This fantastic video by Vi Hart shows you what the math of music looks like in a visual representation — or, should that be "what visual frieze patterns sound like when turned into music"?

Frieze patterns are symmetrical repeating patterns that show up in architecture, art, and even our model of DNA. According to Hart, this video is:

A visual and musical expression of mathematical symmetry groups. The transformations done to the video are equivalent to the transformations done to the notes.

Very cool to watch! Here's the video link.

At Nautilus, which is currently hosting an entire issue on topic of uncertainty, math professor Ayalur Krishnan writes about an idea in set theory that he calls "The Deepest Uncertainty". This is the Continuum Hypothesis — an idea that, paradoxically, can be proven to be unprovable and proven to be something you can't disprove. (And, with that, I've just typed the word "proven" so many times that it has lost all meaning in my brain.)

The uncertainty surrounding the Continuum Hypothesis is unique and important because it is nested deep within the structure of mathematics itself. This raises profound issues concerning the philosophy of science and the axiomatic method. Mathematics has been shown to be “unreasonably effective” in describing the universe. So it is natural to wonder whether the uncertainties inherent to mathematics translate into inherent uncertainties about the way the universe functions. Is there a fundamental capriciousness to the basic laws of the universe? Is it possible that there are different universes where mathematical facts are rendered differently? Until the Continuum Hypothesis is resolved, one might be tempted to conclude that there are.

Read the full story, which explains what set theory and the Continuum Hypothesis actually are. I could that here, but then this link would end up being as long as the story it's trying to link you to. Ahhhh, set theory.

Fabergé Fractals by Tom Beddard, using his WebGL-based fractal engine, Fractal Lab. (via Colossal) ]]> http://boingboing.net/2013/05/21/faberge-fractals.html/feed 5 http://boingboing.net/2013/05/21/unknown-mathematician-makes-hi.html http://boingboing.net/2013/05/21/unknown-mathematician-makes-hi.html#comments Tue, 21 May 2013 12:40:04 +0000 Cory Doctorow http://boingboing.net/?p=231375 Yitang Zhang is a largely unknown mathematician who has struggled to find an academic job after he got his PhD, working at a Subway sandwich shop before getting a gig as a lecturer at the University of New Hampshire. He's just had a paper accepted for publication in Annals of Mathematics, which appears to make a breakthrough towards proving one of mathematics' oldest, most difficult, and most significant conjectures, concerning "twin" prime numbers. According to the Simons Science News article, Zhang is shy, but is a very good, clear writer and lecturer.

For hundreds of years, mathematicians have speculated that there are infinitely many twin prime pairs. In 1849, French mathematician Alphonse de Polignac extended this conjecture to the idea that there should be infinitely many prime pairs for any possible finite gap, not just 2.

Since that time, the intrinsic appeal of these conjectures has given them the status of a mathematical holy grail, even though they have no known applications. But despite many efforts at proving them, mathematicians weren’t able to rule out the possibility that the gaps between primes grow and grow, eventually exceeding any particular bound.

Now Zhang has broken through this barrier. His paper shows that there is some number N smaller than 70 million such that there are infinitely many pairs of primes that differ by N. No matter how far you go into the deserts of the truly gargantuan prime numbers — no matter how sparse the primes become — you will keep finding prime pairs that differ by less than 70 million.

The result is “astounding,” said Daniel Goldston, a number theorist at San Jose State University. “It’s one of those problems you weren’t sure people would ever be able to solve.”

Unknown Mathematician Proves Elusive Property of Prime Numbers [Erica Klarreich/Wired/Simons Science News]

(Photo: University of New Hampshire) ]]> http://boingboing.net/2013/05/21/unknown-mathematician-makes-hi.html/feed 43 http://boingboing.net/2013/05/20/life-of-sally-ride-honored-in.html http://boingboing.net/2013/05/20/life-of-sally-ride-honored-in.html#comments Mon, 20 May 2013 21:27:23 +0000 Xeni Jardin http://boingboing.net/?p=231258 PBS NewsHour.]]>
American astronaut Sally Ride monitors control panels from the pilot's chair on the flight deck in 1983. Photo by Apic/Getty Images, via PBS NewsHour.

Tonight, PBS NewsHour science correspondent Miles O'Brien will serve as master of ceremonies in a Kennedy Center gala honoring the life and legacy of astronaut Sally Ride. The tribute will highlight her impact on the space program and her lifelong commitment to promoting youth science literacy.

Her Sally Ride Science organization reached out to girls, encouraging them to pursue careers in the Science, Technology, Engineering and Math (STEM) fields, where a gender gap persists.

At the PBS NewsHour website, read the column Miles wrote immediately following Ride's death in July 2012, 17 months after she was diagnosed with pancreatic cancer.

The Golden Ratio — that geometric expression of the Fibonacci sequence of numbers (1, 1, 2, 3, 5, etc.) — has influenced the way master painters created art and can be spotted occurring naturally in the seed arrangement on the face of a sunflower. But its serendipitous appearances aren't nearly as frequent as pop culture would have you believe, writes Samuel Arbesman at The Nautilus. In fact, one of the most common examples of mathematical perfection — the chambered nautilus shell — actually isn't. Even math can become part of the myths we tell ourselves as we try to create meaning in the universe.

Image: Golden Ratio, a Creative Commons Attribution (2.0) image from ernestduffoo's photostream

Michael de Podesta has been doing the math on "Grime Dice" -- six sided cubes whose sides average out to 3.5, but whose face values are all radically different:

The interesting thing about these is that the odds of one die beating another are simple to calculate, but shift radically once you start rolling dice in pairs. It's a beautiful piece of counterintuitive probability math:

The amazing property of these dice is discernible when you use them competitively – i.e. you roll one dice against another. If you roll each of them against a normal dice then as you might expect, each dice will win as often as it will lose. But if you roll them against each other something amazing happens.

• Dice A will systematically beat Dice B
• Dice B will systematically beat Dice C

and amazingly

• Dice C will systematically beat Dice A

So the fact that Dice A beats Dice B, and Dice B beats Dice C does not ensure that Dice A will beat Dice C. Wow!

And how about this: If you ‘double up’ and roll 2 Dice  A‘s against 2 Dice B‘s – the odds change around and now the B‘s will beat the A‘s ! Is that really possible? Well yes, and just to convince myself I wrote a Spreadsheet (.xlsx file) and generated the tables at the bottom of the article. If you download it you can change the numbers to try out other combinations.

Amazing Dice: Rediscovering surprise (via Hacker News) ]]> http://boingboing.net/2013/03/15/weird-probabilities-of-non-tra.html/feed 25 http://boingboing.net/2013/03/14/celebrate-pi-day-by-throwi.html http://boingboing.net/2013/03/14/celebrate-pi-day-by-throwi.html#comments Thu, 14 Mar 2013 16:07:37 +0000 Maggie Koerth-Baker http://boingboing.net/?p=218768 Buffon's Needle is an 18th-century experiment in probability mathematics and geometry that can be used as a way to calculate pi through random sampling.]]> Buffon's Needle is an 18th-century experiment in probability mathematics and geometry that can be used as a way to calculate pi through random sampling. This WikiHow posting explains how you can recreate Buffon's Needle at home, by playing with your food. ]]> http://boingboing.net/2013/03/14/celebrate-pi-day-by-throwi.html/feed 3 http://boingboing.net/2013/03/04/calculus-performing-mechanical.html http://boingboing.net/2013/03/04/calculus-performing-mechanical.html#comments Mon, 04 Mar 2013 20:10:01 +0000 Cory Doctorow http://boingboing.net/?p=216399 Dirty Jobs program on tanneries demonstrates the workings of a calculus-performing mechanical calculator. ]]>

A clip from the Discovery Channel's Dirty Jobs program on tanneries demonstrates the workings of a calculus-performing mechanical calculator that measures the surface-area of irregularly shaped hides with a fascinating and clever set of gears, calipers and ratchets.

Dirty Jobs - Tannery Mechanical Surface Integrator (Thanks, Dad!) ]]> http://boingboing.net/2013/03/04/calculus-performing-mechanical.html/feed 35 http://boingboing.net/2013/02/28/why-its-smart-to-be-reckless.html http://boingboing.net/2013/02/28/why-its-smart-to-be-reckless.html#comments Thu, 28 Feb 2013 20:13:36 +0000 Xeni Jardin http://boingboing.net/?p=216009 opinion piece by Chris Arnade on the asymmetry in pay (money for profits, flat for losses), which he describes "the engine behind many of Wall Street’s mistakes" That asymmetry "rewards short-term gains without regard to long-term consequences," Chris writes in a new guest blog at Scientific American.]]> opinion piece by Chris Arnade on the asymmetry in pay (money for profits, flat for losses), which he describes "the engine behind many of Wall Street’s mistakes" That asymmetry "rewards short-term gains without regard to long-term consequences," Chris writes in a new guest blog at Scientific American. "The results? The over-reliance on excessive leverage, banks that are loaded with opaque financial products, and trading models that are flawed." [Scientific American Blog Network]]]> http://boingboing.net/2013/02/28/why-its-smart-to-be-reckless.html/feed 7 http://boingboing.net/2013/02/18/the-worlds-largest-prime-num.html http://boingboing.net/2013/02/18/the-worlds-largest-prime-num.html#comments Mon, 18 Feb 2013 21:25:50 +0000 Maggie Koerth-Baker http://boingboing.net/?p=213798 Philip Bump took the recently discovered 17-million-digit prime number and, six digits at a time, converted it into RGB colors. This is the result.]]>

Philip Bump took the recently discovered 17-million-digit prime number and, six digits at a time, converted it into RGB colors. This is the result.

For instance, suppose our probability space is and is defined by setting for all (here the “experiment” is rolling a single die). Then we are likely interested in more exquisite kinds of outcomes; instead of asking the probability that the outcome is 4, we might ask what is the probability that the outcome is even? This event would be the subset , and if any of these are the outcome of the experiment, the event is said to occur. In this case we would expect the probability of the die roll being even to be 1/2 (but we have not yet formalized why this is the case).

As a quick exercise, the reader should formulate a two-dice experiment in terms of sets. What would the probability space consist of as a set? What would the probability mass function look like? What are some interesting events one might consider (if playing a game of craps)?

Probability Theory — A Primer

(Image: Dice, a Creative Commons Attribution (2.0) image from artbystevejohnson's photostream)]]> http://boingboing.net/2013/01/09/probability-theory-for-program.html/feed 9 http://boingboing.net/2012/12/18/math-fleet-command-a-space-sq.html http://boingboing.net/2012/12/18/math-fleet-command-a-space-sq.html#comments Wed, 19 Dec 2012 00:53:35 +0000 Mark Frauenfelder http://boingboing.net/?p=201388 Kevin Kelly told me about Phil Scale's new iOS application to teach kids arithmetic. It's called Math Fleet and it sounds great.]]>

Kevin Kelly told me about Phil Scale's new iOS application to teach kids arithmetic. It's called Math Fleet and it sounds great. (Phil created Kevin's Asia Grace photobook app, which is also wonderful.)

Phil says:

I've been an independent iOS app developer for four years, and my wife, Jennifer, and I work together from our home in Austin creating games and educational apps. Our newest app is called Math Fleet, an action game set in space where players must use quick math skills to save Earth from invasion, all while dodging asteroids and battling enemy star fighters.

The inspiration for the game came from our sons, Jack, Luke and Dylan (ages 7, 6 and 3) for whom I've downloaded and tried many educational apps and games. At the beginning of the summer we were all home together and I was brainstorming the next game, which I knew I wanted to set in space. I had my sons Jack and Luke playing a math game I had downloaded to earn playing time for the other games they really wanted to play. We all agreed though that the math game I had them playing, which cost me $5, wasn't a very good game and there really wasn't very much math in it. I knew I could write a better game and we started talking about what we would do differently, and in that moment, we decided our mission was to create not just a better game, but the most awesome action math game out there, and that Jack and Luke would be in the driver's seat guiding how the game would take shape.

It became the ultimate geek dad summer for me as I fully committed myself to making their ideas a reality, and some of their ideas were pretty challenging to implement. Such as Luke's idea that the user pilot multiple ships, customizable and upgradable -- the eventual foundation of the game; Or the Patrol Sector concept drawn up by my 7-year-old, Jack.

Throughout the eight month journey, as I coded, they tested prototypes of controls, menus, action sequences, effects, space weapons, and ships. They told me what they liked and what they didn't like, what they'd do differently, and in many cases they would contribute key design concepts delivered as crayon drawings or Lego models. I learned more about game design by watching them, listening, and discussing ideas than I learned from writing my previous two games.

Finally, after eight months and 300,000 lines of code, Math Fleet is officially released, and is hopefully everything we set out to create to create an exciting math game that kids actually want to play, while also challenging their minds by combining fast problem solving with the stress and distraction of piloting a space fleet.

Math Fleet

Secret Science

MP3

(via Schneier) ]]> http://boingboing.net/2012/12/07/crypto-and-bletchley-park-podc.html/feed 1 http://boingboing.net/2012/12/05/momath-more-problems.html http://boingboing.net/2012/12/05/momath-more-problems.html#comments Wed, 05 Dec 2012 20:45:19 +0000 Maggie Koerth-Baker http://boingboing.net/?p=198590 The National Museum of Mathematics (MoMath) will open at a location near the Flatiron Building.]]> The National Museum of Mathematics (MoMath) will open at a location near the Flatiron Building. Opening weekend festivities (and the museum, itself) look really cool. ]]> http://boingboing.net/2012/12/05/momath-more-problems.html/feed 7 http://boingboing.net/2012/12/05/amazing-invisible-work-that-g.html http://boingboing.net/2012/12/05/amazing-invisible-work-that-g.html#comments Wed, 05 Dec 2012 20:00:48 +0000 Cory Doctorow http://boingboing.net/?p=198405
Jeff Moser has a clear, fascinating enumeration of all the incredible math stuff that happens between a server and your browser when you click on an HTTPS link and open a secure connection to a remote end. It's one of the most important (and least understood) parts of the technical functioning of the Internet.

People sometimes wonder if math has any relevance to programming. Certificates give a very practical example of applied math. Amazon's certificate tells us that we should use the RSA algorithm to check the signature. RSA was created in the 1970's by MIT professors Ron *R*ivest, Adi *S*hamir, and Len *A*dleman who found a clever way to combine ideas spanning 2000 years of math development to come up with a beautifully simple algorithm:

You pick two huge prime numbers "p" and "q." Multiply them to get "n = p*q." Next, you pick a small public exponent "e" which is the "encryption exponent" and a specially crafted inverse of "e" called "d" as the "decryption exponent." You then make "n" and "e" public and keep "d" as secret as you possibly can and then throw away "p" and "q" (or keep them as secret as "d"). It's really important to remember that "e" and "d" are inverses of each other.

Now, if you have some message, you just need to interpret its bytes as a number "M." If you want to "encrypt" a message to create a "ciphertext", you'd calculate:

C ≡ Me (mod n)

This means that you multiply "M" by itself "e" times. The "mod n" means that we only take the remainder (e.g. "modulus") when dividing by "n." For example, 11 AM + 3 hours ≡ 2 (PM) (mod 12 hours). The recipient knows "d" which allows them to invert the message to recover the original message:

Cd ≡ (Me)d ≡ Me*d ≡ M1 ≡ M (mod n)

The First Few Milliseconds of an HTTPS Connection (via O'Reilly Radar) ]]> http://boingboing.net/2012/12/05/amazing-invisible-work-that-g.html/feed 7 http://boingboing.net/2012/12/04/free-coursera-calculus-course.html http://boingboing.net/2012/12/04/free-coursera-calculus-course.html#comments Tue, 04 Dec 2012 17:00:33 +0000 Cory Doctorow http://boingboing.net/?p=198158

Robert Ghrist from University of Pennsylvania wrote in to tell us about his new, free Coursera course in single-variable Calculus, which starts on Jan 7. Calculus is one of those amazing, chewy, challenging branches of math, and Ghrist's hand-drawn teaching materials look really engaging.

Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include:

the introduction and use of Taylor series and approximations from the beginning;

* a novel synthesis of discrete and continuous forms of Calculus;

* an emphasis on the conceptual over the computational; and

* a clear, entertaining, unified approach.

They got the OU's Dr Ian Johnston to stress-test a 2X2 Lego in a hydraulic testing machine, increasing the pressure to some 4,000 Newtons, at which point the brick basically melted. Based on this, they calculated the maximum weight a 2X2 brick could bear, and thus the maximum height of a Lego tower:

The average maximum force the bricks can stand is 4,240N. That's equivalent to a mass of 432kg (950lbs). If you divide that by the mass of a single brick, which is 1.152g, then you get the grand total of bricks a single piece of Lego could support: 375,000.

So, 375,000 bricks towering 3.5km (2.17 miles) high is what it would take to break a Lego brick.

"That's taller than the highest mountain in Spain. It's significantly higher than Mount Olympus [tallest mountain in Greece], and it's the typical height at which people ski in the Alps," Ian Johnston says.

"So if the Greek gods wanted to build a new temple on Mount Olympus, and Mount Olympus wasn't available, they could just - but no more - do it with Lego bricks. As long as they don't jump up and down too much."

How tall can a Lego tower get?

More or Less: Opinion polling, Kevin Pietersen, and stacking Lego 30 Nov 2012 [MP3] ]]> http://boingboing.net/2012/12/04/tallest-possible-lego-tower-he.html/feed 71 http://boingboing.net/2012/11/29/some-planets-are-harder-to-lea.html http://boingboing.net/2012/11/29/some-planets-are-harder-to-lea.html#comments Thu, 29 Nov 2012 21:14:00 +0000 Maggie Koerth-Baker http://boingboing.net/?p=197322 At his Psychology Today blog, Michael Chorost delves into a question about exoplanets that I've not really thought much about before — how easy they would be to leave.]]>

At his Psychology Today blog, Michael Chorost delves into a question about exoplanets that I've not really thought much about before — how easy they would be to leave.

Many of the potentially habitable exoplanets that we've found — the ones we call "Earth-like" — are actually a lot bigger than Earth. That fact has an effect — both on how actually habitable those planets would be for us humans and how easily any native civilizations that developed could slip the surly bonds of gravity and make it to outer space.

The good news, says Chorost is that the change in surface gravity wouldn't be as large as you might guess, even for planets much bigger than Earth. The bad news: Even a relatively small increase in surface gravity can mean a big increase in how fast a rocket would have to be going in order to leave the planet. It starts with one equation — SG=M/R^2.

Let’s try it with [exoplanet] HD 40307g, using data from the Habitable Exoplanet Catalog. Mass, 8.2 Earths. Radius, 2.4 times that of Earth. That gets you a surface gravity of 1.42 times Earth.

... it’s amazingly easy to imagine a super-Earth with a comfortable gravity. If a planet had eight Earth masses and 2.83 times the radius, its surface gravity would be exactly 1g. This is the “Fictional Planet” at the bottom of the table. Fictional Planet would be huge by Earth standards, with a circumference of 70,400 miles and an area eight times larger.

Does that mean we could land and take off with exactly the same technology we use here, assuming the atmosphere is similar? Actually, no. Another blogger, who who goes by the moniker SpaceColonizer, pointed out that Fictional Planet has a higher escape velocity than Earth. Put simply, escape velocity is how fast you have to go away from a planet to ensure that gravity can never bring you back. For Earth, escape velocity is about 25,000 miles per hour. Fictional Planet has an escape velocity 68% higher. That’s 42,000 miles per hour.

Read the full story at Psychology Today blogs

Vi Hart is Khan Academy's professional mathemusician. (Yeah, I KNOW, right?) And, this year, she's making the most delightfully nerdy Thanksgiving dinner ever.

It begins with green bean matherole, topped with fried Borromean onion rings. But, besides the fact that it's finished with crispy, delicious hyperbolic geometry, what makes the matherole a matherole?

Vectors. Like the rings, vectors are part of geometry. They've got a magnitude (think: size of the green bean) and they've got a direction (think: which way the green bean is pointing). Most importantly, a single vector can be part of a field of vectors. And that, my friends, is an excellent starting point for a 9 x 13 pan full of beans.

Yes, it's $72. But this 3-D printed metal sculpture/bottle opener is fantastic. And so is its marketing copy.

The problem of beer That it is within a 'bottle', i.e. a boundaryless compact 2-manifold homeomorphic to the sphere. Since beer bottles are not (usually) pathological or "wild" spheres, but smooth manifolds, they separate 3-space into two non-communicating regions: inside, containing beer, and outside, containing you. This state must not remain.

Read the rest of the product description and, you know, maybe buy the bottle opener, too. If you're feeling spendy.

You would think that the 2008 economic meltdown, in which the financial industry broke the entire world when they were blindsided by the fact that housing prices can go down as well as up, might have cut into the idea of Wall Street bankers as geniuses, but evidently not. The weird idea that the titans of investment banking are the smartest people on the planet continues to persist, even among people who ought to know better– another thing that bugged me about Chris Hayes’s Twilight of the Elites was the way he uncritically accepted the line that Wall Street was the very peak of the meritocracy. It’s not hard to see where it originates– Wall Street types can’t go twenty minutes without telling everybody how smart they are– but it’s hard to see why so many people accept such blatant propaganda without question.

Look, Romney was an investment banker and corporate raider at Bain Capital. This is admittedly vastly more quantitative work than, say, being a journalist, but it doesn’t make him a “numbers guy.” The work that they do relies almost as much on luck and personal connections as it does on math– they’re closer to being professional gamblers than mathematical scientists. This is especially true of Bain and Romney, as was documented earlier this year– Bain made some bad bets before Romney got there, and was deep in the hole, and he got them out in large part by exploiting government connections and a sort of hostage-taking brinksmanship, creating a situation in which their well-deserved bankruptcy would’ve created a nightmare for the people they owed money, which bought them enough time for some other bets to pay off.

Romney has no shortage of nerve, and while he creeps me out, he has the sort of faux charm that works well in the finance community. But he’s not a “numbers guy” in any sense that looks meaningful from over here in the land of science. He can do the math needed to add up his personal fortune, but the game that he made his money playing isn’t a rigorously mathematical one– people get rich in finance as much by playing hunches and cutting sharp deals as by crunching numbers. There are people who make their way in that business by taking a rigorously data-driven approach to investing– one of the many things I need to write up for the blog at some point is a review of a forthcoming book called The Physics of Wall Street– but they’re nowhere near a majority of the industry, and Romney’s not one of them.

In particular, you have to remember that, while Nate Silver gives President Obama a 77.4 percent chance of winning the presidential election, that's not the same thing as saying that Obama is going to win.

Suppose a weather forecasting model predicts that the chance of rain in Chicago tomorrow is 75 percent. How do we determine if the model produces accurate assessments of probabilities? After all, the weather in Chicago tomorrow, just like next week’s presidential election, is a “one-off event,” and after the event the probability that it rained will be either 100 percent or 0 percent. (Indeed, all events that feature any degree of uncertainty are one-off events – or to put it another way, if an event has no unique characteristics it also features no uncertainties).

The answer is, the model’s accuracy can be assessed retrospectively over a statistically significant range of cases, by noting how accurate its probabilistic estimates are. If, for example, this particular weather forecasting model predicted a 75 percent chance of rain on 100 separate days over the previous decade, and it rained on 75 of those days, then we can estimate the model’s accuracy in this regard as 100 percent. This does not mean the model was “wrong” on those days when it didn’t rain, any more than it will mean Silver’s model is “wrong” if Romney were to win next week.

What Silver is predicting, in effect, is that as of today an election between a candidate with Obama’s level of support in the polls and one with Mitt Romney’s level of support in those polls would result in a victory for the former candidate in slightly more than three out of every four such elections.

Read the full story at Salon.com

Fibonacci Cabint - 乌托邦建筑设计 - UTOPIA ARCHITECTURE & DESIGN: (via Neatorama) ]]> http://boingboing.net/2012/10/27/fibonacci-drawers-in-a-cabinet.html/feed 16