Posts Tagged ‘optics’
“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.
“I’ve been making a list of the things they don’t teach you at school”*…

In Warsaw’s Gruba Kaśka water plant there are eight clams with sensors attached to their shells. If the clams close because they don’t like the taste of the water, the city’s supply is automatically shut off. [Judita K]
When bar codes were patented in 1952, they were round [Sarah Laskow]
A 70% dilution of isopropyl alcohol is better at killing bacteria, fungi, and viruses than ‘pure’ 99% isopropyl alcohol, for several distinct reasons. [Mitch Walleser]
Epidemiologists at Emory University in Atlanta believe that raising the mimimim wage in the US by $1 would have prevented 27,550 suicides since 1990. [John A Kaufman & Co, via The Economist]
Games Workshop, owner of Warhammer, is worth more than Centrica, owner of British Gas. [Allister Thomas]
Numbers 30-34 of this year’s list from Tom Whitwell of Fluxx: “52 things I learned in 2020.”
* Neil Gaiman, The Kindly Ones
###
As we have fun with facts, we might send fascinatingly-illustrated birthday greetings to David Brewster; he was born on this date in 1781. A physicist, inventor, author, and academic administrator, he is best remembered for his work in optic (especially the phenomenon of polarization). Brewster was a pioneer in photography; he invented an improved stereoscope, which he called “lenticular stereoscope” and which became the first portable 3D-viewing device. He also invented the binocular camera, two types of polarimeters, the polyzonal lens, the lighthouse illuminator, and (perhaps most relevantly to today’s post) the kaleidoscope. For this work, William Whewell dubbed him the “father of modern experimental optics” and “the Johannes Kepler of optics.”
“It’s fine to celebrate success, but it is more important to heed the lessons of failure”*…
If you ever wanted a glimpse into what dooms startups, look no further than autopsy.io, a website that lists the reasons why many newborn tech firms imploded. The website offers entrepreneurs the ability to self-explain why their startup didn’t quite make it; in a bid to separate real-life stories from entertaining fictions, the application form asks for a link to a blog post or medium article “that tells the story of the failure,” along with the founder(s) Twitter handle and Crunchbase or Angel.co profile. Some of the reasons listed for failure are maddeningly opaque, such as UniSport’s “for a number of reasons” or PlayCafe’s “we didn’t reach enough users.” Others are bleakly hilarious; as the founders of Zillionears, self-billed as a “creative pre-sale platform for musicians,” confessed: “People really didn’t really LIKE anything about our product.” If you’re thinking of launching your own company, or you work for a wet-behind-the-ears startup, it’s worth scanning the list to see if any of these potential crises are brewing in your setup.
Via Nerval’s Lobster at Slashdot.
* Bill Gates
###
As we dust ourselves off, we might spare a thought for Roger Bacon; he died on this date in 1292. A philosopher and Franciscan friar, Bacon was one of the first to propose mathematics and experimentation as appropriate methods of science. Working in mathematics, astronomy, physics, alchemy, and languages, he was particularly impactful in optics: he elucidated the principles of refraction, reflection, and spherical aberration, and described spectacles, which soon thereafter came into use. He developed many mathematical results concerning lenses, proposed mechanically propelled ships, carriages, and flying machines, and used a camera obscura to observe eclipses of the Sun. And he was the first European give a detailed description of the process of making gunpowder.
He began his career at Oxford, then lectured for a time at Paris, where his skills as a pedagogue earned him the title Doctor Mirabilis, or “wonderful teacher.” He stopped teaching when he became a Franciscan. But his scientific work continued, despite his Order’s restrictions on activity and publication, as Bacon enjoyed the protection and patronage of Pope Clement… until, on Clement’s death, he was placed under house arrest in Oxford, where he continued his studies, but was unable to publish and communicate with fellow investigators.

Statue of Roger Bacon in the Oxford University Museum








You must be logged in to post a comment.