid hits New York the Metropolitan Museum may survive in a digital copy.

To make a digital image of the Unicorn tapestries was one of the most difficult assignments that Bridgers had ever had. She put together a team to do it, bringing in two consultants, Scott Geffert and Howard Goldstein, and two of the Met’s photographers, Joseph Coscia, Jr., and Oi-Cheong Lee. They built a giant metal scaffolding inside the wet lab, and mounted on it a Leica digital camera, which looked down at the floor. The photographers were forbidden to touch the tapestries; Kathrin Colburn and her team laid each one down, underneath the scaffold, on a plastic sheet. Then the photographers began shooting. The camera had a narrow view; it could photograph only one three-by-three-foot section of tapestry at a time. The photographers took overlapping pictures, moving the camera on skateboard wheels on the scaffolding. Each photograph was a tile that would be used to make a complete, seamless mosaic of each tapestry…

It took two weeks to photograph the tapestries. When the job was done, every thread in every tile was crystal-clear, and the individual twisted strands that made up individual threads were often visible, too. The data for the digital images, which consisted entirely of numbers, filled more than two hundred CDs. With other, smaller works of art, Bridgers and her team had been able to load digital tiles into a computer’s hard drives and memory, and then manipulate them into a complete mosaic—into a seamless image—using Adobe Photoshop software. But with the tapestries that simply wouldn’t work. When they tried to assemble the tiles, they found that the files were too large and too complex to manage. “We had to lower the resolution of the images in order to fit them into the computers we had, and it degraded the images so much that we just didn’t think it was worth doing,” Bridgers said. Finally, they gave up. Bridgers stored the CDs on a shelf and filed the project away as an unsolved problem…

Enter Gregory and David Chudnovsky, brothers whose work was so intertwined that they considered themselves a single mathematician.  Over four months– and after 30 hours of continuous running– their self-designed supercomputer successfully performed the 7.7 quadrillion calculations needed to produce the image for the Met.

Richard Preston tells the genuinely-fascinating story: “Capturing the Unicorn.”

It was a return of sorts for Preston, who, thirteen years earlier, had profiled the brothers and their successful quest to resolve pi to a record number of decimal places: “The Mountains of Pi.”

More on the Chudnovskys here.

* Shakuntala Devi


As we muse of the merger of art and science, we might recall that it was on this date in 1886 that Coca Cola was concocted in an Atlanta, Georgia backyard as a “brain tonic” that could cure hangovers, stomach aches and headaches.  The original formula included caffeine and five ounces of coca leaf (from which cocaine is derived) per gallon.  The creator, pharmacist John Pemberton, took his syrup a few doors down to Jacobs’ Pharmacy, where he mixed it with carbonated water and shared it with customers.  The pharmacy began marketing it on May 8 as a patent medicine for 5¢ a glass.  It spread first through the other Jacobs outlets in Atlanta, and then around the world.

“The valuable tonic and nerve stimulant properties of the coca plant and cola nuts …”

– John Pemberton





“All you really need to know for the moment is that the universe is a lot more complicated than you might think, even if you start from a position of thinking it’s pretty damn complicated in the first place”*…




When you gaze out at the night sky, space seems to extend forever in all directions. That’s our mental model for the universe, but it’s not necessarily correct. There was a time, after all, when everyone thought the Earth was flat, because our planet’s curvature was too subtle to detect and a spherical Earth was unfathomable.

Today, we know the Earth is shaped like a sphere. But most of us give little thought to the shape of the universe. Just as the sphere offered an alternative to a flat Earth, other three-dimensional shapes offer alternatives to “ordinary” infinite space.

We can ask two separate but interrelated questions about the shape of the universe. One is about its geometry: the fine-grained local measurements of things like angles and areas. The other is about its topology: how these local pieces are stitched together into an overarching shape.

Cosmological evidence suggests that the part of the universe we can see is smooth and homogeneous, at least approximately. The local fabric of space looks much the same at every point and in every direction. Only three geometries fit this description: flat, spherical and hyperbolic…

Alternatives to “ordinary” infinite space: “What Is the Geometry of the Universe?

* Douglas Adams, The Hitchhiker’s Guide to the Galaxy


As we tinker with topology, we might recall that it was on this date in 1811 that Percy Bysshe Shelley was expelled from the University of Oxford for publishing the pamphlet The Necessity of Atheism.  Shelley, of course, went on to become a celebrated lyric poet and one of the leaders of the English Romantic movement… one who had a confident (if not to say exalted) sense of his role in society:

Poets are the hierophants of an unapprehended inspiration; the mirrors of the gigantic shadows which futurity casts upon the present; the words which express what they understand not; the trumpets which sing to battle, and feel not what they inspire; the influence which is moved not, but moves. Poets are the unacknowledged legislators of the world.

220px-Percy_Bysshe_Shelley_by_Alfred_Clint source


“It’s exact and indefinite. It’s like pi– you can keep figuring it out and always be right and never be done”*…




It’s Pi Day!  What better way to “prove” 3.14 than with that most perfect of pies– pizza!

Via the ever-illiminating Boing Boing.

See also: “Pi Day: How One Irrational Number Made Us Modern.”

* Peter Schjeldahl, quoting the painter John Currin


As we celebrate the irrational, we might recall that it was on this date in 1958 that “Tequila” hit the top of the pop charts (sales and radio plays, both pop and R&B).



Written by LW

March 14, 2020 at 1:01 am

“Nothing happens until something moves”*…




What determines our fate? To the Stoic Greek philosophers, fate is the external product of divine will, ‘the thread of your destiny’. To transcendentalists such as Henry David Thoreau, it is an inward matter of self-determination, of ‘what a man thinks of himself’. To modern cosmologists, fate is something else entirely: a sweeping, impersonal physical process that can be boiled down into a single, momentous number known as the Hubble Constant.

The Hubble Constant can be defined simply as the rate at which the Universe is expanding, a measure of how quickly the space between galaxies is stretching apart. The slightest interpretation exposes a web of complexity encased within that seeming simplicity, however. Extrapolating the expansion process backward implies that all the galaxies we can observe originated together at some point in the past – emerging from a Big Bang – and that the Universe has a finite age. Extrapolating forward presents two starkly opposed futures, either an endless era of expansion and dissipation or an eventual turnabout that will wipe out the current order and begin the process anew.

That’s a lot of emotional and intellectual weight resting on one small number…

How scientists pinned a single number on all of existence: “Fate of the Universe.”

[Readers might remember that the Big Bang wasn’t always an accepted paradigm— and that on-going research continues to surface challenges.]

* Albert Einstein


As we center ourselves, we might spare a thought for Kurt Friedrich Gödel; he died on this date in 1978.  A  logician, mathematician, and philosopher, he is considered (along with Aristotle, Alfred Tarski— whose birthday this also is– and Gottlob Frege) to be one of the most important logicians in history.  Gödel had an immense impact upon scientific and philosophical thinking in the 20th century.  He is, perhaps, best remembered for his Incompleteness Theorems, which led to (among other important results) Alan Turing’s insights into computational theory.

Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Gödel’s achievement.                  — John von Neumann

kurt_gödel source


“Men of broader intellect know that there is no sharp distinction betwixt the real and the unreal”*…



Colored cubes — known as “Tesseracts” — as depicted in the frontispiece to Hinton’s The Fourth Dimension (1904)


During the period we now call the fin de siècle, worlds collided. Ideas were being killed off as much as being born. And in a sort of Hegelian logic of thesis/antithesis/synthesis, the most interesting ones arose as the offspring of wildly different parents. In particular, the last gasp of Victorian spirituality infused cutting-edge science with a certain sense of old-school mysticism. Theosophy was all the rage; Huysmans dragged Satan into modern Paris; and eccentric poets and scholars met in the British Museum Reading Room under the aegis of the Golden Dawn for a cup of tea and a spot of demonology. As a result of all this, certain commonly-accepted scientific terms we use today came out of quite weird and wonderful ideas being developed at the turn of the century. Such is the case with space, which fascinated mathematicians, philosophers, and artists with its unfathomable possibilities…

In April 1904, C. H. Hinton published The Fourth Dimension, a popular maths book based on concepts he had been developing since 1880 that sought to establish an additional spatial dimension to the three we know and love. This was not understood to be time as we’re so used to thinking of the fourth dimension nowadays; that idea came a bit later. Hinton was talking about an actual spatial dimension, a new geometry, physically existing, and even possible to see and experience; something that linked us all together and would result in a “New Era of Thought.”…

Hinton’s ideas gradually pervaded the cultural milieu over the next thirty years or so — prominently filtering down to the Cubists and Duchamp. The arts were affected by two distinct interpretations of higher dimensionality: on the one hand, the idea as a spatial, geometric concept is readily apparent in early Cubism’s attempts to visualise all sides of an object at once, while on the other hand, it becomes a kind of all-encompassing mystical codeword used to justify avant-garde experimentation. “This painting doesn’t make sense? Ah, well, it does in the fourth dimension…” It becomes part of a language for artists exploring new ideas and new spaces…

By the late 1920s, Einsteinian Space-Time had more or less replaced the spatial fourth dimension in the minds of the public. It was a cold yet elegant concept that ruthlessly killed off the more romantic idea of strange dimensions and impossible directions. What had once been the playground of spiritualists and artists was all too convincingly explained. As hard science continued to rise in the early decades of the twentieth century, the fin-de-siècle’s more outré ideas continued to decline. Only the Surrealists continued to make reference to it, as an act of rebellion and vindication of the absurd. The idea of a real higher dimension linking us together as One sounded all a bit too dreamy, a bit too old-fashioned for a new century that was picking up speed, especially when such vague and multifarious explanations were trumped by the special theory of relativity. Hinton was as much hyperspace philosopher as scientist and hoped humanity would create a more peaceful and selfless society if only we recognised the unifying implications of the fourth dimension. Instead, the idea was banished to the realms of New Age con-artists, reappearing these days updated and repackaged as the fifth dimension. Its shadow side, however, proved hopelessly alluring to fantasy writers who have seen beyond the veil, and bring back visions of horror from an eldritch land outside of time and space that will haunt our nightmares with its terrible geometry, where tentacles and abominations truly horrible sleep beneath the Pacific Ocean waiting to bring darkness to our world… But still we muddle on through.

Hyperspace, tesseracts, ghosts, and colorful cubesJon Crabb, Editor, British Library Publishing, on the work of Charles Howard Hinton and the cultural history of higher dimensions: “Notes on the Fourth Dimension.”

[TotH to MK]

* H.P. Lovecraft, The Tomb


As we get high(er), we might recall that it was on this date in 1946 that Al Gross went public with his invention of the walkie talkie.  Gross had developed it as a top secret project during World War II; he went on to develop the circuitry that opened the way to personal pocket paging systems, CB radio, and patented precursors of the cell phone and the cordless phone.  Sadly for him, his patents expired before they became commercially viable.  ”Otherwise,” Gross said, after winning the M.I.T. lifetime achievement award, ”I’d be as rich as Bill Gates.”

While Gross himself is almost unknown to the general public, he did achieve one-step-removed notoriety in 1948 when he “gifted” his friend Chester Gould the concept of miniaturized radio transceivers, which Gross had just patented.  Gould put it to use as the two-way wrist radio in his comic strip Dick Tracy.

200px-ALGROS2 source


Written by LW

November 20, 2019 at 1:01 am

“It is still an unending source of surprise for me how a few scribbles on a blackboard or on a piece of paper can change the course of human affairs”*…




For the last year, Jessica Wynne, a photographer and professor at the Fashion Institute of Technology in New York, has been photographing mathematicians’ blackboards, finding art in the swirling gangs of symbols sketched in the heat of imagination, argument and speculation. “Do Not Erase,” a collection of these images, will be published by Princeton University Press in the fall of 2020…

This is what thought looks like.

Ideas, and ideas about ideas. Suppositions and suspicions about relationships among abstract notions — shape, number, geometry, space — emerging through a fog of chalk dust, preferably of the silky Hagoromo chalk, originally from Japan, now made in South Korea.

In these diagrams, mysteries are being born and solved…


More (and larger) examples from this photo survey of the blackboards of mathematicians at “Where Theory Meets Chalk, Dust Flies.”

* Stanislaw Ulam


As we scribble “do not erase,” we might spare a thought for Herbert Aaron Hauptman; he died on this date in 2011.  A mathematician, he pioneered and developed a mathematical method that has changed the whole field of chemistry and opened a new era in research in determination of molecular structures of crystallized materials.  Today, Hauptman’s “direct methods,” which he continued to improve and refine, are routinely used to solve complicated structures… work for which he shared the the 1985 Nobel Prize in Chemistry.

R source


Written by LW

October 24, 2019 at 1:01 am

“With the sextant he made obeisance to the sun-god”*…



A practice exam in the navigation workbook of C. J. Boombaar (1727–32)


In 1673, in a North Sea skirmish that killed nearly 150 men, the French privateer Jean-François Doublet took a bullet that tossed him from the forecastle and broke his arm in two places. How did the precocious young second lieutenant choose to spend his convalescence? Doublet repaired to the French port city of Dieppe, where he signed up for three months of navigation lessons…

During the 16th to 18th centuries, Europeans embarked on thousands of long-distance sea voyages around the world. These expeditions in the name of trade and colonisation had irreversible, often deadly, impacts on peoples around the globe. Heedless of those consequences, Europeans focused primarily on devising new techniques to make their voyages safer and faster. They could no longer sail along the coasts, taking their directional cues from prominent landmarks (as had been common in the preceding centuries). Nor did they have sophisticated knowledge of waves and currents, as did their counterparts in the Pacific. They had no choice but to figure out new methods of navigating across the open water. Instead of memorising the shoreline, they looked to the heavens, calculating time and position from the sun and the stars.

Celestial navigation was certainly feasible, but it required real technical skills as well as fairly advanced mathematics. Sailors needed to calculate the angle of a star’s elevation using a cross-staff or quadrant. They needed to track the direction of their ship’s course relative to magnetic north. Trigonometry and logarithms offered the best way to make these essential measurements: for these, a sailor needed to be adept at using dense numerical tables. All of a sudden, a navigator’s main skill wasn’t his memory – it was his mathematical ability.

To help the average sailor with these technical computations, maritime administrators and entrepreneurs opened schools in capital cities and port towns across Europe. Some were less formal arrangements, where small groups of men gathered in the teacher’s home, paying for a series of classes over the course of a winter when they were on shore…

How did the sailors of early modern Europe learn to traverse the world’s seas? By going to school and doing maths problems: “When pirates studied Euclid.”

* “With the sextant he made obeisance to the sun-god, he consulted ancient tomes and tables of magic characters, muttered prayers in a strange tongue that sounded like Indexerrorparallaxrefraction, made cabalistic signs on paper, added and carried one, and then, on a piece of holy script called the Grail – I mean, the Chart – he placed his finger on a certain space conspicuous for its blankness and said, ‘Here we are.’ When we looked at the blank space and asked, “And where is that?” he answered in the cipher-code of the higher priesthood, “31 -15 – 47 north, 133 – 5 – 30 west.” And we said, ‘Oh,’ and felt mighty small.”                           – Jack London, The Cruise of the Snark


As we find our way, we might send carefully-calculated birthday greetings to John Locke; he died on this date in 1856.  A namesake of the famous philosopher, Locke trained as a doctor, but turned to geology– and to the invention of scientific, surveying, and navigational instruments, including a surveyor’s compass, a collimating level (Locke’s Hand Level), and a gravity escapement for regulator clocks.  The electro-chronograph he constructed (1844-48) for the United States Coast Survey was installed in the Naval Observatory, Washington, in 1848.  It improved determination of longitudes, as it was able to make a printed record on a time scale of an event to within one one-hundredth of a second.  When connected via the nation’s telegraph system, astronomers could record the time of events they observed from elsewhere in the country, by pressing a telegraph key.

Locke,_John source


Written by LW

July 10, 2019 at 1:01 am

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