Throughout my years playing around with fractals, the Sierpinski triangle has been a consistent staple. The triangle is named after Wacław Sierpiński and as fractals are wont the pattern appears in many places, so there are many different ways of constructing the triangle on a computer.
All of the methods are fundamentally iterative. The most obvious method is probably the triangle-in-triangle approach. We start with one triangle, and at every step we replace each triangle with 3 subtriangles:
The discussion even veers into cows at some point…but zero mentions of the Menger sponge though? (via hacker news)
People are also emotional, and it turns out an unhappy truck driver can be trouble. Modern routing models incorporate whether a truck driver is happy or not — something he may not know about himself. For example, one major trucking company that declined to be named does “predictive analysis” on when drivers are at greater risk of being involved in a crash. Not only does the company have information on how the truck is being driven — speeding, hard-braking events, rapid lane changes — but on the life of the driver. “We actually have built into the model a number of indicators that could be surrogates for dissatisfaction,” said one employee familiar with the program.
This could be a change in a driver’s take-home pay, a life event like a death in the family or divorce, or something as subtle as a driver whose morning start time has been suddenly changed. The analysis takes into account everything the company’s engineers can think of, and then teases out which factors seem correlated to accident risk. Drivers who appear to be at highest risk are flagged. Then there are programs in place to ensure the driver’s manager will talk to a flagged driver.
In other words, the traveling salesman problem grows considerably more complex when you actually have to think about the happiness of the salesman. And, not only do you have to know when he’s unhappy, you have to know if your model might make him unhappy. Warren Powell, director of the Castle Laboratory at Princeton University’s Department of Operations Research and Financial Engineering, has optimized transportation companies from Netjets to Burlington Northern. He recalls how, at Yellow Freight company, “we were doing things with drivers — they said, you just can’t do that.” There were union rules, there was industry practice. Tractors can be stored anywhere, humans like to go home at night. “I said we’re going to need a file with 2,000 rules. Trucks are simple; drivers are complicated.”
From a new site called Stupid Calculations, here’s what an iPhone consisting of all the iPhone displays ever built would look like plopped down in the midst of Manhattan. Behold the Monophone:
What if Marissa preferred instead to thumb off hundred-dollar bills into an ecstatic crowd of Tumblr owners? Using the stack of hundreds kept handy around the house, I conducted a test that worked out to a rate of 90 bills per minute. It could certainly go faster, but it’s important to make a little flourish with each flick, a self-satisfied grin spread across the face. 90 bills per minute x $100= $9000. $1.1 billion / $9000 per minute = 122,222 minutes or 2037 hours or 84.87 continuous, no-bathroom, no-sleep days.
And what will she be getting for all this generosity? In addition to the office, it buys 175 Six Million Dollar Men; with 175 employees as of May, the acquisition works out to $6,285,714 per employee. That’s $41,904 per pound in livestock terms (175 employees @ an average of 150 lbs= 26,250 lbs total).
Editors of prominent mathematics journals are used to fielding grandiose claims from obscure authors, but this paper was different. Written with crystalline clarity and a total command of the topic’s current state of the art, it was evidently a serious piece of work, and the Annals editors decided to put it on the fast track.
Just three weeks later — a blink of an eye compared to the usual pace of mathematics journals — Zhang received the referee report on his paper.
“The main results are of the first rank,” one of the referees wrote. The author had proved “a landmark theorem in the distribution of prime numbers.”
Rumors swept through the mathematics community that a great advance had been made by a researcher no one seemed to know — someone whose talents had been so overlooked after he earned his doctorate in 1992 that he had found it difficult to get an academic job, working for several years as an accountant and even in a Subway sandwich shop.
“Basically, no one knows him,” said Andrew Granville, a number theorist at the Universite de Montreal. “Now, suddenly, he has proved one of the great results in the history of number theory.”
Reminds me of a certain patent clerk and his theories about time and space. History doesn’t repeat itself, but it does rhyme. (via @daveg)
Erica Klarreich, a Berkeley-based science writer who has a Ph.D. in mathematics and has written about Zhang, says his proof demonstrates the remarkable balance between order and randomness within the prime numbers. “Prime numbers are anything but random — they are completely determined,” Klarreich says. “Nevertheless, they seem to behave in many respects like randomly-sprinkled numbers that eventually display all possible clumps and clusters. Zhang’s work helps to put this conjectured picture of the primes on a solid footing.”
Update: Alec Wilkinson has a profile of Zhang in the Feb 2, 2015 issue of the New Yorker: The Pursuit of Beauty.
Zhang, who also calls himself Tom, had published only one paper, to quiet acclaim, in 2001. In 2010, he was fifty-five. “No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man’s game,” Hardy wrote. He also wrote, “I do not know of an instance of a major mathematical advance initiated by a man past fifty.” Zhang had received a Ph.D. in algebraic geometry from Purdue in 1991. His adviser, T. T. Moh, with whom he parted unhappily, recently wrote a description on his Web site of Zhang as a graduate student: “When I looked into his eyes, I found a disturbing soul, a burning bush, an explorer who wanted to reach the North Pole.” Zhang left Purdue without Moh’s support, and, having published no papers, was unable to find an academic job. He lived, sometimes with friends, in Lexington, Kentucky, where he had occasional work, and in New York City, where he also had friends and occasional work. In Kentucky, he became involved with a group interested in Chinese democracy. Its slogan was “Freedom, Democracy, Rule of Law, and Pluralism.” A member of the group, a chemist in a lab, opened a Subway franchise as a means of raising money. “Since Tom was a genius at numbers,” another member of the group told me, “he was invited to help him.” Zhang kept the books. “Sometimes, if it was busy at the store, I helped with the cash register,” Zhang told me recently. “Even I knew how to make the sandwiches, but I didn’t do it so much.” When Zhang wasn’t working, he would go to the library at the University of Kentucky and read journals in algebraic geometry and number theory. “For years, I didn’t really keep up my dream in mathematics,” he said.
“You must have been unhappy.”
He shrugged. “My life is not always easy,” he said.
In August of 2012, mathematician Shinichi Mochizuki posted a series of four papers online that purported to prove the ABC Conjecture, “a famed, beguilingly simple number theory problem that had stumped mathematicians for decades”. Then, nothing. Or nearly nothing.
The problem, as many mathematicians were discovering when they flocked to Mochizuki’s website, was that the proof was impossible to read. The first paper, entitled “Inter-universal Teichmuller Theory I: Construction of Hodge Theaters,” starts out by stating that the goal is “to establish an arithmetic version of Teichmuller theory for number fields equipped with an elliptic curve…by applying the theory of semi-graphs of anabelioids, Frobenioids, the etale theta function, and log-shells.”
This is not just gibberish to the average layman. It was gibberish to the math community as well.
“Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” wrote Ellenberg on his blog.
But seeming jibberish by a genius might just be solid mathematics, but Mochizuki isn’t doing much to help other mathematicians confirm or refute his assertions. Which raises an interesting point: mathematics isn’t all just logic and truth…there’s a social element to it as well.
“You don’t get to say you’ve proved something if you haven’t explained it,” she says. “A proof is a social construct. If the community doesn’t understand it, you haven’t done your job.”
N Is a Number is an hour-long documentary about Hungarian mathematician Paul Erdős.
Erdős was famously a prolific mathematician who collaborated widely….he coauthored over 1500 papers with 500 different collaborators. He was also a homeless methamphetamine user.
In 2006, Garth Sundem and John Tierney published an equation in the NY Times that attempted to predict celebrity marriage crackups using a few metrics: age, fame, sexiness, etc. The pair recently modified the equation based on the evidence of the last five years and surprisingly, the equation is simpler.
What went right with them — and wrong with our equation? Garth, a self-professed “uber-geek,” has crunched the numbers and discovered a better way to gauge the toxic effects of celebrity. Whereas the old equation measured fame by counting the millions of Google hits, the new equation uses a ratio of two other measures: the number of mentions in The Times divided by mentions in The National Enquirer.
“This is a major improvement in the equation,” Garth says. “It turns out that overall fame doesn’t matter as much as the flavor of the fame. It’s tabloid fame that dooms you. Sure, Katie Holmes had about 160 Enquirer hits, but she had more than twice as many NYT hits. A high NYT/ENQ ratio also explains why Chelsea Clinton and Kate Middleton have better chances than the Kardashian sisters.”
Garth’s new analysis shows that it’s the wife’s fame that really matters. While the husband’s NYT/ENQ ratio is mildly predictive, the effect is so much weaker than the wife’s that it’s not included in the new equation. Nor are some variables from the old equation, like the number of previous marriages and the age gap between husband and wife.
Now available in its entirety on YouTube, a 95-minute documentary on physicist Richard Feynman called No Ordinary Genius.
The excellent film on Andrew Wiles’ search for the solution to Fermat’s Last Theorem is available as well (watch the first two minutes and you’ll be hooked).
You are comfortable with feeling like you have no deep understanding of the problem you are studying. Indeed, when you do have a deep understanding, you have solved the problem and it is time to do something else. This makes the total time you spend in life reveling in your mastery of something quite brief. One of the main skills of research scientists of any type is knowing how to work comfortably and productively in a state of confusion.
The distance between the metal bands holding the cylindrical structure together decreases from top to bottom because the pressure the water exerts increases with depth. The top band only needs to fight against the water at the very top of the tower but the bottom bands have to hold the entire volume from bursting out.
And in celebration, this is my new favorite fact about pi: we have calculated pi out to over 6.4 billion digits but only 39 of them are needed to calculate the circumference of a circle as big as the universe “with a precision comparable to the radius of a hydrogen atom”. (via @santheo)
This equation’s initial purpose, he wrote, was to put meaningful prices on the terrestrial exoplanets that Kepler was bound to discover. But he soon found it could be used equally well to place any planet-even our own-in a context that was simultaneously cosmic and commercial. In essence, you feed Laughlin’s equation some key parameters — a planet’s mass, its estimated temperature, and the age, type, and apparent brightness of its star — and out pops a number that should, Laughlin says, equate to cold, hard cash.
At the time, the exoplanet Gliese 581 c was thought to be the most Earth-like world known beyond our solar system. The equation said it was worth a measly $160. Mars fared better, priced at $14,000. And Earth? Our planet’s value emerged as nearly 5 quadrillion dollars. That’s about 100 times Earth’s yearly GDP, and perhaps, Laughlin thought, not a bad ballpark estimate for the total economic value of our world and the technological civilization it supports.
Nothing in the news media yet, but many folks on Twitter and colleague Nassim Taleb are reporting that the father of fractal geometry is dead at age 85. We’re not there yet, but someday Mandelbrot’s name will be mentioned in the same breath as Einstein’s as a genius who fundamentally shifted our perception of how the world works.
This 45-minute documentary on Andrew Wiles’ proof of Fermat’s Last Theorem is surprisingly powerful and emotional. Give it until 1:45 or so and you’ll want to watch the whole thing. The film is not really about math; it’s about all of those movie trailer cliches — “one man!”, “finds the truth!”, “fights the odds!”, etc. — except that this is actually true and poignant.
With Friedman’s work, it seems Gödel’s delayed triumph has arrived: the final proof that if there is a universal grammar of numbers in which all facets of their behaviour can be expressed, it lies beyond our ken.
But don’t worry…”the most severe implications are philosophical”. Phew?
Here’s the entire text of a talk given at math, magic, and puzzle gathering (attendees included Stephen Wolfram and John Horton “Game of Life” Conway) by Gary Foshee:
I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?
The first thing you think is “What has Tuesday got to do with it?” Well, it has everything to do with it.
The key word in the puzzle is “probability”, which is not a very well understood term outside of the mathematics community. The full answer is at the end of the article.
After analyzing dozens of Hollywood films, a team of researchers has found evidence that the visual rhythm of movies at the shot level matches a pattern called the 1/f fluctuation, the same pattern that is found in dozens of natually occurring phenomena, including the length of the human attention span.
These results suggest that Hollywood film has become increasingly clustered in packets of shots of similar length. For example, action sequences are typically a cluster of relatively short shots, whereas dialogue sequences (with alternating shots and reverse-shots focused sequentially on the speakers) are likely to be a cluster of longer shots. In this manner and others, film editors and directors have incrementally increased their control over the visual momentum of their narratives, making the relations among shot lengths more coherent over a 70-year span.
Modern action movies are particularly adept at matching the audience’s attention span in this manner. The full paper is available here.
I was really into fractals in college (I know…) when I was making rave flyers (I know!) for a friend’s parties in Iowa (I know! I know! Shut up already!). Anyway, the thing that I really used to love doing with this fractal application that I had on my computer was zooming in to different parts of the familiar Mandelbrot set as far as I could. I never got very far…between 5 or 6 zooms in, my Packard Bell 486/66 (running Windows 3.11) would buckle under the computational pressure and hang. Therefore, I absolutely love this extremely deep HD zoom into the Mandelbrot set:
Just how deep is this computational rabbit hole?
The final magnification is e.214. Want some perspective? a magnification of e.12 would increase the size of a particle to the same as the earths orbit! e.21 would make a particle look the same size as the milky way and e.42 would be equal to the universe. This zoom smashes all of them all away. If you were “actually” traveling into the fractal your speed would be faster than the speed of light.
After awhile, the self-similarity of the thing is almost too much to bear; I think I went into a coma around 5:00 but snapped to in time for the exciting (but not unexpected) conclusion. Full-screen in a dark room is recommended.
Update: This 46-minute video seems to be the deepest fractal zoom out there right now, with a zoom level of 10^10000.
The magnification factor is so much less in the video above but that one’s more fun/artistic. And 10^10000 is such an absurdly large number1 that there’s no way to think about it in physical terms…the zoom factor from the size of the universe to the smallest measurable distance (the Planck length) is only about 10^60.
I’ll be writing about the elements of mathematics, from pre-school to grad school, for anyone out there who’d like to have a second chance at the subject — but this time from an adult perspective. It’s not intended to be remedial. The goal is to give you a better feeling for what math is all about and why it’s so enthralling to those who get it.
More subject blogs like this, please. There are lots of art, politics, technology, fashion, economics, typography, photography, and physics blogs out there, but almost none of them appeal to the beginner or interested non-expert. (thx, steve)
The ham sandwich theorem is also sometimes referred to as the “ham and cheese sandwich theorem”, again referring to the special case when n = 3 and the three objects are
1. a chunk of ham, 2. a slice of cheese, and 3. two slices of bread (treated as a single disconnected object).
The theorem then states that it is possible to slice the ham and cheese sandwich in half such that each half contains the same amount of bread, cheese, and ham. It is possible to treat the two slices of bread as a single object, because the theorem only requires that the portion on each side of the plane vary continuously as the plane moves through 3-space.
No idea how this is related to the I Cut You Choose conundrum.
A round pizza with radius ‘z’ and thickness ‘a’ has the volume pi*z*z*a. That and other math jokes are available on Wikipedia. Don’t you love it when people explain jokes:
In this case, DEAD refers to a hexadecimal number (57005 base 10), not the state of being no longer alive.
If you and someone else hate the same third person, but like each other, balance theory says you’re golden — all three can persist without changing their opinions. On the other hand, if all three of you despise the others, it’s an unstable triad, as well as a wildly common plot point for crime movies. While there are numerous resolutions — one person changes his preference toward another, a relationship tie is cut — another route back to stability, albeit a messy one, is the gunning down of at least one person.
Arbesman has some videos and stills on his web site from the movies mentioned in the article as well as the relevant mathematical materials.
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