Posts Tagged ‘calculus’
“The bigger, the better”*…
Thea Applebaum Licht with a reminder that, when it comes to size, Texas has got nothing on California…
Between about 1905 and 1915, the United States entered a golden age of postcards. Cheaper and faster mail service, the advent of “divided back” cards (freeing the entire front for images), and improved commercial printing all drove a new mass market for collectible communication. It was at this same moment that a craze for “tall-tale” or “exaggeration” postcards reached its peak. By cutting, collaging, and re-photographing images, artists created out-of-proportion illusions. One of the most popular genres was agricultural goods of fantastic dimensions.
Nowhere were such postcards more popular than in the western states. There, in the heart of the tough business of agriculture, illustrations of folkloric American abundance were understandable favorites. Pride and place were tied up with the prodigious crops. Supersized fruits and vegetables were often accompanied by brief captions: “How We Do Things at Attica, Wis.”, “The Kind We Raise in Our State”, or “The Kind We Grow in Texas”. Photographers like William “Dad” H. Martin and Alfred Stanley Johnson Jr. captured farmers harvesting furniture-sized onions and stacking corn cobs like timber, fisherman reeling in leviathans, and children sharing canoe-like slices of watermelon.
In the series of exaggeration postcards [produced in the run-up to the postcard boom, then published during it] collected [here], it is California that takes center stage. Produced by the prolific San Francisco–based publisher Edward H. Mitchell, each card features a single rail car rolling through lush farmland. Aboard are gargantuan, luminous fruits and vegetables: dimpled navel oranges, a dusky bunch of grapes, and mottled walnuts. Placed end-to-end, the cards would make a colorful train crossing California’s fertile valleys. Unlike other, more action-packed “tall-tale” cards — filled with farmers, fisherman, and children for scale — Mitchell’s series is restrained. Sharply illuminated, the colossal cargo lean toward artwork rather than gag. “A Carload of Mammoth Apples”[here], green-yellow and gleaming, could have been plucked from Rene Magritte’s The Son of Man [here].
Fabulous fruit and vegetables: “Calicornication: Postcards of Giant Produce (1909),” from @publicdomainrev.bsky.social.
In other art-related news: (very) long-term readers might recall that, back in 2008, (R)D reported that London’s Daily Mail believed that it had tracked him down, and that he is Robin Gunningham. Now as Boing Boing reports:
Anyone reading Banksy’s Wikipedia article at any point since a famous Mail on Sunday exposé in 2008 would likely get the impression the secretive stenciler is probably Robin Gunningham or Robert Del Naja, artists who came from the Bristol Underground. Reuters, having conducted extensive research into their movements, finds both men present at critical moments, but only one at all of them: an arrest report from New York City puts Gunningham firmly in the frame, and recent public records from Ukraine put it beyond doubt.
We later unearthed previously undisclosed U.S. court records and police reports. These included a hand-written confession by the artist to a long-ago misdemeanor charge of disorderly conduct – a document that revealed, beyond dispute, Banksy’s true identity. … Reuters presented that man with its findings about his identity and detailed questions about his work and career. He didn’t reply. Banksy’s company, Pest Control, said the artist “has decided to say nothing.”
His long-time lawyer, Mark Stephens, wrote to Reuters that Banksy “does not accept that many of the details contained within your enquiry are correct.” He didn’t elaborate. Without confirming or denying Banksy’s identity, Stephens urged us not to publish this report, saying doing so would violate the artist’s privacy, interfere with his art and put him in danger.
Del Naja (better known for other work) evidently participates in painting the murals and is perhaps the stencil draftsman (Banksy: “he can actually draw”). Banksy’s former manager, Steve Lazarides, organized a legal name change for Gunningham after the Mail on Sunday item, which successfully ended records for Banksy’s movements under his birth name and stymied researchers—until Reuters figured out the new one by poring through Ukrainian public records on days Del Naja was there. Gunningham used the name David Jones, among the most common in the U.K. If it rings a bell, you might be thinking of another famous British artist was who obliged by his record company to find something more unique.
* common idiom
###
As we live large, we might spare a thought for Isaac Newton; he died on this date (O.S.) in 1727. A polymath who was a key figure in the Scientific Revolution and the Enlightenment that followed, Newton was a mathematician, physicist, astronomer, alchemist, theologian, author, and inventor. He contributed to and refined the scientific method, and his work is considered the most influential in bringing forth modern science. His book Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1687, achieved the first great unification in physics and established classical mechanics. He also made seminal contributions to optics, and shares credit with the German mathematician Gottfried Wilhelm Leibniz for formulating infinitesimal calculus. (Newton developed calculus a couple of years before Leibniz, but published a couple of years after.) Newton spent the last three decades of his life in London, serving as Warden (1696–1699) and Master (1699–1727) of the Royal Mint, a role in which he increased the trustworthiness/accuracy and security of British coinage in a way crucial to the rise of Great Britain as a commercial and colonial power.
Newton, of course, had a famous relationship with fruit:
Newton often told the story that he was inspired to formulate his theory of gravitation by watching the fall of an apple from a tree. The story is believed to have passed into popular knowledge after being related by Catherine Barton, Newton’s niece, to Voltaire. Voltaire then wrote in his Essay on Epic Poetry (1727), “Sir Isaac Newton walking in his gardens, had the first thought of his system of gravitation, upon seeing an apple falling from a tree.” – source
Newton’s apple is thought to have been the green skinned ‘Flower of Kent’ variety.
“If geometry is dressed in a suit coat, topology dons jeans and a T-shirt”*…
Paulina Rowińska on how, in the mid-19th century, Bernhard Riemann conceived of a new way to think about mathematical spaces, providing the foundation for modern geometry and physics…
Standing in the middle of a field, we can easily forget that we live on a round planet. We’re so small in comparison to the Earth that from our point of view, it looks flat.
The world is full of such shapes — ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds. Introduced by Bernhard Riemann in the mid-19th century, manifolds transformed how mathematicians think about space. It was no longer just a physical setting for other mathematical objects, but rather an abstract, well-defined object worth studying in its own right.
This new perspective allowed mathematicians to rigorously explore higher-dimensional spaces — leading to the birth of modern topology, a field dedicated to the study of mathematical spaces like manifolds. Manifolds have also come to occupy a central role in fields such as geometry, dynamical systems, data analysis and physics.
Today, they give mathematicians a common vocabulary for solving all sorts of problems. They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi, a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.”
So what are manifolds, and what kind of vocabulary do they provide?…
[Rowińska explains manifolds and the history of the development of our understanding of them, concentrating on the pivotal role of Riemann…]
… Manifolds are crucial to our understanding of the universe… In his general theory of relativity, Einstein described space-time as a four-dimensional manifold, and gravity as that manifold’s curvature. And the three-dimensional space we see around us is also a manifold — one that, as manifolds do, appears Euclidean to those of us living within it, even though we’re still trying to figure out its global shape.
Even in cases where manifolds don’t seem to be present, mathematicians and physicists try to rewrite their problems in the language of manifolds to make use of their helpful properties. “So much of physics comes down to understanding geometry,” said Jonathan Sorce, a theoretical physicist at Princeton University. “And often in surprising ways.”
Consider a double pendulum, which consists of one pendulum hanging from the end of another. Small changes in the double pendulum’s initial conditions lead it to carve out very different trajectories through space, making its behavior hard to predict and understand. But if you represent the configuration of the pendulum with just two angles (one describing the position of each of its arms), then the space of all possible configurations looks like a doughnut, or torus — a manifold. Each point on this torus represents one possible state of the pendulum; paths on the torus represent the trajectories the pendulum might follow through space. This allows researchers to translate their physical questions about the pendulum into geometric ones, making them more intuitive and easier to solve. This is also how they study the movements of fluids, robots, quantum particles and more.
Similarly, mathematicians often view the solutions to complicated algebraic equations as a manifold to better understand their properties. And they analyze high-dimensional datasets — such as those recording the activity of thousands of neurons in the brain — by looking at how those data points might sit on a lower-dimensional manifold.
Asking how scientists use manifolds is akin to asking how they use numbers, Sorce said. “They are at the foundation of everything.”…
“What Is a Manifold?” from @quantamagazine.bsky.social.
Apposite: Rowińska in conversation with Ira Flatow on Science Friday: “How Math Helps Us Map The World.”
* David S. Richeson, Euler’s Gem: The Polyhedron Formula and the Birth of Topology (Riemann’s work was an advance on the foundation that Euler laid in his 1736 paper on the Seven Bridges of Königsberg, which led to his polyhedron formula)
###
As we get down with geometry, we might spare a thought for John Wallis; he died on this date in 1703. A clergyman and mathematician, he served as chief cryptographer for Parliament (decoding Royalist messages during the Civil War) and, later (as Savilian Chair of geometry at Oxford after the hostilities), for the the royal court. Wallis is credited with introducing the symbol ∞ to represent the concept of infinity, and used 1/∞ for an infinitesimal… which earned him (along with his contemporaries Isaac Newton and Gottfried Wilhelm Leibniz) a share of the credit for the development of infinitesimal calculus. He was a founding member of the Royal Society and one of its first Fellows.
“We must not forget that the wheel is reinvented so often because it is a very good idea”*…
… but when was it first discovered? And, and given its obvious and ubiquitous utility, why there (and not somewhere else)? Kai James offers an answer…
Imagine you’re a copper miner in southeastern Europe in the year 3900 B.C.E. Day after day you haul copper ore through the mine’s sweltering tunnels.
You’ve resigned yourself to the grueling monotony of mining life. Then one afternoon, you witness a fellow worker doing something remarkable.
With an odd-looking contraption, he casually transports the equivalent of three times his body weight on a single trip. As he returns to the mine to fetch another load, it suddenly dawns on you that your chosen profession is about to get far less taxing and much more lucrative.
What you don’t realize: You’re witnessing something that will change the course of history – not just for your tiny mining community, but for all of humanity.
Despite the wheel’s immeasurable impact, no one is certain as to who invented it, or when and where it was first conceived. The hypothetical scenario described above is based on a 2015 theory that miners in the Carpathian Mountains – in present-day Hungary – first invented the wheel nearly 6,000 years ago as a means to transport copper ore.
The theory is supported by the discovery of more than 150 miniaturized wagons by archaeologists working in the region. These pint-sized, four-wheeled models were made from clay, and their outer surfaces were engraved with a wickerwork pattern reminiscent of the basketry used by mining communities at the time. Carbon dating later revealed that these wagons are the earliest known depictions of wheeled transport to date.
This theory also raises a question of particular interest to me, an aerospace engineer who studies the science of engineering design. How did an obscure, scientifically naive mining society discover the wheel, when highly advanced civilizations, such as the ancient Egyptians, did not?…
Read on to find out: “How was the wheel invented? Computer simulations reveal the unlikely birth of a world-changing technology nearly 6,000 years ago,” from @us.theconversation.com.
* “We must not forget that the wheel is reinvented so often because it is a very good idea; I’ve learned to worry more about the soundness of ideas that were invented only once.” – David Parnas
###
As we roll along, we might we might send a “Alles Gute zum Geburtstag” to man at the center of the question of the invention of another foundational “technology”: the polymathic Gottfried Wilhelm Leibniz, the philosopher, mathematician, inventor (of, among other things, an early calculator) and political adviser.
Leibnitz was important both as a metaphysician and as a logician, but who is probably best remembered for his independent invention of the calculus; he was born on this date in 1646. Leibniz independently discovered and developed differential and integral calculus, which he published in 1684; but he became involved in a bitter priority dispute with Isaac Newton, whose ideas on the calculus were developed earlier (1665), but published later (1687). Scholars largely agree that, in fact, Leibnitz and Newton independently developed “the greatest advance in mathematics that had taken place since the time of Archimedes.”
“Great minds think alike”*…
Brian Potter on the (perhaps surprising) frequency with which “heroic” inventors are in fact better understood as the winners of close races…
When Alexander Graham Bell filed a patent for the telephone on February 14th, 1876, he beat competing telephone developer Elisha Gray to the patent office by just a few hours. The resulting legal dispute between Bell Telephone and Western Union (which owned the rights to Gray’s invention) would consume millions of dollars before being resolved in Bell’s favor in 1879.
Such cases of multiple invention are common, and some of the most famous and important modern inventions were invented in parallel. Both Thomas Edison and Joseph Swan patented incandescent lightbulbs in 1880. Jack Kilby and Robert Noyce patented integrated circuits in 1959. Hans von Ohain and Frank Whittle independently invented the jet engine in the 1930s. In a 1922 paper, William Ogburn and Dorothy Thomas documented 150 cases of multiple discovery in science and technology. Robert Merton found 261 examples in 1961, and observed that the phenomenon of multiple discovery was itself a multiple discovery, having been described over and over again since at least the early 19th century.
But exactly how common is multiple invention? The frequency of examples suggests that it can’t be particularly rare, but that doesn’t tell us the rate at which it occurs. In “How Common is Independent Discovery?,” Matt Clancy catalogues several attempts to estimate the frequency of multiple discovery, and tentatively comes up with a frequency of around 2-3% for simultaneous scientific discoveries, and perhaps an 8% chance that a given invention will be reinvented in the next decade. But the evidence for inventions is somewhat inconsistent, and varies greatly between studies. Clancy estimates a reinvention rate of around 8% per decade, but another study he found that looked at patent interference lawsuits between 1998 and 2014 suggests an independent invention rate of only around 0.02% per year.
The frequency of multiple invention is a useful thing to know, because it can give us clues about the nature of technological progress. A very low rate of multiple invention suggests that progress might be driven by a small number of “genius” inventors (what we might call the Great Man Theory of technological progress), and that it might be highly historically contingent (if you re-rolled the dice of history, maybe you get a totally new set of inventions and a different technological palette). A high rate of multiple invention suggests that progress is more a function of broad historical forces (that inventions appear when the conditions are right), and that progress is less contingent (if you re-rolled the dice of history, you’d get a similar progression of inventions). And if the rate of multiple invention is changing over time, perhaps the nature of technological progress is changing as well…
[Potter reviews the history and concludes that “multiple invention was extremely common”…]
… My main takeaway is that the ideas behind inventions are often in some sense “obvious,” or at least not so surprising or unexpected that many people won’t think of them. In some cases, this is probably because once some new possibility comes along, lots of people think of similar things that could be done with it. Once the properties of electricity began to be understood, many people came up with the idea of using it to send signals (telephone, telegraph), or to create motion (engines and generators), or to generate light (arc lamps, incandescent lights). Once the steam engine came along, lots of people had the idea to use it to power various types of vehicles.
In other cases, multiple invention probably occurs because important problems will attract many people trying to solve them. Steel corrosion was a large problem inspiring many folks to look for ways to create a steel that didn’t rust, or notice the potential value if they stumbled across such a material. Lamps causing mine fires were a major problem, inspiring many people to come up with ideas for safety lamps. The smoke produced by gunpowder was a major problem, inspiring many efforts to develop smokeless powders. And because would-be inventors will all draw from the same pool of available technologies, materials, and capabilities when coming up with a solution, there will be a large degree of convergence in the solutions they come up with…
Fascinating: “How Common is Multiple Invention?” from @const-physics.blogsky.venki.dev.
* common idiom
###
As we reconsider credit, we might recall that it was on this date in 1661 that Isaac Newton— a key figure in the Scientific Revolution and the Enlightenment that followed– entered Trinity College, Cambridge. Soon after Newton obtained his BA degree at Cambridge in August 1665, the university temporarily closed as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student, his private studies and the years following his bachelor’s degree have been described as “the richest and most productive ever experienced by a scientist.”
Relevantly to the piece above, Newton was party to a dispute with Gottfried Wilhelm Leibniz (who started, at age 14, at the University of Leipzig the same year that Newton matriculated at Cambridge) over which of them developed calculus– called “the greatest advance in mathematics that had taken place since the time of Archimedes.” The modern consensus is that the two men independently developed their ideas.
“Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things”*…
From Kim (Scott) Morrison‘s and Dror Bar-Natan‘s, The Knot Atlas, “a complete user-editable knot atlas, in the wiki spirit of Wikipedia“– a marvelous example of a wide-spread urge in mathematics to find order through classification. As Joseph Howlett explains, that quest continues, even as it proves vexatious…
Biology in the 18th century was all about taxonomy. The staggering diversity of life made it hard to draw conclusions about how it came to be. Scientists first had to put things in their proper order, grouping species according to shared characteristics — no easy task. Since then, they’ve used these grand catalogs to understand the differences among organisms and to infer their evolutionary histories. Chemists built the periodic table for the same purpose — to classify the elements and understand their behaviors. And physicists made the Standard Model to explain how the fundamental particles of the universe interact.
In his book The Order of Things, the philosopher Michel Foucault describes this preoccupation with sorting as a formative step for the sciences. “A knowledge of empirical individuals,” he wrote, “can be acquired only from the continuous, ordered and universal tabulation of all possible differences.”
Mathematicians never got past this obsession. That’s because the menagerie of mathematics makes the biological catalog look like a petting zoo. Its inhabitants aren’t limited by physical reality. Any conceivable possibility, whether it lives in our universe or in some hypothetical 200-dimensional one, needs to be accounted for. There are tons of different classifications to try — groups, knots, manifolds and so on — and infinitely many objects to sort in each of those classifications. Classification is how mathematicians come to know the strange, abstract world they’re studying, and how they prove major theorems about it.Take groups, a central object of study in math. The classification of “finite simple groups” — the building blocks of all groups — was one of the grandest mathematical accomplishments of the 20th century. It took dozens of mathematicians nearly 100 years to finish. In the end, they figured out that all finite simple groups fall into three buckets, except for 26 itemized outliers. A dedicated crew of mathematicians has been working on a “condensed” proof of the classification since 1994 — it currently comprises 10 volumes and several thousand pages, and still isn’t finished. But the gargantuan undertaking continues to bear fruit, recently helping to prove a decades-old conjecture that you can infer a lot about a group by examining one small part of it.
Mathematics, unfettered by the typical constraints of reality, is all about possibility. Classification gives mathematicians a way to start exploring that limitless potential…[Howlett reviews attempts to classify numbers by “type” (postive/negative, rational/irrational), and mathematical objects by “equivalency” (shapes that can be stretched or squeezed into the other without breaking or tearing, like a doughnut and and coffee cup (see here)…]
… Similarly, classification has played an important role in knot theory. Tie a knot in a piece of string, then glue the string’s ends together — that’s a mathematical knot. Knots are equivalent if one can be tangled or untangled, without cutting the string, to match the other. This mundane-sounding task has lots of mathematical uses. In 2023, five mathematicians made progress on a key conjecture in knot theory that stated that all knots with a certain property (being “slice”) must also have another (being “ribbon”), with the proof ruling out a suspected counterexample. (As an aside, I’ve often wondered why knot theorists insist on using nouns as adjectives.)
Classifications can also get more meta. Both theoretical computer scientists and mathematicians classify problems about classification based on how “hard” they are.
All these classifications turn math’s disarrayed infinitude into accessible order. It’s a first step toward reining in the deluge that pours forth from mathematical imaginings…
“The Never-Ending Struggle to Classify All Math,” from @quantamagazine.bsky.social.
* Isaac Newton
###
As we sort, we might spare a thought for the author of our title quote, Sir Isaac Newton; he died in this date in 1727. A polymath, Newton excelled in– and advanced– mathematics, physics, and astronomy; he was a theologian and a government offical (Master of the Mint)… and a dedicated alchemist. He was key to the Scientific Revolution and the Enlightenment that followed.
Newton’s book Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1687, achieved the first great unification in physics and established classical mechanics (e.g., the Laws of Motion and the principle of universal gravitation). He also made seminal contributions to optics, and shares credit with German mathematician Gottfried Wilhelm Leibniz for formulating infinitesimal calculus. Indeed, Newton contributed to and refined the scientific method to such an extent that his work is considered the most influential in the development of modern science.
You must be logged in to post a comment.