Two approachable visual presentations of simple neural networks: one showing how a soft activation function allows the successive layers of a neural network to distort the input until the different classes are separable, and the other showing how a hard step activation function can be represented as carving out polygons in the space of inputs. Don't be intimidated by the rather condensed summaries above- the actual articles are very readable.
Trio for Three Angles (1968) is one of many beautiful acclaimed visually-oriented short films with music by mathematical filmmakers Bruce and Katharine Cornwell, some animated by hand and some using early digital technology. It inspired three sequels: Similar Triangles (1975), Congruent Triangles (1976), and Journey to the Center of a Triangle (1978) (previously). [more inside]
Forensic Topology. "In his 2003 memoir Where The Money Is: True Tales from the Bank Robbery Capital of the World, co-authored with Gordon Dillow, retired Special Agent William J. Rehder briefly suggests that the design of a city itself leads to and even instigates certain crimes—in Los Angeles’s case, bank robberies. Rehder points out that this sprawling metropolis of freeways and its innumerable nondescript banks is, in a sense, a bank robber’s paradise. Crime, we could say, is just another way to use the city."
Mathematicians Henry Segerman and Saul Schleimer have produced a triple gear, three linked gears in space that can rotate together. A short writeup of the topology and geometry behind the triple gear on the arXiv.
And collapsible hexagons are, I suppose, cool enough to at least amuse you a little bit during your class...
To keep yourself amused during your math class, you start playing with [all these strips of paper]. And by you, I mean Arthur H. Stone in 1939. (SLYT)
Robert MacPherson interviewed as part of the Simons Foundation's Science Lives series. MacPherson is among the founders of the modern theory of singularities, points like a kink in a curve where the geometry of a space stops being smooth and starts behaving badly. In the interview, MacPherson talks about cultural differences between math and music, his frustration with high school math, growing up gay in the South and life as a gay man in the scientific community, smuggling $23,000 in cash into post-Soviet Russia to help mathematicians there keep the lights on, catastrophe theory, perverse sheaves, how to be a successful graduate student, stuttering, and of course the development of the intersection homology theory for which he is most well-known.
The fine people over at the International Guild of Knot Tyers Forum talk knots. On Mars.
"The real satisfaction from mathematics is in learning from others and sharing with others." William Thurston, one of the greatest mathematicians of the 20th century, has died. He revolutionized topology and geometry, insisting always that geometric intuition and understanding played just as important a role in mathematical discovery as did the austere formalism championed by the school of Grothendieck. Thurston's views on the relation between mathematical understanding and formal proof are summed up in his essay "On Proof and Progress in Mathematics." [more inside]
Science through yarn: Wooly Thoughts. The Home of Mathematical Knitting, including knitted klein bottles and hyperbolic planes. The Museum of Scientifically Accurate Fabric Brain Art (previously). Much, much, more on knitting, crochet and quilting used to visualize complex theories in topology, probability, chaos and fractals. [more inside]
So you're me and you're in math class and you're learning about graph theory, a subject too interesting to be included in most grade school's curricula so maybe you're in some special program or maybe you're in college and were somehow not scarred for life by your grade school math teachers. [more inside]
Borromean rings consist of three circles linked as a group, with no two circles interlinked; removal of one ring results in the separation of all three. Named for the Borromeo family of 15th century Italy which featured the rings on its coat of arms, the symbol has had a long and varied history. The rings have appeared everywhere from medieval Christian iconography to Norse rune stones to the pillars of Hindu temples. In more recent times, Borromean rings have attracted formal study in the fields of topology, chemistry and (strangely enough) quantum mechanics. [more inside]
Measure-theoretic probability: Why it should be learnt and how to get started. The clickable chart of distribution relationships. Just two of the interesting and informative probability resources I've learned about, along with countless other tidbits of information, from statistician John D. Cook's blog and his probability fact-of-the-day Twitter feed ProbFact. John also has daily tip and fact Twitter feeds for Windows keyboard shortcuts, regular expressions, TeX and LaTeX, algebra and number theory, topology and geometry, real and complex analysis, and beginning tomorrow, computer science and statistics.
There's always been hyperbole in fashion; but fashion became truly hyperbolic this week when mathematican William Thurston, winner of a 1982 Fields Medal for his revolutionary re-envisioning of low-dimensional topology and geometry, teamed up with designer Dai Fujiwara (of the house of Issey Miyake) to produce a Paris runway show based on the fundamental geometries of 3-dimensional spaces. Thurston and Fujiwara briefly interviewed. Thurston's famous essay "Proof and Progress in Mathematics" concerns, among other things, Thurston's belief that the production of mathematical understanding can be carried out by means other than the writing down of formal proofs (though fashion shows are not specifically mentioned.) Previously in wearable non-Euclidean geometry: Daina Taimina's hyperbolic skirt.
"the scale-free network modeing paradigm is largely inconsistent with the engineered nature of the Internet..." For a decade it's been conventional wisdom that the Internet has a scale-free topology, in which the number of links emanating from a site obeys a power law. In other words, the Internet has a long tail; compared with a completely random network, its structure is dominated by a few very highly connected nodes, while the rest of the web consists of a gigantic list of sites attached to hardly anything. Among its other effects, this makes the web highly vulnerable to epidemics. The power law on the internet has inspired a vast array of research by computer scientists, mathematicians, and engineers. According to an article in this month's Notices of the American Math Society, it's all wrong. How could so many scientists make this kind of mistake? Statistician Cosma Shalizi explains how people see power laws when they aren't there: "Abusing linear regression makes the baby Gauss cry."
Topology and Geometry Software by Jeff Weeks.
Whether you want to learn to lace shoes, tie shoelaces, stop shoelaces from coming undone, calculate shoelace lengths or even repair aglets, Ian's Shoelace Site has the answer!
Inside Out A topographical bedtime story. (Warning, contains spheres!)
Can you cut a hole in a 3x5 card that's large enough to crawl through? Topological trickery and some other classic science experiments.
Dr. Jeannine Mosely finishes building a level-3 Menger sponge from business cards. You can also build your own, though Dr. Mosely warns, "[a] level 4 sponge would require almost a million cards and weigh over a ton. I do not believe it could support its own weight — so a level 3 is the biggest sponge we can hope to build." (related)
Grigory Perelman, awarded the Fields Medal for his work on the Poincare Conjecture, talks to the New Yorker.
Bending a soccer ball - mathematically. Found via Ivars Peterson's short exposition on Braungardt and Kotschick's The Classification of Football Patterns [pdf, technical].
Grisha Perelman, where are you? Perelman has quite possibly solved one of mathematics biggest mysteries, Poincaré’s conjecture, but has since disappeared.
Pretty and pretty interesting: unrooted haplotype networks -- diagrams showing the relation and mutational distance between different sets of DNA, with haplotypes represented by circles proportional to haplotype frequency, joined by lines proportional to mutational difference between haplotypes -- in cichlid fish (on page 3 ) [pdf], in stone loach fish ( on page 3) [pdf], in lesser prairie chickens (on page 6) [pdf] and in a ring species! (on page 2) [pdf]
Beautiful Mathematical Surfaces : "Conceptual Forms," a series of photographs by Hiroshi Sugimoto, conceived as an hommage to Marcel Duchamp (English summary) and as an un-Man Ray-like treatment of the subject, consisting of (English summary) "Mathematical Forms" ("Curves" and "Surfaces") and Mechanical Forms.
The Shapes of Space [note : pdf, sciam, poincaré conjecture]
After getting the inside story (ha?) on the inventor of everyone's favorite non-orientable surface, the Klein Bottle; and perhaps playing a few games inside of one, you can check out a few 3-dimensional immersions of klein bottles: in Lego, knitted fabric, paper, or glass.