tag:tagteam.harvard.edu,2005:/hubs/13/tag/atom/mathematicsItems tagged with mathematics in Blogs.law Aggregation Hub2015-04-20T14:35:05-04:00TagTeam social RSS aggregratortag:tagteam.harvard.edu,2005:FeedItem/20972672015-04-20T14:34:58-04:002015-04-20T14:35:05-04:00▶ Stephanie Dick | Harvard Horizons Symposium - YouTubetag:tagteam.harvard.edu,2005:FeedItem/15416952014-08-16T12:41:33-04:002015-07-01T03:49:24-04:00Helen R<p><img src="https://www.sciencenews.org/sites/default/files/main/articles/notebook_fonts_main.jpg" alt="Martin Demaine and Erik Demaine"></p><p>A mathematician once posed a deceptively simple question. Can a single 2-D conveyor belt be stretched around a set of wheels such that the belt is taut and touches every wheel without crossing itself?</p><p>MIT computer scientist Erik Demaine pondered the problem. For just a few wheels, the solution is easy: Arrange four into a square, wrap the belt around the outside, and the problem is solved — one version of it, at least.</p><p>“The question is whether it’s always possible to solve no matter how you draw the wheels,” Demaine says. A complete solution would lay out a set of rules that applies to every possible wheel arrangement and number. “But so far every algorithm we’ve come up with has been foiled.”</p><p>One day Demaine was working on the problem with his dad, who happens to be an artist and mathematician, and a colleague. The trio got stuck. So they decided to take a break with another activity the Demaines enjoy: designing new fonts. The team stuck thumbtacks into poster board to represent wheels, and wrapped them with rubber band “conveyor belts” to form letters.</p><div><img title="<b>SECRET MESSAGE</b> By erasing the black lines wrapped around the pink dots in the conveyor belt font (left), puzzle enthusiasts can craft hidden messages from seemingly random clusters of circles. The glass-squishing font (right) can also hide words; users reveal individual letters by imagining what happens after a set of balls and thin rods of glass are pushed or “squished” from the sides." src="https://www.sciencenews.org/sites/default/files/images/notebook_font_diagram.png" alt="science news created in conveyor belt and glass font"></div><p><span><b>SECRET MESSAGE</b><span> </span>By erasing the black lines wrapped around the pink dots in the conveyor belt font (left), puzzle enthusiasts can craft hidden messages from seemingly random clusters of circles. The glass-squishing font (right) can also hide words; users reveal individual letters by imagining what happens after a set of balls and thin rods of glass are pushed or “squished” from the sides.</span></p><div>E. DEMAINE</div><p>“It became a game,” Demaine says. “One of us would put in some thumbtacks, and the other would say, “Oh, I see, it’s a ‘K’!”</p><p>Demaine and his father, MIT artist-in residence Martin Demaine, published the complete alphabet of conveyor belt letters in the <em>Proceedings of the 7th International Conference on Fun with Algorithms</em> in July along with four other typeface ideas sparked by math and computational geometry. The Demaines’ interest in geometric folding spurred creation of three fonts, one of which — the “origami maze” typeface — uses a computer algorithm to create crease patterns that can fold into 3-D letters.</p><p>Several of the Demaines’ fonts can be turned into geometry or math puzzles. In the conveyor belt font, for example, take the belt away from a letter and all that’s left is a cryptic arrangement of wheels. “You can hide secret messages this way,” Erik Demaine says.</p><p>Another puzzle font, called the glass-squishing typeface, drew inspiration from their passion for glass blowing. After inventing a software program that helps glass blowers design pieces, the Demaines wanted to mathematically describe how pieces of glass squish together when heated. They started experimenting by making actual glass letters.</p><p>The duo arranged blue glass sticks around clear discs, popped the patterns into a volcano-hot oven and then pushed the softened pieces together. “We’d say, ‘Ah, I think this will make an ‘A,’ ” Demaine says, “Then we’d squish it and it would come out looking nothing like an ‘A’.”</p><p>After a week of experiments, they posted videos of the full alphabet on Demaine’s website (see<span> </span><a href="http://erikdemaine.org/fonts/">erikdemaine.org/fonts</a>). Now, they’re hoping to use what they have learned with the letters to build new software for their virtual glass program.</p><p><a></a></p><p>Demaine thinks the fonts are a fun way to introduce people to the world of computational geometry.</p><p>“We want people to play with the fonts,” he says. “We really love puzzles — now anyone can participate.”</p><p>A mishmash of what looks like sticks and bubbles morphs into letters that spell the words “Science News” in a font inspired by glass blowing. Once the glass sticks heat up, metal bars squish the pieces together around glass discs, forming letters.<span> </span><em>Credit: E. Demaine and M. Demaine</em></p><p> </p> <a href="http://feeds.wordpress.com/1.0/gocomments/sciencepublication.wordpress.com/570/"><img alt="" src="http://feeds.wordpress.com/1.0/comments/sciencepublication.wordpress.com/570/"></a> <img alt="" src="http://pixel.wp.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=570&subd=sciencepublication&ref=&feed=1" width="1" height="1">Father-son mathematicians fold math into fonts: MIT’s Erik and Martin Demaine create puzzle typefaces to test new ideasA mathematician once posed a deceptively simple question. Can a single 2-D conveyor belt be stretched around a set of wheels such that the belt is taut and touches every wheel without crossing itself? MIT computer scientist Erik Demaine pondered the problem. For just a few wheels, the solution is easy: Arrange four into a […]<img alt="" src="http://pixel.wp.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=570&subd=sciencepublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/15416992014-08-16T12:41:38-04:002014-08-16T12:41:38-04:00Helen RFather-son mathematicians fold math into fonts: MIT’s Erik and Martin Demaine create puzzle typefaces to test new ideasA mathematician once posed a deceptively simple question. Can a single 2-D conveyor belt be stretched around a set of wheels such that the belt is taut and touches every wheel without crossing itself? MIT computer scientist Erik Demaine pondered … <a href="http://journalpublication.wordpress.com/2014/08/16/father-son-mathematicians-fold-math-into-fonts-mits-erik-and-martin-demaine-create-puzzle-typefaces-to-test-new-ideas/">Continue reading <span>→</span></a><img alt="" src="http://pixel.wp.com/b.gif?host=journalpublication.wordpress.com&blog=65010642&post=449&subd=journalpublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/10511232014-06-15T23:51:12-04:002014-06-15T23:51:12-04:00Helen R‘Angry alien’ in packing puzzle shocks mathematiciansWhen it comes to packing, you can’t beat a mathematician. Centuries of work has gone into finding the most efficient ways to pack identical objects into the densest possible arrangements. But the latest experiment shows that a surprising amount of chaos lurks … <a href="http://journalpublication.wordpress.com/2014/06/16/angry-alien-in-packing-puzzle-shocks-mathematicians/">Continue reading <span>→</span></a><img alt="" src="http://stats.wordpress.com/b.gif?host=journalpublication.wordpress.com&blog=65010642&post=424&subd=journalpublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/10511202014-06-15T23:51:08-04:002015-07-01T03:47:01-04:00Helen R<p>When it comes to packing, you can’t beat a mathematician. Centuries of work has gone into finding the<span> </span><a href="http://www.newscientist.com/article/dn16716-recordbreaking-algorithm-really-packs-them-in.html">most efficient ways to pack identical objects</a><span> </span>into the densest possible arrangements. But the latest experiment shows that a surprising amount of chaos lurks within our attempts to create order.</p><p><img src="http://www.newscientist.com/data/images/ns/cms/dn25163/dn25163-1_300.jpg" alt="Easier packing <i>(Image: Sipa Press/Rex)</i>"></p><p><a href="http://www.engin.umich.edu/college/about/people/profiles/f-to-j/sharon-glotzer">Sharon Glotzer</a><span> </span>at the University of Michigan in Ann Arbor and her colleagues started with three distinct shapes: a cube, a 12-sided dodecahedron and a<span> </span><a href="http://www.newscientist.com/article/mg20627604.100-pyramids-are-the-best-shape-for-packing.html">4-sided pyramid</a>, or tetrahedron. First they created variations of each by slicing off corners, edges or both, until they had more than 55,000 shapes. “We wanted to look at a large enough set of shapes that we could start to see some trends that you can’t get from looking at a couple of shapes,” says Glotzer.</p><p>The team then used computer simulations to create identical copies of each new shape and pack them in the most efficient way possible. Finally, they displayed the results for the three shape “families” as three-dimensional landscapes, with changes in height corresponding to packing efficiency.</p><p>The team thought small changes to a shape would only gradually affect how it packs, meaning each landscape should be made up of gentle hills. That is the case for the dodecahedron family, but the landscape is bumpy and chaotic for variants of cubes and tetrahedrons. The 3D tetrahedron landscape, shown below from multiple angles, is so weird that the researchers nicknamed it the angry alien, seen best at the bottom left.</p><p><img src="http://www.newscientist.com/data/images/ns/cms/dn25163/dn25163-2_1200.jpg" alt=""><i>(Image: Elizabeth R. Chen, Daphne Klotsa et al)</i></p><p>Real-world objects are likely to have minor defects that change their shapes in similar ways. That means a better understanding of packing variations could be important for<span> </span><a href="http://www.newscientist.com/article/dn21821-spacefilling-solution-could-boost-wifi-security.html">nanotech materials built from small particles</a>, or for pharmaceuticals in which the density of a drug matters.</p><p>“I believe this research has tremendous value for designing new materials based on nanoparticles,” says Oleg Gang at Brookhaven National Laboratory in Upton, New York. “It gives us a complete understanding of how particles of different shapes pack in larger-scale organisations. Many practical applications, from catalysis to batteries, depend on particle packing.”</p><p><a href="http://www.cs.ucl.ac.uk/staff/tomaso_aste/">Tomaso Aste</a><span> </span>at University College London also thinks the results could prove useful, but he adds that they need to be double-checked. Packing is a difficult problem, he says, and different computer simulations might find better or worse packing densities for particular shapes, which could alter the landscape.</p> <a href="http://feeds.wordpress.com/1.0/gocomments/sciencepublication.wordpress.com/547/"><img alt="" src="http://feeds.wordpress.com/1.0/comments/sciencepublication.wordpress.com/547/"></a> <img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=547&subd=sciencepublication&ref=&feed=1" width="1" height="1">‘Angry alien’ in packing puzzle shocks mathematiciansWhen it comes to packing, you can’t beat a mathematician. Centuries of work has gone into finding the most efficient ways to pack identical objects into the densest possible arrangements. But the latest experiment shows that a surprising amount of chaos lurks within our attempts to create order. Sharon Glotzer at the University of Michigan in Ann Arbor […]<img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=547&subd=sciencepublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/7912482014-05-14T14:32:03-04:002015-07-01T03:45:44-04:00Helen R<p><img src="http://www.newscientist.com/data/images/ns/cms/mg22229680.700/mg22229680.700-1_1200.jpg" alt="">Deep connections join up many mathematical fields<span> </span><i>(Image: Jan Michalko/Picturetank)</i></p><p><i>From the appeal of seven to a shadowy textbook-writing cabal,<span> </span></i>Alex Through The Looking Glass<i><span> </span>is a whistle-stop tour of entertaining areas of mathematics</i></p><p>WHAT is your favourite number? I don’t have one, but if pressed I’d probably go for 7. I’m not exactly sure why, but there’s something about 7 that makes it stand out from the infinity of alternatives.</p><p><img src="http://www.newscientist.com/data/images/ns/cms/mg22229680.700/mg22229680.700-2_1200.jpg" alt="">It turns out I’m not alone. In an<span> </span><a href="http://www.newscientist.com/article/mg21028186.300-alex-bellos-tell-me-all-about-your-favourite-number.html">online survey</a><span> </span>conducted by Alex Bellos for his book,<span> </span><i>Alex Through the Looking-Glass: How life reflects numbers and numbers reflect life</i>, nearly 10 per cent of more than 30,000 respondents also chose 7.</p><p>Bellos isn’t surprised, given how often the number crops up in our culture – wonders of the world, deadly sins and dwarves are just a few of the cultural references that come in sevens. We are also driven towards 7 in an effort to be spontaneous. When asked to think of a random digit between 1 and 10, we tend to avoid the first and eliminate those divisible by 2, 3 and 5 as being too obvious, leaving only good old 7. So maybe, in a sense, my favourite number has been chosen for me.</p><p>Bellos’s book is sprinkled with similarly surprising revelations about familiar mathematical objects. Take the trigonometric functions sine, cosine and tangent, which you might remember from school thanks to the phrase SOH-CAH-TOA. What you probably didn’t know is that the word sine comes from multiple mistranslations, from Sanskrit to Latin via a made-up word that sounds a bit like the Arabic term for bosom.</p><p>Unfortunately, this light dusting of quotable facts only serves to highlight the book’s weak structure – its 10 chapters feel more like a loose collection of articles than a cohesive whole. A trilogy of chapters in the middle of the book on<span> </span><i>π</i>,<span> </span><i>e</i><span> </span>and<span> </span><i>i</i>, three important numbers linked by the single equation<span> </span><i>e<sup>iπ</sup><span> </span>+ 1 = 0</i>, hints at the deeper connections between different areas of mathematics, before moving swiftly onwards.</p><p>In part that is by design: Bellos avoids overwhelming readers by resetting the level of mathematical complexity at the start of each chapter. But it makes for a disjointed read on a subject in which unexpected links are the key to major breakthroughs.<span> </span><a href="http://www.newscientist.com/article/mg13918802.800-science-thousandpage-proof-vindicates-fermat.html">Fermat’s last theorem, for example, was only solved</a><span> </span>after 350 years because University of Oxford mathematician Andrew Wiles was able to make connections no one else could.</p><p>In fact, one area where Bellos does show the powerful ability of mathematics to join disparate concepts is seriously overblown. A section on power laws – equations that can explain the distributions of city sizes, the frequency of word usage and the magnitude of earthquakes – brushes over the fact that many<span> </span><a href="http://aperiodical.com/2013/01/log-log-whos-there-not-a-power-law/">of these laws are nothing but torturous curve-fitting</a><span> </span>with little evidence to back them up. “It’s an important debate, but not one for this book,” says Bellos.</p><p>That aside, the mathematical areas he guides us through on this whistle-stop tour are some of the most entertaining around. There is the detective who uncovers fraud with the crime-fighting<span> </span><a href="http://www.newscientist.com/article/dn24668-mathematical-crimefighter-helps-hunt-for-alien-worlds.html">Benford’s law</a>, fractal explorers discovering<span> </span><a href="http://www.newscientist.com/article/dn18171-the-mandelbulb-first-true-3d-image-of-famous-fractal.html">uncharted infinite worlds</a><span> </span>and a group of mathematical ecologists cataloguing self-replicating creatures in the<span> </span><a href="http://www.newscientist.com/article/mg20627653.800-first-replicating-creature-spawned-in-life-simulator.html">Game of Life</a><img title="Contains video content" src="http://www.newscientist.com/img/icon/artx_video.gif" alt="Movie Camera">, a cellular automaton in which patterns form and evolve on a grid according to a few simple mathematical rules.</p><p>These characters help Bellos inject personality into even the most notoriously dry topics such as calculus. Take the flamboyant French mathematician<a href="http://www.newscientist.com/article/dn20742-mathematical-medallist-seducing-ceos-and-socialists.html">Cédric Villani</a>, who compares his area of study, the Boltzmann equation, to a Michelangelo sculpture: “Not pure and ethereal and elegant, but very human, very tortured, with the strength of the energy of the world.”</p><p>My favourite episode is when Bellos meets a member of a shadowy mathematical cabal that for nearly 80 years has authored highly rigorous textbooks, all under the pseudonym<span> </span><a href="http://www.newscientist.com/article/mg14519624.400-fighting-against-the-dying-of-the-light.html">Nicolas Bourbaki</a>. Bellos can’t give the member’s name, saying only that he has a beard and wears a purple shirt and a straw hat.</p><p>Bellos has a fantastic knack of making you feel as if you’re sharing a room with these mathematical explorers, but I can’t help feel his choice of locations is less inspired than in his previous book,<span> </span><a href="http://www.newscientist.com/blogs/culturelab/2010/04/big-numbers-brilliant-minds-mind-boggling-concepts.html"><i>Alex’s Adventures in Numberland</i></a>, which journeyed through India and Japan, as well as the European and US locales of his new book.</p><p>If you haven’t read<span> </span><i>Numberland</i>, I urge you to pick it up, and come back to<i>Looking-Glass</i><span> </span>afterwards if you have a desire to fall even further down the mathematical rabbit hole.</p> <a href="http://feeds.wordpress.com/1.0/gocomments/sciencepublication.wordpress.com/471/"><img alt="" src="http://feeds.wordpress.com/1.0/comments/sciencepublication.wordpress.com/471/"></a> <img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=471&subd=sciencepublication&ref=&feed=1" width="1" height="1">Alex Bellos wanders in his mathematical wonderlandDeep connections join up many mathematical fields (Image: Jan Michalko/Picturetank) From the appeal of seven to a shadowy textbook-writing cabal, Alex Through The Looking Glass is a whistle-stop tour of entertaining areas of mathematics WHAT is your favourite number? I don’t have one, but if pressed I’d probably go for 7. I’m not exactly sure why, but there’s […]<img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=471&subd=sciencepublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/7912732014-05-14T14:32:13-04:002014-05-14T14:32:13-04:00Helen RAlex Bellos wanders in his mathematical wonderlandDeep connections join up many mathematical fields (Image: Jan Michalko/Picturetank) From the appeal of seven to a shadowy textbook-writing cabal, Alex Through The Looking Glass is a whistle-stop tour of entertaining areas of mathematics WHAT is your favourite number? I don’t have one, … <a href="http://journalpublication.wordpress.com/2014/05/13/alex-bellos-wanders-in-his-mathematical-wonderland/">Continue reading <span>→</span></a><img alt="" src="http://stats.wordpress.com/b.gif?host=journalpublication.wordpress.com&blog=65010642&post=354&subd=journalpublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/7912502014-05-14T14:32:04-04:002015-07-01T03:45:44-04:00Helen R<div><img src="http://www.iflscience.com/sites/www.iflscience.com/files/styles/ifls_large/public/blog/%5Bnid%5D/Energy-Spiral.jpg?itok=6WwrmSTt" alt=""></div><div>When two thugs bashed Jason Padgett outside a bar they weren’t trying to release skills he never knew he had, sless till conduct one of the most groundbreaking neuroscience experiments of the century. But as it turned out, that’s what they did. Hopefully the events will never be repeated, but they opened up new worlds for Padgett and lines of inquiry for neuroscientists.</div><div> </div><div>Pre-bashing Padgett not only had no particular mathematical skill, he had no interest. “I cheated on everything and I never cracked a book” is how the self-confessed former “jock” describes his approach to math. After recovering from concussion resulting from being knocked to the ground and repeatedly kicked in the head Padgett saw the world in an entirely different way. </div><div> </div><div>“Everything has a pixilated look,” Padgett says. He claims, for example, to see the way fractal way water drains in real time. Moreover, he can represent the world he sees with astonishing drawings. It was while he was demonstrating this skill in a local mall that he was spotted by a physicist, who encouraged him to study math at university. Once there his talents became apparent to his teachers, unleashing a blizzard of study, and now his memoir<span> </span><em><a href="http://www.barnesandnoble.com/s/struck-by-genius?store=allproducts&keyword=struck+by+genius">Struck By Genius</a></em>.<span> </span><em><a href="http://www.struckbygenius.com/">Struck By Genius</a></em><span> </span>is co-written with Maureen Seaberg, an author who specializes in writing about<span> </span><a href="http://onlinelibrary.wiley.com/doi/10.1348/000712610X528305/abstract;jsessionid=F13799507F1E20A35833386EE0C906A3.f02t01">synesthesia</a>, where pathways in the brain are mixed so that stimulation of one generates an automatic response from another, such as letters or numbers being color-coded.</div><div> </div><div>Padgett’s brain has been scanned with a functional magnetic resonance imaging (fMRI) machines by Associate Professor Berit Brogaard of the University of Miami. She reported in<span> </span><a href="http://www.tandfonline.com/doi/abs/10.1080/13554794.2012.701646#.U29c5YGSyvI"><em>Neurocase<span> </span></em></a>that when Padgett was shown images that brought out his skills his left parietal cortex lit up most strongly as drawing blood flow. The area integrates information from different senses. Using transcranial magnetic stimulation (TMS) to turn this part of Padgett’s brain down temporarily deprived him of his capacities.</div><div> </div><div>Savants are very<span> </span><a href="http://www.psy.dmu.ac.uk/drhiles/Savant%20Syndrome.htm">rare</a>, but cases where extraordinary mental capacities are acquired as a result of brain injury or illness, rather than from birth, are<span> </span><a href="http://journals.lww.com/neurotodayonline/Citation/2010/02040/Autistic_Savant_Made_Famous_by__Rain_Man__Dies__.8.aspx">rarer still</a>.</div><div> </div><div>The feats of memory, perception and calculation that savants are capable of almost always<span> </span><a href="http://rstb.royalsocietypublishing.org/content/364/1522/1351">come with a price</a>. In Padgett’s case the hard-partying individual of his youth also developed PTSD, social anxiety and obsessive-compulsive disorder. It is hard to tell to what extent these are coincidental products of the manner in which his injury came about or necessary features for his skills to emerge. For Padgett, this was a price worth paying. “It’s so good, I can’t even describe it,” he told<span> </span><a href="http://www.livescience.com/45349-brain-injury-turns-man-into-math-genius.html">Live Science</a>.</div><div> </div><div>Just as Brogaard has used TMS to turn Padgett’s talents down on a temporary basis, some<span> </span><a href="http://www.worldscientific.com/doi/abs/10.1142/S0219635203000287">dream </a>of being able to use the same method to temporarily give anyone exceptional creativity, breathtaking memory or barely imaginable computational power. Professor Allan Snyder of the University of Sydney has been claiming<span> </span><a href="http://www.sciencedirect.com/science/article/pii/S0304394012003618">some success</a> with experiments along these lines. While such work may also affect the capacity of other parts of the brain to do the things we need for everyday living, TMS’ boosters hope it will allow us to turn different functions on and off at will.</div><p> </p> <a href="http://feeds.wordpress.com/1.0/gocomments/sciencepublication.wordpress.com/469/"><img alt="" src="http://feeds.wordpress.com/1.0/comments/sciencepublication.wordpress.com/469/"></a> <img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=469&subd=sciencepublication&ref=&feed=1" width="1" height="1">Brain Injury Releases Astonishing Mathematical PowersWhen two thugs bashed Jason Padgett outside a bar they weren’t trying to release skills he never knew he had, sless till conduct one of the most groundbreaking neuroscience experiments of the century. But as it turned out, that’s what they did. Hopefully the events will never be repeated, but they opened up new worlds […]<img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=469&subd=sciencepublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/7912752014-05-14T14:32:14-04:002014-05-14T14:32:14-04:00Helen RBrain Injury Releases Astonishing Mathematical PowersWhen two thugs bashed Jason Padgett outside a bar they weren’t trying to release skills he never knew he had, sless till conduct one of the most groundbreaking neuroscience experiments of the century. But as it turned out, that’s what … <a href="http://journalpublication.wordpress.com/2014/05/11/brain-injury-releases-astonishing-mathematical-powers/">Continue reading <span>→</span></a><img alt="" src="http://stats.wordpress.com/b.gif?host=journalpublication.wordpress.com&blog=65010642&post=352&subd=journalpublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/4361862014-03-27T18:11:24-04:002015-07-01T03:43:41-04:00Helen R<p><img src="http://cdn.phys.org/newman/gfx/news/2014/after400year.jpg" alt="After 400 years, mathematicians find a new class of solid shapes" width="409" height="309"></p><p>The work of the Greek polymath Plato has kept millions of people busy for millennia. A few among them have been mathematicians who have obsessed about Platonic solids, a class of geometric forms that are highly regular and are commonly found in nature.</p><p>Since Plato’s work, two other classes of equilateral convex polyhedra, as the collective of these shapes are called, have been found: Archimedean solids (including truncated icosahedron) and Kepler solids (including rhombic polyhedra). Nearly 400 years after the last class was described, researchers claim that they may have now invented a new, fourth class, which they call Goldberg polyhedra. Also, they believe that their rules show that an infinite number of such classes could exist. Platonic love for geometry Equilateral convex polyhedra need to have certain characteristics. First, each of the sides of the polyhedra needs to be of the same length. Second, the shape must be completely solid: that is, it must have a well-defined inside and outside that is separated by the shape itself. Third, any point on a line that connects two points in a shape must never fall outside the shape. Platonic solids, the first class of such shapes, are well known. They consist of five different shapes: tetrahedron, cube, octahedron, dodecahedron and icosahedron. They have four, six, eight, twelve and twenty faces, respectively. These highly regular structures are commonly found in nature. For instance, the carbon atoms in a diamond are arranged in a tetrahedral shape. Common salt and fool’s gold (iron sulfide) form cubic crystals, and calcium fluoride forms octahedral crystals. The new discovery comes from researchers who were inspired by finding such interesting polyhedra in their own work that involved the human eye. Stan Schein at the University of California in Los Angeles was studying the retina of the eye when he became interested in the structure of protein called clathrin. Clathrin is involved in moving resources inside and outside cells, and in that process it forms only a handful number of shapes. These shapes intrigued Schein, who ended up coming up with a mathematical explanation for the phenomenon.</p><p>Platonic solids in ascending order of number of faces. nasablueshift<img src="http://cdn.phys.org/newman/gfx/news/2014/1-after400year.jpg" alt="After 400 years, mathematicians find a new class of solid shapes"></p><p>During this work, Schein came across the work of 20th century mathematician Michael Goldberg who described a set of new shapes, which have been named after him, as Goldberg polyhedra. The easiest Goldberg polyhedron to imagine looks like a blown-up football, as the shape is made of many pentagons and hexagons connected to each other in a symmetrical manner (see image to the left).</p><p> </p><p>However, Schein believes that Goldberg’s shapes – or cages, as geometers call them – are not polyhedra. “It may be confusing because Goldberg called them polyhedra, a perfectly sensible name to a graph theorist, but to a geometer, polyhedra require planar faces,” Schein said. Instead, in a new paper in the Proceedings of the National Academy of Sciences, Schein and his colleague James Gayed have described that a fourth class of convex polyhedra, which given Goldberg’s influence they want to call Goldberg polyhedra, even at the cost of confusing others. A crude way to describe Schein and Gayed’s work, according to David Craven at the University of Birmingham, “is to take a cube and blow it up like a balloon” – which would make its faces bulge (see image to the right). The point at which the new shapes breaks the third rule – which is, any point on a line that connects two points in that shape falls outside the shape – is what Schein and Gayed care about most.</p><p><img src="http://cdn.phys.org/newman/gfx/news/2014/2-after400year.jpg" alt="After 400 years, mathematicians find a new class of solid shapes"></p><p>Goldberg polyhedron.</p><p>Craven said, “There are two problems: the bulging of the faces, whether it creates a shape like a saddle, and how you turn those bulging faces into multi-faceted shapes. The first is relatively easy to solve. The second is the main problem. Here one can draw hexagons on the side of the bulge, but these hexagons won’t be flat. The question is whether you can push and pull all these hexagons around to make each and everyone of them flat.” During the imagined bulging process, even one that involves replacing the bulge with multiple hexagons, as Craven points out, there will be formation of internal angles. These angles formed between lines of the same faces – referred to as dihedral angle discrepancies – means that, according to Schein and Gayed, the shape is no longer a polyhedron. Instead they claimed to have found a way of making those angles zero, which makes all the faces flat, and what is left is a true convex polyhedron (see image below). Their rules, they claim, can be applied to develop other classes of convex polyhedra. These shapes will be with more and more faces, and in that sense there should be an infinite variety of them.</p><p><img src="http://cdn.phys.org/newman/gfx/news/2014/3-after400year.jpg" alt="After 400 years, mathematicians find a new class of solid shapes"></p><p>Only the one in the right bottom corner is a convex polyhedra. </p><p>Playing with shapes Such mathematical discoveries don’t have immediate applications, but often many are found. For example, dome-shaped buildings are never circular in shape. Instead they are built like half-cut Goldberg polyhedra, consisting of many regular shapes that give more strength to the structure than using round-shaped construction material. However, there may be some immediate applications. The new rules create polyhedra that have structures similar to viruses or fullerenes, a carbon allotrope. The fact that there has been no “cure” against influenza, or common flu, shows that stopping viruses is hard. But if we are able to describe the structure of a virus accurately, we get a step closer to finding a way of fighting them. If nothing else, Schein’s work will invoke mathematicians to find other interesting geometric shapes, now that equilateral convex polyhedra may have been done with. </p> <a href="http://feeds.wordpress.com/1.0/gocomments/sciencepublication.wordpress.com/353/"><img alt="" src="http://feeds.wordpress.com/1.0/comments/sciencepublication.wordpress.com/353/"></a> <img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=353&subd=sciencepublication&ref=&feed=1" width="1" height="1">After 400 years, mathematicians find a new class of solid shapesThe work of the Greek polymath Plato has kept millions of people busy for millennia. A few among them have been mathematicians who have obsessed about Platonic solids, a class of geometric forms that are highly regular and are commonly found in nature. Since Plato’s work, two other classes of equilateral convex polyhedra, as the […]<img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=353&subd=sciencepublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/4361872014-03-27T18:11:28-04:002014-03-27T18:11:28-04:00Helen RAfter 400 years, mathematicians find a new class of solid shapesThe work of the Greek polymath Plato has kept millions of people busy for millennia. A few among them have been mathematicians who have obsessed about Platonic solids, a class of geometric forms that are highly regular and are commonly … <a href="http://journalpublication.wordpress.com/2014/03/27/after-400-years-mathematicians-find-a-new-class-of-solid-shapes/">Continue reading <span>→</span></a><img alt="" src="http://stats.wordpress.com/b.gif?host=journalpublication.wordpress.com&blog=65010642&post=224&subd=journalpublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/4346652014-03-27T13:51:17-04:002015-07-01T03:43:40-04:00Helen R<p><img src="http://cdn.phys.org/newman/gfx/news/2014/magicandsymm.jpg" alt="Magic and symmetry in mathematics"></p><p>Associate professor of mathematics Ivan Loseu was named a Sloan 2014 Research Fellow for his contributions to the field of representation theory, a sophisticated branch of algebra. </p><p>We live in a three-dimensional world. Despite the many benefits this presents, it also makes for a complicated math problem, according to Northeastern associate professor of mathematics Ivan Loseu. The best a path to a solution, he said, is reducing the number of variables we’re dealing with.</p><p>Consider the Earth moving around the sun. Three variables are needed to describe the position of the Earth because the motion occurs in this three-dimensional space. Newton’s laws in physics allow you to reduce the number of variables even further to two, since the Earth never leaves a certain plane. But, hey, one variable is even better than two. That’s why physicists use the properties of gravitational force to track the Earth’s elliptical trajectory. “The formula for gravity depends only on the distance between the sun and Earth,” Loseu said. “You rotate the picture, but the physical law remains the same.” The reason this problem can be solved using only one variable, he said, can be described in one word: symmetry. “A symmetry is any transformation that preserves your problem,” Loseu explained. The symmetry people typically imagine involves reflecting an image over a single plane to reveal the exact same image—like looking in a mirror. But that’s only one type of symmetry. There are plenty of others. For instance, rotational symmetry describes the fact that rotating an object—say the Earth’s orbital pattern—around an axis doesn’t change its properties. Loseu explained that symmetries allow for reducing the dimension of a system because they can be used to produce preserved quantities; in other words, properties that do not change no matter how much the system changes. This idea of using symmetries to reduce the number of variables is the critical element in Loseu’s research toolbox. He uses it not to solve problems in physics, but rather to solve problems in representation theory, a sophisticated branch of algebra. He was recently named a Sloan 2014 Research Fellow for his contributions to this field. And just as numbers are used in algebra, so too are symmetries. “Take two symmetrical transformations, apply them consequently, and the composition of the two is again a symmetry,” Loseu explained. The more symmetric a system, he continued, the easier the system is to solve. Therefore, identifying symmetries can help simplify a problem and transform it from an unsolvable one to a solvable one. This idea serves as the foundation for what he’ll be focusing on in the first year of his two-year fellowship, presented by the Alfred P. Sloan Foundation. “If all of this is a tree, I’ve told you about only its roots,” he said, noting that a mole may think it has a full picture of an oak or a maple, but until it pops its head through the soil, its perspective is limited. Loseu’s interest in mathematics took shape in elementary school in Belarus. His parents were engineers whose work revolved around the applied sciences, and he often played with the many math books and calculators he could find around the house. He quickly learned that mathematics was “the thing I loved most of all.” As an undergraduate student at Belarusian State University, Loseu initially thought he would pursue work in applied mathematics, but the field didn’t retain the beauty that he appreciated about pure mathematics. “Any scientific discovery involves some kind of magic,” he said. That is, various pieces that may seem to be completely unrelated eventually start to fit together through the fruits of one’s labor. “Since pure math is pure, all this magic is much more clearly seen.</p><p> </p> <a href="http://feeds.wordpress.com/1.0/gocomments/sciencepublication.wordpress.com/352/"><img alt="" src="http://feeds.wordpress.com/1.0/comments/sciencepublication.wordpress.com/352/"></a> <img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=352&subd=sciencepublication&ref=&feed=1" width="1" height="1">Magic and symmetry in mathematicsAssociate professor of mathematics Ivan Loseu was named a Sloan 2014 Research Fellow for his contributions to the field of representation theory, a sophisticated branch of algebra. We live in a three-dimensional world. Despite the many benefits this presents, it also makes for a complicated math problem, according to Northeastern associate professor of mathematics Ivan […]<img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=352&subd=sciencepublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/4346662014-03-27T13:51:19-04:002014-03-27T13:51:19-04:00Helen RMagic and symmetry in mathematicsAssociate professor of mathematics Ivan Loseu was named a Sloan 2014 Research Fellow for his contributions to the field of representation theory, a sophisticated branch of algebra. We live in a three-dimensional world. Despite the many benefits this presents, it … <a href="http://journalpublication.wordpress.com/2014/03/27/magic-and-symmetry-in-mathematics/">Continue reading <span>→</span></a><img alt="" src="http://stats.wordpress.com/b.gif?host=journalpublication.wordpress.com&blog=65010642&post=223&subd=journalpublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/4345942014-03-27T13:41:23-04:002014-03-27T13:41:23-04:00Helen RRussia’s Yakov Sinai wins Abel mathematics prizeRussian mathematician Yakov Sinai won the prestigious Abel mathematics prize for his work in dynamical systems and mathematical physics, Yakov, 78, teaches at US Princeton University and at the Landau Institute for Theoretical Physics in Moscow.“Yakov Sinai is one of … <a href="http://journalpublication.wordpress.com/2014/03/27/russias-yakov-sinai-wins-abel-mathematics-prize/">Continue reading <span>→</span></a><img alt="" src="http://stats.wordpress.com/b.gif?host=journalpublication.wordpress.com&blog=65010642&post=220&subd=journalpublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/4345932014-03-27T13:41:20-04:002015-07-01T03:43:40-04:00Helen R<p>Russian mathematician Yakov Sinai won the prestigious Abel mathematics prize for his work in dynamical systems and mathematical physics,</p><p><img src="http://cdn.phys.org/newman/gfx/news/2014/thisprinceto.jpg" alt="This Princeton University handout photo shows Yakov Sinai of Princeton University, USA, and the Landau Institute for Theoretical"></p><p>Yakov, 78, teaches at US Princeton University and at the Landau Institute for Theoretical Physics in Moscow. “Yakov Sinai is one of the most influential mathematicians of the twentieth century, the Norwegian academy said in a statement. “He has achieved numerous groundbreaking results in the theory of dynamical systems, in mathematical physics and in probability theory.” Crown Prince Haakon will hand Yakov the prize, worth six million kroner (720,000 euros, almost one million dollars) in Oslo on May 20. The Abel prize was created by Norway’s government in 2002 to mark the 200th anniversary of the birth of the great Norwegian mathematician Niels Henrik Abel, in order to make up for the absence of mathematics in the Nobel categories.</p><p> </p> <a href="http://feeds.wordpress.com/1.0/gocomments/sciencepublication.wordpress.com/350/"><img alt="" src="http://feeds.wordpress.com/1.0/comments/sciencepublication.wordpress.com/350/"></a> <img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=350&subd=sciencepublication&ref=&feed=1" width="1" height="1">Russia’s Yakov Sinai wins Abel mathematics prizeRussian mathematician Yakov Sinai won the prestigious Abel mathematics prize for his work in dynamical systems and mathematical physics, Yakov, 78, teaches at US Princeton University and at the Landau Institute for Theoretical Physics in Moscow.“Yakov Sinai is one of the most influential mathematicians of the twentieth century, the Norwegian academy said in a statement.“He […]<img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=350&subd=sciencepublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/4182112014-03-25T13:31:45-04:002015-07-01T03:43:33-04:00Helen R<p><img src="http://images.sciencedaily.com/2014/03/140306130048-large.jpg" alt="" width="505" height="285"></p><p>Millions of high school and college algebra students are united in a shared agony over solving for x and y, and for those to whom the answers don’t come easily, it gets worse: Most preschoolers and kindergarteners can do some algebra before even entering a math class.</p><p>In a just-published study in the journal Developmental Science, lead author and post-doctoral fellow Melissa Kibbe and Lisa Feigenson, associate professor of psychological and brain sciences at Johns Hopkins University’s Krieger School of Arts and Sciences, find that most preschoolers and kindergarteners, or children between 4 and 6, can do basic algebra naturally. “These very young children, some of whom are just learning to count, and few of whom have even gone to school yet, are doing basic algebra and with little effort,” Kibbe said. “They do it by using what we call their ‘Approximate Number System:’ their gut-level, inborn sense of quantity and number.” The “Approximate Number System,” or ANS, is also called “number sense,” and describes humans’ and animals’ ability to quickly size up the quantity of objects in their everyday environments. Humans and a host of other animals are born with this ability and it’s probably an evolutionary adaptation to help human and animal ancestors survive in the wild, scientists say. Previous research has revealed some interesting facts about number sense, including that adolescents with better math abilities also had superior number sense when they were preschoolers, and that number sense peaks at age 35. Kibbe, working in Feigenson’s lab, wondered whether preschool-age children could harness that intuitive mathematical ability to solve for a hidden variable, or in other words, to do something akin to basic algebra before they ever received formal classroom mathematics instruction. The answer was “yes,” at least when the algebra problem was acted out by two furry stuffed animals — Gator and Cheetah — using “magic cups” filled with objects like buttons, plastic doll shoes and pennies. In the study, children sat down individually with an examiner who introduced them to the two characters, each of whom had a cup filled with an unknown quantity of items. Children were told that each character’s cup would “magically” add more items to a pile of objects already sitting on a table. But children were not allowed to see the number of objects in either cup: they only saw the pile before it was added to, and after, so they had to infer approximately how many objects Gator’s cup and Cheetah’s cup contained. At the end, the examiner pretended that she had mixed up the cups, and asked the children — after showing them what was in one of the cups — to help her figure out whose cup it was. The majority of the children knew whose cup it was, a finding that revealed for the researchers that the pint-sized participants had been solving for a missing quantity, which is the essence of doing basic algebra. “What was in the cup was the x and y variable, and children nailed it,” said Feigenson, director of Johns Hopkins Laboratory for Child Development. “Gator’s cup was the x variable and Cheetah’s cup was the y variable. We found out that young children are very, very good at this. It appears that they are harnessing their gut level number sense to solve this task.” If this kind of basic algebraic reasoning is so simple and natural for 4, 5 and 6-year-olds, the question remains why it is so difficult for teens and others. “One possibility is that formal algebra relies on memorized rules and symbols that seem to trip many people up,” Feigenson said. “So one of the exciting future directions for this research is to ask whether telling teachers that children have this gut level ability — long before they master the symbols — might help in encouraging students to harness these skills. Teachers may be able to help children master these kind of computations earlier, and more easily, giving them a wedge into the system.” While the ANS helps children in solving basic algebra, more sophisticated concepts and reasoning are needed to master the complex algebra problems that are taught later in the school age years. Another finding from the research was that an ANS aptitude does not follow gender lines. Boys and girls answered questions correctly in equal proportions during the experiments, the researchers said. Although other research shows that even young children can be influenced by gender stereotypes about girls’ versus boys’ math prowess, “we see no evidence for gender differences in our work on basic number sense,” Feigenson said. Parents with numerically challenged kids shouldn’t worry that not showing a strong aptitude with numbers is a sign that Bobby or Becky will be bad at math. The psychologists say it’s more important to nurture and support young children’s use of the ANS in solving problems that will later be introduced more formally in school. “We find links at all ages between the precision of people’s Approximate Number System and their formal math ability,” Feigenson said. “But this does not necessarily mean that children with poorer precision grow up to be bad at math. For example, children with poorer number sense may need to rely on other strategies, besides their gut sense of number, to solve math problems. But this is an area where much future research is needed.” This research was supported by the National Science Foundation (NSF Living Lab 1113648) and the National Institute of Child Health and Human Development (NIH R01 HD057258). Video: <a href="http://www.youtube.com/watch?v=3rddHtACub8">http://www.youtube.com/watch?v=3rddHtACub8</a> Story Source: The above story is based on materials provided by Johns Hopkins. Note: Materials may be edited for content and length. Journal Reference: Melissa M. Kibbe, Lisa Feigenson. Young children ‘solve forx’ using the Approximate Number System. Developmental Science, 2014; DOI: 10.1111/desc.12177</p> <a href="http://feeds.wordpress.com/1.0/gocomments/sciencepublication.wordpress.com/329/"><img alt="" src="http://feeds.wordpress.com/1.0/comments/sciencepublication.wordpress.com/329/"></a> <img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=329&subd=sciencepublication&ref=&feed=1" width="1" height="1">Are you smarter than a 5-year-old? Preschoolers can do algebraMillions of high school and college algebra students are united in a shared agony over solving for x and y, and for those to whom the answers don’t come easily, it gets worse: Most preschoolers and kindergarteners can do some algebra before even entering a math class. In a just-published study in the journal Developmental […]<img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=329&subd=sciencepublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/4182182014-03-25T13:31:50-04:002014-03-25T13:31:50-04:00Helen RAre you smarter than a 5-year-old? Preschoolers can do algebraMillions of high school and college algebra students are united in a shared agony over solving for x and y, and for those to whom the answers don’t come easily, it gets worse: Most preschoolers and kindergarteners can do some … <a href="http://journalpublication.wordpress.com/2014/03/25/are-you-smarter-than-a-5-year-old-preschoolers-can-do-algebra/">Continue reading <span>→</span></a><img alt="" src="http://stats.wordpress.com/b.gif?host=journalpublication.wordpress.com&blog=65010642&post=200&subd=journalpublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/4182132014-03-25T13:31:46-04:002015-07-01T03:43:33-04:00Helen R<p> </p><p><img src="http://www.newscientist.com/data/images/ns/cms/dn24915/dn24915-1_300.jpg" alt="Becoming a millionaire mathematician? <i>(Image: http://astana.gov.kz)</i>"></p><p>Mathematics is a universal language. Even so, a Kazakh mathematician’s claim to have solved a problem worth a million dollars is proving hard to evaluate – in part because it is not written in English.</p><p>Mukhtarbay Otelbayev of the Eurasian National University in Astana, Kazakhstan, says he has proved the Navier-Stokes existence and smoothness problem, which concerns equations that are used to model fluids – from airflow over a plane’s wing to the crashing of a tsunami. The equations work, but there is no proof that solutions exist for all possible situations, and won’t sometimes “blow up”, producing unrealistic answers.</p><p>In 2000, the Clay Mathematics Institute, now in Providence, Rhode Island, named this one of seven Millennium Prize problems offering $1 million to anyone who could devise a proof.</p><p>Otelbayev claims to have done just that in a paper published in the Mathematical Journal, also based in Kazakhstan. “I worked on the problem on and off, for 30 years,” he told New Scientist, in Russian – he does not speak English.</p><p>Mathematical Babel fish</p><p>However, the combination of the Russian text and the specialist knowledge needed to understand the Navier-Stokes equations means the international mathematical community, which usually communicates in English, is having difficulty evaluating it. Although mathematics is expressed through universal symbols, mathematics papers also contain large amounts of explanatory text.</p><p>“Over the years there have been several alleged solutions to the Navier-Stokes problem that turned out to be wrong,” says Charles Fefferman of Princeton University, who wrote the official formulation of the problem for Clay. “Since I don’t speak Russian and the paper is not yet translated, I’m afraid I can’t say more right now.”</p><p>Otelbayev is a professional, so mathematicians are paying more attention to his proof than is typical for amateur efforts to solve Millennium Prize problems, which are regularly posted online.</p><p>The Russian-speaking Misha Wolfson, a computer scientist and chemist at the Massachusetts Institute of Technology is attempting to spark an online, group effort to translate the paper. “While my grasp on the math is good enough to enable translation up to this point, I am not qualified to say anything about whether or not the solution is any good,” he says.</p><p>Stephen Montgomery-Smith of the University of Missouri in Columbia, who is working with Russian colleagues to study the paper, is hopeful.”What I have read so far does seem valid,” he says “but I don’t feel that I have yet got to the heart of the proof.”</p><p>Otelbayev says that three colleagues in Kazakhstan and another in Russia agree that the proof is correct.</p><p>Burden of proof</p><p>Understandably, a high burden of proof is required to claim the $1 million prize. Clay’s rules say the solution must be published in a journal of “worldwide repute” and remain unchallenged for two years before it can even be considered. Nick Woodhouse, president of the Clay Mathematics Institute, declined to comment on Otelbayev’s proof.</p><p>“It is currently being translated by my students, and will be available soon,” says Otelbayev. He says that he will publish it again once it is translated into English – initially in a second Kazakh journal, and then perhaps abroad.</p><p>To date, only one Millennium Prize problem has been officially solved. In 2002, Grigori Perelman proved the Poincaré conjecture, but later withdrew from the mathematical community and refused the $1 million prize.</p><p>A possible solution for another problem, known as P vs NP, caught mathematicians’ attentions in 2010, but later proved to be flawed. Whether Otelbayev’s proof will share the same fate remains to be seen.</p> <a href="http://feeds.wordpress.com/1.0/gocomments/sciencepublication.wordpress.com/327/"><img alt="" src="http://feeds.wordpress.com/1.0/comments/sciencepublication.wordpress.com/327/"></a> <img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=327&subd=sciencepublication&ref=&feed=1" width="1" height="1">Kazakh mathematician may have solved $1 million puzzle Mathematics is a universal language. Even so, a Kazakh mathematician’s claim to have solved a problem worth a million dollars is proving hard to evaluate – in part because it is not written in English. Mukhtarbay Otelbayev of the Eurasian National University in Astana, Kazakhstan, says he has proved the Navier-Stokes existence and smoothness […]<img alt="" src="http://stats.wordpress.com/b.gif?host=sciencepublication.wordpress.com&blog=64821633&post=327&subd=sciencepublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/4182202014-03-25T13:31:50-04:002014-03-25T13:31:50-04:00Helen RKazakh mathematician may have solved $1 million puzzleMathematics is a universal language. Even so, a Kazakh mathematician’s claim to have solved a problem worth a million dollars is proving hard to evaluate – in part because it is not written in English. Mukhtarbay Otelbayev of the Eurasian … <a href="http://journalpublication.wordpress.com/2014/03/25/kazakh-mathematician-may-have-solved-1-million-puzzle/">Continue reading <span>→</span></a><img alt="" src="http://stats.wordpress.com/b.gif?host=journalpublication.wordpress.com&blog=65010642&post=198&subd=journalpublication&ref=&feed=1" width="1" height="1">tag:tagteam.harvard.edu,2005:FeedItem/2735652013-09-17T11:10:18-04:002015-07-01T03:37:09-04:00phillipsacademy<div><p>These are my live-blogging notes from morning sessions — especially Sal Khan’s keynote — at the New York Times Schools Conference on September 17, 2013 at the Times Center. Here are some high-points from Sal Khan’s keynote, which expand on the basics about reach that you may already know (reaches 200 countries, 8 million registered users, 1.2 billion problems completed):</p><ul><li>Khan Academy (KA) is implementing game mechanics, badges, leveling-up, lots of experimentation, assessment of big data, testing education theory – including growth mindset theory of Carol Dweck at Stanford, e.g. (early returns suggest that she is right).</li><li>What KA is super-focused on is common core alignment, deep mathematics, and real mastery.</li><li>If you or child go to KA today, you will get asked to take an 8-question pretest for math, starting personalization & pathways.</li><li>“We are a tool, but it’s really about the teacher.”</li><li>Blended learning: promising results by teacher Peter McIntosh at Oakland Unity in implementing the KA model in a classroom.</li><li>KA is not about putting kids in front of a computer, but rather to free up time for teachers and learners to do better things with time off the computer.</li><li>There is a great deal of work underway around the world to take KA into communities, via non-profits and schools.</li><li>The #1 creator of content in Mongolian is a 17-year-old girl in an orphanage who just started with KA content at 15 when Cisco engineers spent their vacation setting up tech in Mongolian orphanages.</li><li>Last week: launch of the full Spanish language KA. Brazilian Portugese is next, and on from there.</li><li>“We are at a special moment in history” for education, Khan claims. It’s not a cheap approximation of a good education that we want to provide for kids who are not otherwise able to afford it; we ought to provide a world-class education for anyone. Education is not scarce and only for the few.</li><li>The advanced placement tests will be going up on the site, with Phillips Academy faculty (yay!) working with Khan Academy team members on advanced mathematics, e.g.</li></ul><p><strong>Good questions for Sal Khan from the audience:</strong></p><ul><li>Tension between two statements: 1) teachers matter and 2) any child can get a world-class education for free. How can those both be true?</li><li>Worries about data privacy (as a non-profit, we are careful about that, says Khan).</li><li>Is it ever going to be possible to get a high-school degree just on KA, without ever going to a school? Maybe, says Khan. We should have a mastery-based model rather than a time-based model. Perhaps community colleges will be involved; maybe employers; but in any event, it will be a competency-based model.</li><li>Two questions address the role of community college. Sal likes the combination of a competency-based online assessment with an in-person component, perhaps at community colleges.</li></ul><p><strong>From the “debate” on “whether the university has had its day” (which no one on the panel seems to think has had its day):</strong></p><ul><li>Residential education can be improved. Online education is causing hard questions to be asked about both online and in-person teaching, which are only to the good.</li><li>There’s no “one” single higher-ed experience, as teachers or learners.</li><li>President Martin of Amherst stresses things that can only happen in person, on college campuses: certain important intergenerational relationships, life in the most socio-economically diverse communities that exist anywhere, and the value of the company of attentive others, which should not be foregone, for instance.</li><li>There’s already a disruptive set of innovations underway at the hands of institutions on their own.</li><li>There’s a changing high school demographic that will cause enrollment to flatten in higher ed, says Chancellor Zimpher of the State University of New York.</li><li>We’re not good enough at measuring success in higher ed (possibilities thrown out by panelists and Tweeters: completion; mastery; education-for-education’s sake; learning things like ethics and morality?).</li></ul><a href="http://feeds.wordpress.com/1.0/gocomments/jpalfrey.wordpress.com/10954/"><img alt="" src="http://feeds.wordpress.com/1.0/comments/jpalfrey.wordpress.com/10954/"></a> <img alt="" height="1" src="http://stats.wordpress.com/b.gif?host=jpalfrey.andover.edu&blog=38935766&post=10954&subd=jpalfrey&ref=&feed=1" width="1"></div>Live-Blogged Notes from New York Times Schools for Tomorrow Conference 2013<div>These are my live-blogging notes from morning sessions — especially Sal Khan’s keynote — at the New York Times Schools Conference on September 17, 2013 at the Times Center. Here are some high-points from Sal Khan’s keynote, which expand on … <a href="http://jpalfrey.andover.edu/2013/09/17/live-blogged-notes-from-new-york-times-schools-for-tomorrow-conference-2013/">Continue reading <span>→</span></a><img alt="" height="1" src="http://stats.wordpress.com/b.gif?host=jpalfrey.andover.edu&blog=38935766&post=10954&subd=jpalfrey&ref=&feed=1" width="1"></div>tag:tagteam.harvard.edu,2005:FeedItem/2735652013-09-17T11:10:18-04:002015-07-01T03:37:09-04:00phillipsacademy<div><p>These are my live-blogging notes from morning sessions — especially Sal Khan’s keynote — at the New York Times Schools Conference on September 17, 2013 at the Times Center. Here are some high-points from Sal Khan’s keynote, which expand on the basics about reach that you may already know (reaches 200 countries, 8 million registered users, 1.2 billion problems completed):</p><ul><li>Khan Academy (KA) is implementing game mechanics, badges, leveling-up, lots of experimentation, assessment of big data, testing education theory – including growth mindset theory of Carol Dweck at Stanford, e.g. (early returns suggest that she is right).</li><li>What KA is super-focused on is common core alignment, deep mathematics, and real mastery.</li><li>If you or child go to KA today, you will get asked to take an 8-question pretest for math, starting personalization & pathways.</li><li>“We are a tool, but it’s really about the teacher.”</li><li>Blended learning: promising results by teacher Peter McIntosh at Oakland Unity in implementing the KA model in a classroom.</li><li>KA is not about putting kids in front of a computer, but rather to free up time for teachers and learners to do better things with time off the computer.</li><li>There is a great deal of work underway around the world to take KA into communities, via non-profits and schools.</li><li>The #1 creator of content in Mongolian is a 17-year-old girl in an orphanage who just started with KA content at 15 when Cisco engineers spent their vacation setting up tech in Mongolian orphanages.</li><li>Last week: launch of the full Spanish language KA. Brazilian Portugese is next, and on from there.</li><li>“We are at a special moment in history” for education, Khan claims. It’s not a cheap approximation of a good education that we want to provide for kids who are not otherwise able to afford it; we ought to provide a world-class education for anyone. Education is not scarce and only for the few.</li><li>The advanced placement tests will be going up on the site, with Phillips Academy faculty (yay!) working with Khan Academy team members on advanced mathematics, e.g.</li></ul><p><strong>Good questions for Sal Khan from the audience:</strong></p><ul><li>Tension between two statements: 1) teachers matter and 2) any child can get a world-class education for free. How can those both be true?</li><li>Worries about data privacy (as a non-profit, we are careful about that, says Khan).</li><li>Is it ever going to be possible to get a high-school degree just on KA, without ever going to a school? Maybe, says Khan. We should have a mastery-based model rather than a time-based model. Perhaps community colleges will be involved; maybe employers; but in any event, it will be a competency-based model.</li><li>Two questions address the role of community college. Sal likes the combination of a competency-based online assessment with an in-person component, perhaps at community colleges.</li></ul><p><strong>From the “debate” on “whether the university has had its day” (which no one on the panel seems to think has had its day):</strong></p><ul><li>Residential education can be improved. Online education is causing hard questions to be asked about both online and in-person teaching, which are only to the good.</li><li>There’s no “one” single higher-ed experience, as teachers or learners.</li><li>President Martin of Amherst stresses things that can only happen in person, on college campuses: certain important intergenerational relationships, life in the most socio-economically diverse communities that exist anywhere, and the value of the company of attentive others, which should not be foregone, for instance.</li><li>There’s already a disruptive set of innovations underway at the hands of institutions on their own.</li><li>There’s a changing high school demographic that will cause enrollment to flatten in higher ed, says Chancellor Zimpher of the State University of New York.</li><li>We’re not good enough at measuring success in higher ed (possibilities thrown out by panelists and Tweeters: completion; mastery; education-for-education’s sake; learning things like ethics and morality?).</li></ul><a href="http://feeds.wordpress.com/1.0/gocomments/jpalfrey.wordpress.com/10954/"><img alt="" src="http://feeds.wordpress.com/1.0/comments/jpalfrey.wordpress.com/10954/"></a> <img alt="" height="1" src="http://stats.wordpress.com/b.gif?host=jpalfrey.andover.edu&blog=38935766&post=10954&subd=jpalfrey&ref=&feed=1" width="1"></div>Live-Blogged Notes from New York Times Schools for Tomorrow Conference 2013<div>These are my live-blogging notes from morning sessions — especially Sal Khan’s keynote — at the New York Times Schools Conference on September 17, 2013 at the Times Center. Here are some high-points from Sal Khan’s keynote, which expand on … <a href="http://jpalfrey.andover.edu/2013/09/17/live-blogged-notes-from-new-york-times-schools-for-tomorrow-conference-2013/">Continue reading <span>→</span></a><img alt="" height="1" src="http://stats.wordpress.com/b.gif?host=jpalfrey.andover.edu&blog=38935766&post=10954&subd=jpalfrey&ref=&feed=1" width="1"></div>tag:tagteam.harvard.edu,2005:FeedItem/630742012-11-23T01:46:04-05:002012-11-23T01:46:04-05:00metasj<p>This is a project I’ve had in mind for some time. From where do you draw your favorite problems? For a bit of inspiration, here is an excellent and insightful essay on why math education is so much stronger in Russia (for instance) than in the US and Brazil (for instance), focusing on the appreciation for and use of word problems. </p>
<p><a href="http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pdf">Word Problems in Russia and America</a> by Andrei Toom (↬ Jacob Rus)</p>The Million Problems project: the world’s best problems in each disciplineThis is a project I’ve had in mind for some time. From where do you draw your favorite problems? For a bit of inspiration, here is an excellent and insightful essay on why math education is so much stronger in Russia (for instance) than in the US and Brazil (for instance), focusing on the appreciation [...]tag:tagteam.harvard.edu,2005:FeedItem/257902012-07-19T02:26:07-04:002012-07-19T02:26:07-04:00metasj<p>An idea born in 2010, by <a href="http://www.ams.org/notices/201201/rtx120100005p.pdf">the American Mathematical Society and friends</a>, now bearing fruit at a beautifully burgeoning <a href="http://mpe2013.org/">MPE 2013</a> website.</p>
<p>The mathematics of interest includes everything related to four themes: Discovering the planet, Supporting life on the planet, Human organization on the planet, and Risks to the future of the planet.</p>The Mathematics that Matter for Planet Earth, in 2013An idea born in 2010, by the American Mathematical Society and friends, now bearing fruit at a beautifully burgeoning MPE 2013 website. The mathematics of interest includes everything related to four themes: Discovering the planet, Supporting life on the planet, Human organization on the planet, and Risks to the future of the planet.tag:tagteam.harvard.edu,2005:FeedItem/11582012-03-09T05:32:33-05:002015-07-01T02:56:45-04:00metasj<p><em>From <a href="http://en.wikipedia.org/wiki/J._B._S._Haldane">Haldane</a>‘s 1941 essay in </em>Eureka<em> #6 on “</em><a href="http://www.archim.org.uk/eureka/27/faking.html">The Faking of Genetical Results</a><em>“, reproduced here with appropriate corrections and hyperlinks.</em></p><div><a href="http://en.wikipedia.org/wiki/John_Scott_Haldane">MY FATHER</a> published a number of papers on blood analysis. In the proofs of one of them the following sentence, or something very like it, occurred: “<em>Unless the blood is very thoroughly faked, it will be found that duplicate determinations rarely agree.</em>” Every biochemist will sympathise with this opinion. I may add that the verb “to lake,” when applied to blood, means to break up the corpuscles so that it becomes transparent.<p>In genetical work also, duplicates rarely agree unless they are faked. Thus I may mate two brother black mice, both sons of a black father and a white mother, with two white sisters, and one will beget 10 black and 15 white young; the other 15 black and 10 white. To the ingenuous biologist this appears to be a bad agreement. A mathematician will tell him that where the same ratio of black to white is expected in each family, so large a discrepancy (though how best to compare discrepancies is not obvious) will occur in about 26 percent of all cases. If the mathematician is a rigorist he will say the same thing a little more accurately in a great many more words.</p><p>A biologist who has no mathematical knowledge, and, what is vastly more serious, no scientific honour, will be tempted to fake his results, and say that he got 12 black and 13 white in one family, and 13 black and 12 white in the other. The temptation is generally more subtle. In one of a number of families where equality is expected he gets 19 black and 6 white mice. It looks much more like a ratio of 3 black to 1 white. How is he to explain it? Wasn’t that the cage whose door once seemed to be insecurely fastened? Perhaps the female got out for a while or some other mouse got in. Anyway he had better reject the family. The total gives a better fit to expectation if he does so, by the way. Our poor friend has forgotten the binomial theorem. A study of the expansion of <strong>(1+x/2)<sup>25</sup></strong> would have shown him that as bad a fit or worse would be obtained with a probability of 122753 x 2<sup>-23</sup>, or <strong>.0146</strong>. There is nothing at all surprising in getting one family as aberrant as this in a set of 20. But he is now on a slippery slope.</p><p>He gets his Ph.D. He wants a fellowship, and time is short. But he has been reading <em>Nature</em> and noticed two letters* to that journal of which I was joint author, in which I might appear to have hinted at faking by my genetical colleagues. Thoroughly alarmed, he goes to a venal mathematician. Cambridge is full of mathematicians who have been so corrupted by quantum mechanics that they use series which are clearly <a href="http://en.wikipedia.org/wiki/Divergent_series">divergent</a>, and not even proved to be summable. Interrupting such a one in the midst of an orgy of <a href="http://en.wikipedia.org/wiki/Homi_J._Bhabha">Bhabha</a> and benzedrine, our villain asks for a treatise on faking.</p><div>“I am trying to reconcile <a href="http://en.wikipedia.org/wiki/Edward_Arthur_Milne">Milne</a>, <a href="http://en.wikipedia.org/wiki/Max_Born">Born</a> and <a href="http://en.wikipedia.org/wiki/Paul_Dirac">Dirac</a>, not to mention some facts which don’t seem to agree with any of them, or with Eddington,” replies the debauchee, “and I feel discontinuous in every interval; but here goes.<p>“I suppose you know the hypothesis you want to prove. It wouldn’t be a bad thing to grow a few mice or flies or parrots or cucumbers or whatever you’re supposed to be working on, to see if your hypothesis is anywhere near the facts. Suppose in a given series of families you expect to get four classes of hedgehogs or whatnot with frequencies <em>p</em><sub>1</sub>, <em>p</em><sub>2</sub>, <em>p</em><sub>3</sub>, <em>p</em><sub>4</sub>, and your total is <strong><em>S</em></strong>, I shouldn’t advise you to say you got just <em>Sp</em><sub>1</sub>, <em>Sp</em><sub>2</sub>, <em>Sp</em><sub>3</sub> and <em>Sp</em><sub>4</sub>, or even the nearest whole number. Here is what you’d better do. Say you got <em>A</em><sub>1</sub>, <em>A</em><sub>2</sub>, <em>A</em><sub>3</sub> and <em>A</em><sub>4</sub>, and evaluate</p><p><img src="http://www.archim.org.uk/eureka/27/faking-eq2.png" alt="\chi^2 = ((A_1 - Sp_1)^2 / Sp_1) + ((A_2 - Sp_2)^2 / Sp_2) + ..." width="290" height="42"></p><p>Your <img src="http://www.archim.org.uk/eureka/27/faking-chi2.png" alt="\chi^2" width="21" height="19"> has three degrees of freedom. That is to say you can say you got <em>A</em><sub>1</sub> red, <em>A</em><sub>2</sub> green and <em>A</em><sub>3</sub> blue hedgehogs. But you will then have to say you got <strong><em>S</em></strong>-<em>A</em><sub>1</sub>-<em>A</em><sub>2</sub>-<em>A</em><sub>3</sub> purple ones. Hence the expected value of <img src="http://www.archim.org.uk/eureka/27/faking-chi2.png" alt="\chi^2" width="21" height="19"> is 3, and its standard error is <img src="http://www.archim.org.uk/eureka/27/faking-sqrt6.png" alt="\sqrt{6}" width="29" height="19">; so choose your <em>A</em>‘s so as to give a <img src="http://www.archim.org.uk/eureka/27/faking-chi2.png" alt="\chi^2" width="21" height="19"> anywhere between 1 and 6. This is called faking of the first order. It isn’t really necessary. You might have <img src="http://www.archim.org.uk/eureka/27/faking-eq3.png" alt="p_1 = 9/16" width="61" height="36">, <img src="http://blogs.law.harvard.edu/sj/files/2012/03/p2-equals-p3.png" alt="p_2 = p_3 = 3/16" width="92" height="36">, <img src="http://www.archim.org.uk/eureka/27/faking-eq5.png" alt="p_4 = 1/16" width="61" height="36"> and <strong><em>A</em><sub>1</sub>=9, <em>A</em><sub>2</sub>=<em>A</em><sub>3</sub>=3, <em>A</em><sub>4</sub>=1</strong>. The probability of getting this is <img src="http://www.archim.org.uk/eureka/27/faking-eq6.png" alt="(16! 3^24) / (9! (3!)^2 1! 16^16)" width="99" height="43">, which is only just under <strong>.04</strong>. However, it looks better not to get the exact numbers expected, and if you do it on a population of hundreds or thousands you may be caught out.“Your second order faking is the same sort of thing. Supposing your total is made up of <strong><em>n</em></strong> families, and you say the <strong><em>r</em></strong>th consisted of <em>a</em><sub><em>r</em>1</sub>, <em>a</em><sub><em>r</em>2</sub>, <em>a</em><sub><em>r</em>3</sub>, <em>a</em><sub><em>r</em>4</sub> members of the four classes, <strong><em>s</em><sub><em>r</em></sub></strong> in all: you take</p><p><img src="http://www.archim.org.uk/eureka/27/faking-eq7.png" alt="((a_{r1} - s_r p_1)^2 / s_r p_1) + ((a_{r2} - s_r p_2)^2 / s_r p_2) + ..." width="260" height="42"></p><p>and sum for all values of <strong><em>r</em></strong>. Your total ought to be somewhere near <strong>3<em>n</em></strong>. The standard error is <img src="http://www.archim.org.uk/eureka/27/faking-sqrt6n.png" alt="\sqrt{6n}" width="40" height="19">, and it’s better to be too high than too low. A chap called <a href="http://en.wikipedia.org/wiki/Franz_Moewus">Moewus</a> in Berlin who counted different types of <a href="http://en.wikipedia.org/wiki/Chlamydomonas">algae</a> (or so he said), got such a magnificent agreement between observed and theoretical results, that if every member of the human race had repeated his work once a month for <strong>10<sup>12</sup></strong> years, they might expect as good a fit on one occasion (though not with great confidence). So Moewus certainly hadn’t done any second order faking. Of course I don’t suggest that he did any faking at all. He <a href="http://books.google.com/books/about/Where_the_Truth_Lies.html?id=1rDCaJ25KRAC">may have run into</a> one of those theoretically possible miracles, like the monkey typing out the text of Hamlet by mere luck. But I shouldn’t have a miracle like that in your fellowship dissertation.</p><p>“There is also third order faking. The <strong>4<em>n</em></strong> different components of <img src="http://www.archim.org.uk/eureka/27/faking-chi2.png" alt="\chi^2" width="21" height="19"> should be distributed round their mean in the proper way. That is to say, not merely their mean, but their mean square, cube and so on, should be near the expected values (but not too near). But I shouldn’t worry too much about the higher orders. The only examiner who is likely to spot that you haven’t done them is <a href="http://en.wikipedia.org/wiki/J._B._S._Haldane">Haldane</a>, and he’ll probably be interned as a Red before you send your thesis in. Of course you might get <a href="http://en.wikipedia.org/wiki/Ronald_Fisher">R. A. Fisher</a>, which would be quite as bad. So if you are worried about it you’d better come back and see me later.”</p></div><p>Man is an orderly animal. He finds it very hard to imitate the disorder of nature. In fact the situation is the exact opposite of what the reader of <a href="http://en.wikipedia.org/wiki/William%20Paley">Paley</a>‘s <em>Evidences</em> might expect. But the problem is an interesting one, because it raises in a sharp and concrete way the question of what is meant by randomness, a question which, I believe, has not been fully worked out. The number of independent numerical criteria of randomness which can be applied increases with the number of observations, but much more slowly, perhaps as its logarithm. The criteria now in use have been developed to search for excessive irregularity, that is to say, unduly bad fit between observation and hypothesis. It does not follow that they are so well adapted to a search for an unduly good fit. Here, I believe, is a real problem for students of probability. Its solution might lead to a better set of axioms for that very far from rigorous but none the less fascinating branch of mathematics.</p><p>* see U. Philip and J. B. S. Haldane (1939). <em>Nature</em>, <strong>143</strong>, p. 334. and Hans Grüneberg and J. B. S. Haldane (1940). <em>Nature</em>, <strong>145</strong>, p. 704.</p></div><p>Two closing comments by <a href="http://en.wikipedia.org/wiki/Thomas_William_K%C3%B6rner">T. W. Körner</a>, who found Haldane’s essay worth reprinting in his brilliant <a href="http://www.amazon.com/Fourier-Analysis-T-246-rner/dp/0521389917">textbook</a> on Fourier analysis: </p><blockquote><p>“<em>The reluctance of the scientific community to accept the possibility of fraud is illustrated by the fact that Moewus was still cited in the literature (and even spoken of as a possible Nobel prize winner) until 1953. However, no one else ever succeeded in repeating his experiments…</em></p><p>Unfortunately the statistical war against fraud is now over and the cheaters have won. The kind of tests proposed by Haldane depended on the fact that ‘higher order faking’ required a great deal of computational work. The invention and accessibility of the computer means that the computational work involved has ceased to be a problem for the dishonest scientist. In the physical and biological sciences the possibility that others will attempt to replicate experiments may act as a sufficient deterrent but in purely statistical subjects like sociology and experimental psychology the poblems raised by potential fraud have still to be faced.”</p></blockquote>J. B. S. Haldane on Statistical FraudFrom Haldane‘s 1941 essay in Eureka #6 on “The Faking of Genetical Results“, reproduced here with appropriate corrections and hyperlinks. MY FATHER published a number of papers on blood analysis. In the proofs of one of them the following sentence, or something very like it, occurred: “Unless the blood is very thoroughly faked, it will be found that [...]tag:tagteam.harvard.edu,2005:FeedItem/8952012-03-08T11:02:34-05:002012-06-16T09:16:37-04:00Ali B<p>I just came out of a four-day conference (which shall remain nameless), and it was such a life-affirming, mind-expanding, invigorating experience that I thought I would share my notes. I got doused by a downpour of novel ideas from disparate fields in the many talks I attended. Here’s a sampling, in no particular order:<span></span></p>
<p><strong>Religion</strong></p>
<p>There are no diacritical marks (representing vowels) in the Hebrew Torah – it’s all consonants. As a result, <em>any</em> reading of the Torah is effectively an act of interpretation.</p>
<p><strong>Mathematics </strong></p>
<ul>
<li>The most efficient way to tile a 2-dimensional surface is a hexagon. It has the best ratio of boundary to surface, or uses the least amount of ‘ink’ per surface area as you draw the cells on paper. The honeycomb exemplifies this in nature. Although for hundreds of years mathematicians have intuitively known this to be true, it was only in 1998 that Thomas Hales proved it mathematically.</li>
<li>Until recently, the most efficient way to tile a 3-dimensional space was a Kelvin cell – a modified 3-D hexagon. Then in 1994, Weaire-Phelan cells came along and beat the Kelvin cell’s efficiency by 0.03%.</li>
</ul><p><strong>Problem-solving and creativity</strong></p>
<ul>
<li>Improve your thinking and creativity by making deliberate mistakes. Then, ask yourself, “If this is the wrong way, what’s the <em>right</em> way?”</li>
<li>Second, exaggerate: how would you go about solving this problem if you had no constraints of time or money? That will yield some useful insights.</li>
<li>Third, invert the problem to come up with novel solutions. If traffic’s problem is that the destination is more fun than the traffic, what if we were to make the traffic more fun than the destination? What if you had great books on tape that you really enjoyed listening to in traffic? Then it wouldn’t be as much of a problem.</li>
<li>Fourth, use all ideas. A mistaken answer for one problem may be the correct answer for another. As one wise person said, “The time to work on a problem is after you solve it.” What are the chances that your insight only applies to that one measly problem you were working on? So go forth and find some applications.</li>
</ul><p><strong>Psychology</strong></p>
<ul>
<li>After battle exposure, officers get post-traumatic stress disorder (PTSD) at dramatically lower rates than enlisted soldiers.</li>
<li>Experimentally, hypnosis can alter your dominant sense of time orientation and take it from future-oriented (ant) to present-oriented (grasshopper).</li>
<li>There are at least four types of charisma: authority (exemplified by Obama), kindness (e.g. Dalai Lama), visionary (Steve Jobs) and focus (Bill Clinton). Each can be learned, and you can turn them on or off depending on the situation.</li>
</ul><p><strong>Psychology of love and relationships </strong></p>
<ul>
<li>Adrenaline mediates feelings of attraction. As a result, many women end up going for dominant, intimidating men who scare them a little, mistaking the feeling for attraction. This isn’t always good. Instead, it’s useful for a couple to be aware of this phenomenon and use it to recalibrate and rejuvenate their relationship by engaging in exciting activities together.</li>
<li>Marrying for love is a relatively recent phenomenon in human history.</li>
<li>John Gottman’s studies show that successful responses to a spouse’s <em>bids for attention</em> are the best indicator of marital success. Example of a bid: wife’s reading the paper and says, “Wow, that’s interesting!” (the bid) and the husband says, “What’s interesting, honey?” (the response). Couples who have an 80-85% positive response rate basically don’t get divorced. Even a 50% bid response rate augurs a rapidly rising divorce rate.</li>
<li>Marital satisfaction for men correlates with how much sex he gets and how little criticism he gets. For women, it correlates with how sensitive he is to her emotional cues and how much housework he does. There’s a certain complementarity to this, in that the more housework a man does, the less her wife nags him.</li>
</ul><p><strong>Keeping your brain young </strong></p>
<ul>
<li>Playing first-person shooter video games can reduce mistakes by surgeons.</li>
<li>Travel, new language acquisition, taking a different route to work, and brushing your teeth with the non-dominant hand are ways of waking up your brain and keeping it cognitively fit.</li>
<li>Women should let their husbands talk more because men are most at risk for losing their verbal abilities as they age. Men should let their wives drive more because women are most at risk of losing their visuospatial abilities as they age.</li>
<li>The best thing you can do to keep your brain sharp is to walk briskly 30min per day. You gain the maximum benefit at around 40min/day. There’s no additional benefit beyond 60min/day of exercise.</li>
<li>Social contact is the second most protective agent against cognitive decline. People who have 5 or more close social ties have half the cognitive decline of those who have fewer than five.</li>
<li>Smoking doubles the rate of dementia later in life. Even a casual cigarette or cigar is highly deleterious.</li>
<li>Water and cosmetics can contain lead. Use a PUR water filter (better than Brita), and check the cosmetic products you use against the <a href="http://cosmeticsdatabase.com/">Cosmetics Database</a>.</li>
<li>A Mediterranean diet has been shown to be the best for protecting against cognitive decline. Think 7-9 servings of colorful fruits and vegetables a day. Juices don’t count; eat the whole fruit or vegetable. Wild fish and sardines are good.</li>
<li>Green tea is the best for improving brain function. Oolong tea is second.</li>
<li>Texting while driving increases accident rates by a factor of 23! You may as well be driving while completely wasted off your ass.</li>
<li>A person on average fails to notice a sudden event while driving – eg kid jumping in front of car – around 30% of the time. That goes up to 90% when you’re on the phone, so you’re better off not talking at all, hands-free or not.</li>
<li>Relaxation is a key to optimal brain function. This means meditating or doing absolutely nothing for at least 10min.</li>
<li>Getting slimmer improves brain function. Measure progress by waist size, not weight.</li>
<li>Have a positive emotional outlook. Read <em>Learned Optimism</em> by Martin Seligman.</li>
</ul><p><strong>Mycology (the study of mushrooms)</strong></p>
<ul>
<li>Fungi have ruled the earth twice: once during the Permian-Triassic (P-T) extinction 250 million years ago, and once again at the Cretaceous-Tertiary (K-T) extinction 65 million years ago. At these extinctions, sunlight could not get through the atmosphere and most organisms perished while fungi flourished.</li>
<li>Oyster mushrooms can break down petroleum-based oils (like motor oil) embedded in soil, taking levels from 20,000ppm to 200 ppm</li>
<li>Other fungi have shown to be hyperaccumulators of heavy metals, sequestering heavy isotopes like Cs-137 from the ecosystem.</li>
<li>A mycelium network visually resembles a network of neurons.</li>
<li>Agarikon, red reishi and chaga mushrooms have high activity against viruses. In combination, they are more active than ribavirin against influenza virus</li>
<li>
<em>Cordyceps subsessilis</em> and <em>Cordyceps sinensis</em> have immune-modulating activities that make them candidates for use in organ transplantation and multiple sclerosis treatment, respectively.</li>
<li>Turkey tail mushroom capsules as an adjunct to traditional breast cancer chemotherapy mediated dramatic remissions and multiyear survival for some late-stage cancer cases with otherwise bleak prognoses (3 month survival or less).</li>
</ul><p>This represents but a fraction of what transpired, and I’m deeply grateful to the organizers for creating such a lovely environment conducive to the free exchange of ideas and connections.</p>Notes from a great conferenceI just came out of a four-day conference (which shall remain nameless), and it was such a life-affirming, mind-expanding, invigorating experience that I thought I would share my notes. I got doused by a downpour of novel ideas from disparate fields in the many talks I attended. Here’s a sampling, in no particular order: Religion [...]