26 posts tagged with topology. (View popular tags)

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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.

posted by Jpfed on Aug 1, 2014 - 4 comments

posted by Jpfed on Aug 1, 2014 - 4 comments

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]

posted by beryllium on Jul 6, 2014 - 5 comments

posted by beryllium on Jul 6, 2014 - 5 comments

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."

posted by homunculus on Jul 13, 2013 - 14 comments

posted by homunculus on Jul 13, 2013 - 14 comments

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.

posted by escabeche on Apr 26, 2013 - 36 comments

posted by escabeche on Apr 26, 2013 - 36 comments

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)

posted by 256 on Oct 1, 2012 - 35 comments

posted by 256 on Oct 1, 2012 - 35 comments

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.

posted by escabeche on Sep 12, 2012 - 5 comments

posted by escabeche on Sep 12, 2012 - 5 comments

The fine people over at the International Guild of Knot Tyers Forum talk knots. On Mars.

posted by Foci for Analysis on Sep 4, 2012 - 33 comments

posted by Foci for Analysis on Sep 4, 2012 - 33 comments

"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]

posted by escabeche on Aug 22, 2012 - 32 comments

posted by escabeche on Aug 22, 2012 - 32 comments

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]

posted by Bora Horza Gobuchul on Nov 6, 2011 - 8 comments

posted by Bora Horza Gobuchul on Nov 6, 2011 - 8 comments

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]

posted by achmorrison on Feb 22, 2011 - 32 comments

posted by achmorrison on Feb 22, 2011 - 32 comments

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]

posted by dephlogisticated on Jan 8, 2011 - 24 comments

posted by dephlogisticated on Jan 8, 2011 - 24 comments

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.

posted by grouse on Dec 5, 2010 - 17 comments

posted by grouse on Dec 5, 2010 - 17 comments

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.

posted by escabeche on Mar 8, 2010 - 19 comments

posted by escabeche on Mar 8, 2010 - 19 comments

"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."

posted by escabeche on Apr 23, 2009 - 30 comments

posted by escabeche on Apr 23, 2009 - 30 comments

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!

posted by Blazecock Pileon on Jun 27, 2008 - 22 comments

posted by Blazecock Pileon on Jun 27, 2008 - 22 comments

Inside Out A topographical bedtime story. (Warning, contains spheres!)

posted by loquacious on Aug 28, 2007 - 20 comments

posted by loquacious on Aug 28, 2007 - 20 comments

Can you cut a hole in a 3x5 card that's large enough to crawl through? Topological trickery and some other classic science experiments.

posted by Wolfdog on Jul 2, 2007 - 40 comments

posted by Wolfdog on Jul 2, 2007 - 40 comments

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)

posted by Blazecock Pileon on Feb 2, 2007 - 19 comments

posted by Blazecock Pileon on Feb 2, 2007 - 19 comments

Grigory Perelman, awarded the Fields Medal for his work on the Poincare Conjecture, talks to the New Yorker.

posted by Gyan on Aug 29, 2006 - 17 comments

posted by Gyan on Aug 29, 2006 - 17 comments

Bending a soccer ball - mathematically. Found via Ivars Peterson's short exposition on Braungardt and Kotschick's The Classification of Football Patterns [pdf, technical].

posted by Wolfdog on Aug 17, 2006 - 18 comments

posted by Wolfdog on Aug 17, 2006 - 18 comments

Grisha Perelman, where are you? Perelman has quite possibly solved one of mathematics biggest mysteries, Poincaré’s conjecture, but has since disappeared.

posted by kliuless on Aug 15, 2006 - 32 comments

posted by kliuless on Aug 15, 2006 - 32 comments

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]

posted by orthogonality on Mar 25, 2005 - 14 comments

posted by orthogonality on Mar 25, 2005 - 14 comments

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.

posted by Zurishaddai on Dec 12, 2004 - 10 comments

posted by Zurishaddai on Dec 12, 2004 - 10 comments

The Shapes of Space [note : *pdf, sciam, poincaré conjecture*]

posted by kliuless on Aug 1, 2004 - 4 comments

posted by kliuless on Aug 1, 2004 - 4 comments

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.

posted by kidsplateusa on Oct 30, 2000 - 5 comments

posted by kidsplateusa on Oct 30, 2000 - 5 comments

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