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593 posts tagged with Math.

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## 15 bits of crypto should be enough for anybody

On May 13, security advisories published by Debian and Ubuntu revealed that, for over a year, their OpenSSL libraries have had a major flaw in their CSPRNG, which is used by key generation functions in many widely-used applications, which caused the "random" numbers produced to be extremely predictable. [lolcat summary] [more inside]

## Abstract concepts vs. concrete examples for teaching math

A new study in Science claims that teaching math is better done by teaching the abstract concepts rather than using concrete examples. From an article by the study authors in Science Mag (requires subscription):

*If a goal of teaching mathematics is to produce knowledge that students can apply to multiple situations, then presenting mathematical concepts through generic instantiations, such as traditional symbolic notation, may be more effective than a series of "good examples." This is not to say that educational design should not incorporate contextualized examples. What we are suggesting is that grounding mathematics deeply in concrete contexts can potentially limit its applicability. Students might be better able to generalize mathematical concepts to various situations if the concepts have been introduced with the use of generic instantiations.*## A Mathematician's Lament

“…if I had to design a mechanism for the express purpose of

*destroying*a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done — I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.” [more inside]## Behind Door Number One...

The Monty Hall Problem has struck again, and this time it’s not merely embarrassing mathematicians. If the calculations of a Yale economist are correct, there’s a sneaky logical fallacy in some of the most famous experiments in psychology." The NY Times' John Tierney reports on new research into cognitive dissonance as examined through the famous Monty Hall Problem. [A previous MetaFilter thread about the Monty Hall Problem: Let's Make A Deal!]

## Matchbox Computing

"Once you've constructed your Hexapawn opponent, it's time to start playing against it... If you play this game repeatedly ... you'll quickly notice that your matchbox opponent plays better and better until it is unbeatable!" Martin Gardner created a game called Hexapawn, and also devised an artificially intelligent opponent you can build yourself out of matchboxes and colored beads. Bonus link: An interview with Martin Gardner.

## A road is a road is a road

Did you know that you can create a simple set of directions to your house that works

*no matter where the recipient starts from?*After 38 years this remarkable conjecture has now been proved by a 63-year old former security guard.## The Geometry of Music

The connection between mathematics and music is often touted in awed, mysterious tones, but it is grounded in hard-headed science. For example, mathematical principles underlie the organization of Western music into 12-note scales. And even a beginning piano student encounters geometry in the "circle of fifths" when learning the fundamentals of music theory. ...according to Dmitri Tymoczko, a composer and music theorist at Princeton University, these well-known connections reveal only a few threads of the hefty rope that binds music and math.The Geometry of Music

See also The Geometry of Musical Chords - Dmitri Tymoczko, Science 7 July 2006: Abstract

See also Dmitri Tymoczko, Composer and Music Theoristvia [more inside]

## golden ratio in the amen break

The Amen Break and the Golden Ratio by mathematics educator and author, Michael S. Schneider. Schneider, having already researched and written about the golden ratio extensively, noticed it right away when hearing the the amen break for the first time (amen break previously on the blue). While some composers have been known to intentionally incorporate fibonacci numbers and the golden ratio into their works, perhaps this is just another one of the many instances of the ratio showing up in nature.

## Big book of algorithms

If you could use a great big free handbook of discrete math and algorithms, Jörg Arndt's fxtbook wants to be your friend. Plain text table of contents to whet your appetite.

## learning math online

## Would you like to play a game?

Fun and games with mathematics and mathematical puzzles (e.g. heart basket, Rubik's Cube, Rubik's Magic, hypercubes, and more) in both English and (with yet more content in) German.

## Men from Nantucket need not apply

The man who runs xkcd

has created the LimerickDB.

Though often quite dirty

There are more that are nerdy;

If you check out the best ones, you'll see.

has created the LimerickDB.

Though often quite dirty

There are more that are nerdy;

If you check out the best ones, you'll see.

## Basic Concepts in Science: A List

Basic Concepts in Science: A List A regularly updated list of blog entries explaining the basics of science and mathematics.

## A Visual Dictionary of Famous Plane Curves

A Visual Dictionary of Famous Plane Curves is an outstanding resource for curves found in nature, man-made objects, and mathematics. Other websites that list exotically named curves also animate how they are created. One of the most unusually named curves, the “Witch of Agnesi”, has an unusual etymology. A number of these curves will be familiar to anyone who has used a Spirograph. Previously.

## Wheel me out

MATSYS Based on the idea that architecture can be understood as a material body with its own intrinsic and extrinsic forces relating to form, growth, and behavior, the studio investigates methodologies of performative integration through geometric and material differentiation.

B_Complex, N_Table, Endless Ocean, Endless Sky (more), P_Wall. more.

B_Complex, N_Table, Endless Ocean, Endless Sky (more), P_Wall. more.

## math for dummies

## There's definitely a method to my madness. Definitely.

Lightning calculator and "mathemagician" Art Benjamin goes through his paces in a 15 minute video.

## 3.14159265itwasthebestoftimesitwastheworstofti...

Ever wondered if and where a specific set of numbers could be found in pi? Maybe you'd like to know where your birthday is? Or maybe just something funny. [prev. here, here] [more inside]

## Squaring the square

Nowhere-neat tilings are actually pretty neat. We all know you can't "square the circle", but do you know the story of squaring the square? (And by the way, even if you can't construct π with a ruler and compass, you can come awfully close without too much work.)

## Cut The Knot

Interactive mathematics miscellany and puzzles, including 75 proofs of the Pythagorean Theorem, an interactive column using Java applets, and eye-opening demonstrations. (Actually, much more.)

## “It's one minute before 12.”

“The greatest shortcoming of the human race is our inability to understand the exponential function.” (RealVideo) Al Bartlett, retired University of Colorado Professor of Physics, gives a stunning hour-long old-school lecture (overhead projector!) on exponential growth and its inevitable results. [more inside]

## Symmetry. Shakespeare. Islamic medicine. Creative writing challenges.

Symmetry. Shakespeare. Islamic medicine. Creative writing challenges. Four podcast series from University of Warwick.

## A Study in the Trickyness Involved in Rocking a Rhyme

## Freely-available textbooks

## .99999...=1

No, I'm sorry, it does. There are some arguments that never end. John or Paul? "Another thing coming" or "Another think coming?" But none has the staying power of "Is 0.999999...., with the 9s repeating forever, equal to 1?" A high school math teacher takes on all doubters. Round 2. Round 3. Refutations of some popular "They're not equal" arguments. Refutations, round 2. You don't have to a mathematician to get in on the fun: .99999=1 discussed on a conspiracy theory website, an Ayn Rand website (where it is accused to violating the "law of identity"), and a World of Warcraft forum. But never, as far as I can tell, on MetaFilter.

## Scholarpedia

Worried about inaccuracies in Wikipedia? Try Scholarpedia, a peer-reviewed encyclopedia, with articles written by experts in their field. [more inside]

## Life is complex: it has both real and imaginary components

More than fifty selected articles from

*The Princeton Companion of Mathematics*(username:*Guest*, password:*PCM*) — a thematically-organized compendium of mathematics and mathematicians from Fields Medal-winner Tim Gowers. [via, previously]## Mathematics vs. Democracy: A Clear Winner or a Tie Game?

The Marquis de Condorcet and Admiral Jean-Charles de Borda were two men of the French Enlightenment who struggled with how to design voting systems that accurately reflected voters' preferences. Condorcet favored a method that required the winner in a multiparty election to win a series of head-to-head contests, but he also discovered that his method easily led to a paradoxes that produced no clear winners. The Borda method avoids the Condorcet paradox by requiring voters to rank choices numerically in order of preference, but this method is flawed because the withdrawal of a last-place candidate can reverse the election results. Mathematicians in the 19th century attempted to design better voting systems, including Lewis Carroll, who favored an early form of proportional representation. Economist Kenneth Arrow argued that designing a perfect voting system was futile, because his "impossibility theorem" proved that it's impossible to design a non-dictatorial voting system that fulfills five basic criteria of fairness. (more inside)

## "Not without my formula sheets, sir."

Even experts don't know what (3 and 3/16ths) times 20 is. But it has something to do with square roots and kinetic energy. [single link to excellent youtube deposition]

## Möbius strip unravelled

## 3 is an odd prime, 5 is an odd prime, 7 is an odd prime, 9 is a very odd prime...

The Prime Game is not really much of a game, but it

*is*a neat & little-known fact about the decimal representation of prime numbers.## ...the models live in the curved space of the hypersphere...

Here are some beautifully rendered views of polytopes, and a few more. The rendering program, Jenn 3D, is free and downloadable, (OS X, Linux, Win) and includes some really dazzling fly-about and camera effects as well as tons of high-dimensional models to explore. There's also a mind-boggling possibility of playing Go on boards in projective space. Via the Math Paint blog, which leads to other interesting places...

## Snakes, in my heart-blood warm'd, that sting my heart!

SlitherLink - a little spatial-numerical puzzle. Here's a better exposition of the rules from the puzzle's inventors, and another collection of puzzles. Oh, and a little survey of other sneaky, snaky puzzles.

## Math + Vishnu = Really Fast Math

*What is the square of 85? In an instant, a 17-year-old boy said without blinking, "7,225."*

*Kamlesh Shetty had used a trick from a quaint concept called Vedic math, a compilation of arithmetic shortcuts believed to have been written by ancient Indians who lived centuries before Christ, during a glorious period in Indian history called the Vedic Age.*More on Vedic math. Still more. And there's a similar system called the Trachtenberg system, invented in a Nazi concentration camp. Where were these guys when I was in the third grade struggling with my times tables?

## It's math set to music!

The Klein Four is a group of math students at the Northwestern University who delight in bringing you various lovely, well-sung A Capella songs infused with their very own and very nerdy flavour. They're not the newest of the web, having released their first CD in 2005, but witty lyrics and five-part harmonizing are definitely worth checking out. I did do a search for this and didn't find anything. Please don't kill me.

## Mathematics in Movies

## Aptitude Schmaptitude!: innumeracy in America

Aptitude Schmaptitude!

*While the state of mathematical incompetence in this country has been much lamented, most famously in Paulos's brilliant 1988 book Innumeracy, it is still tacitly accepted . . . Being incompetent in math has become not only acceptable in this widely innumerate culture, it has almost become a matter of pride. No one goes around showing off that he is illiterate, or has no athletic ability, but declarations of innumeracy are constantly made without any embarrassment or shame.*## Stonehenge math

Solstice/equinox calculations Been hankering to build your own Stonehenge but got stumped at the planning stage? Paul Doherty shows you the math to construct a modern ancient observatory with angles and facings correct for your latitude.

## The Narrow Road

The Narrow Road : in which a professional mathematician guides you through pure mathematics (and touches on tangential issues).

## by the numbers

Pi to 1,000 places on piano is just one of the many catchy tunes on math sonifications. And check out more interesting things on on artist Tom Dukich's site.

## Math is congruent with fun!

You have spacial skills. Apply them in Building Houses 2, on mathsnet.net. Or freestyle in Building Houses 1. Or at night! Oh and also there's like a hundred more puzzles over there too. Some java required.

## Golden Ratios

Did the roof of the Pantheon influence Copernicus? Are the planets of the solar system aligned in accordance with a nearly-forgotten hypothesis known (unfairly) as Bode's Law? A fascinating wide-ranging discussion on BLDGBLOG with Walter Murch, the visionary editor and sound designer for such films as

*The Conversation, Apocalypse Now, The English Patient, THX1138,*and many others. [Murch's film work has previously been discussed here and here.]## E8 Structure Decoded

Math Team Solves the Unsolvable E8

"If you thought writing calculations to describe 3-D objects in math class was hard, consider doing the same for one with 248 dimensions. Mathematicians call such an object E

"If you thought writing calculations to describe 3-D objects in math class was hard, consider doing the same for one with 248 dimensions. Mathematicians call such an object E

_{8}, a symmetrical structure whose mathematical calculation has long been considered an unsolvable problem. Yet an international team of math whizzes cracked E_{8}'s symmetrical code in a large-scale computing project, which produced about 60 gigabytes of data. If they were to show their handiwork on paper, the written equation would cover an area the size of Manhattan."## Impossible Crystals

"This is a story of how the impossible became possible. How, for centuries, scientists were absolutely sure that solids (as well as decorative patterns like tiling and quilts) could only have certain symmetries - such as square, hexagonal and triangular - and that most symmetries, including five-fold symmetry in the plane and icosahedral symmetry in three dimensions (the symmetry of a soccer ball), were strictly forbidden. Then, about twenty years ago, a new kind of pattern, known as a "quasicrystal," was envisaged that shatters the symmetry restrictions and allows for an infinite number of new patterns and structures that had never been seen before, suggesting a whole new class of materials...."

Physicist Paul J. Steinhardt delivers a fascinating lecture (WMV) on tilings and quasicrystals. However, it turns out science was beaten to the punch: a recent paper (PDF) suggests Islamic architecture developed similar tilings centuries earlier.

Physicist Paul J. Steinhardt delivers a fascinating lecture (WMV) on tilings and quasicrystals. However, it turns out science was beaten to the punch: a recent paper (PDF) suggests Islamic architecture developed similar tilings centuries earlier.

## My blog is smarter than your blog.

Alain Connes has a blog. Terry Tao also has a blog. Two Fields medalists blog on open problems, their views on mathematics, and Tomb Raider. Timothy Gowers doesn't have a blog, but does have a compendium of informal essays on topics like Why is multiplication commutative? If you prefer pictures to words: Faces of Mathematics.

## Integrals!

The Integrator is Mathematica's integration capabilities, available over the web. Other online resources from Wolfram include Tones, an automatic music generator, and the venerable Mathworld, an extensive collection of math terms and theorems. (which, yes, has been mentioned previously.)

## Graphs

## Seb Przd's mind-bending photos

Seb Przd's photos specialising in delightful and mind-bending spherical panoramas and conformal mappings.

## Physics simulators. Lots of physics simulators.

PhET - Physics Education Technology offers this astoundingly large library of online physics simulations. Play orbital billiards. Land on a cheesy moon. Experiment with sound. Or try more advanced quantum physics simulators. Still bored? Try the "cutting edge" catagory. Here's the complete index. (Warnings: Frames, Flash, Javascript, Java applets, graphics, sound, quantum timesuck.)

## In Soviet Russia, sponge soaks you

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)