kottke.org posts about mathematics

Fantastic recursive book coverJun 23 2014


Naive Set Theory Cover




Cheese chartsJun 18 2014

Camembert chart


The infinitely large .zip fileJun 16 2014


Louis C.K. seeks cure for the Common CoreMay 01 2014

"My kids used to love math! Now it makes them cry." So tweeted Louis C.K. earlier this week. His opinion of the new math and standardized tests is echoed by a lot of parents who "have found themselves puzzled by the manner in which math concepts are being presented to this generation of learners as well as perplexed as to how to offer the most basic assistance when their children are struggling with homework." Rebecca Mead in the The New Yorker: Louis C.K. Against the Common Core.

So, what exactly does pi mean?Mar 13 2014


Pi Calc

Always buy the bigger pizzaFeb 26 2014

Planet Money: always buy the bigger pizza because geometry.




The sum of all positive integersJan 16 2014





Short Errol Morris film on Benoit MandelbrotNov 18 2013



How to win at The Price is RightNov 15 2013

It turns out that for many of the games on The Price is Right, a simple application of game theory is all you need to greatly increase your chances of winning. You don't even need to know any of the prices.

In one instance, when Margie was the last contestant to bid, she guessed the retail price of an oven was $1,150. There had already been one bid for $1,200 and another for $1,050. She therefore could only win if the actual price was between $1,150 and $1,200. Since she was the last to bid, she could have guessed $1051, expanding her range by almost $100 (any price from $1051 to $1199 would have made her a winner), with no downside. What she really should have done, however, is bid $1,201. Game theory says that when you are last to bid, you should bid one dollar more than the highest bidder. You obviously won't win every time, but in the last 1,500 Contestants' Rows to have aired, had final bidders committed to this strategy, they would have won 54 percent of the time.

See also how a man named Terry Kniess solved The Price is Right.

The Sierpinski triangleOct 09 2013

Sierpinski Curved

More than you've ever wanted to know about the Sierpinski triangle.




The unhappy traveling salesman problemSep 06 2013

Solving the traveling salesman problem is difficult enough without having to consider the happiness of the salesman. But Tom Vanderbilt reports that's essentially what UPS, FedEx, and the like have had to do.

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."

Stupid calculationsMay 31 2013

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:




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).

Unknown mathematician hits a home runMay 20 2013

Yitang Zhang, an unknown mathematician who worked at Subway while trying to find an academic position earlier in his career, has written a paper that makes significant progress towards understanding the twin prime conjecture, "one of mathematics' oldest problems".

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)

The proof "from outer space"May 13 2013

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."


"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."

(via @dunstan)

N Is a Number: A Portrait of Paul ErdosOct 16 2012



The celebrity marriage duration equationApr 17 2012

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.

Richard Feynman, No Ordinary GeniusApr 03 2012



How to find Waldo with MathematicaMar 21 2012

Someone asked on Stack Overflow how one might go about finding Waldo using Mathematica and someone replied with a solution.

Here's Waldo

(via mlkshk)

The thing about 998,001 is...Jan 30 2012

If you divide 1 by the number 998,001, you get a list of all the three digit numbers in order except 998. Like so:


Math! (via mlkshk)

Neat multiplication visualizationJan 20 2012

According to this YouTube video, Japanese do multiplication by drawing lines like this:

(via ★vuokko)

What's it like to deeply understand math?Jan 05 2012

Another Quora gem: an answer to the question "what is it like to have an understanding of very advanced mathematics?"


(via @pomeranian99)

NYC water towersOct 20 2011


Water Tower


Menger sponge built from Post-It notesSep 28 2011


Menger sponge Post It


Today is pi dayMar 14 2011


How much is a planet worth?Feb 04 2011




Mathematical doodlingDec 07 2010

This is a wonderfully whimsical introduction to doodling by way of graph theory, snakes, Oroborous and mobius strips. Oh, and the Mobiaboros.

(via vulture)

Benoit Mandelbrot, RIPOct 15 2010



Habits of mindSep 10 2010

Originally written for mathematics students, this list of useful habits of mind is applicable to nearly anyone doing anything.

Fermat's Last TheoremAug 27 2010

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.

Basic rules of arithmetic may be brokenAug 18 2010

And not just broken but unrepairable without the addition of uncertainty. Gödel's incompleteness theorems aren't even the half of it.

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?

Monday puzzle time!May 25 2010





The formula for Hollywood moviesMar 09 2010

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.

Insanely deep fractal zoomFeb 11 2010


Just how deep is this computational rabbit hole?



Found functionsFeb 05 2010


Found Functions

(via snarkmarket)

Math for non-expertsFeb 01 2010

Mathematician Steven Strogatz is doing what sounds like a fascinating series of posts on mathematics for adults. From the initial post:

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 theoremDec 11 2009



1. a chunk of ham,
2. a slice of cheese, and



Have you seen this fractal?Nov 03 2009

Circle gasket



Pizza piOct 15 2009



High larious. (via reddit)

Mathematics at the moviesSep 11 2009

Sam Arbesman highlights the use of mathematics in movies, including game theory (The Dark Knight), epidemiology (zombie movies), and balance theory (Reservoir Dogs).

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.

Gambling strategyAug 13 2009

A relatively short article on the mathematics of gambling.




Mathematics in Infinite JestAug 11 2009




Cicadas matingAug 05 2009




Radiolab on randomnessJun 19 2009

Newish episode of Radiolab about randomness: Stochasticity.

How big a role does randomness play in our lives? Do we live in a world of magic and meaning or ... is it all just chance and happenstance? To tackle this question, we look at the role chance and randomness play in sports, lottery tickets, and even the cells in our own body. Along the way, we talk to a woman suddenly consumed by a frenzied gambling addiction, two friends whose meeting seems purely providential, and some very noisy bacteria.

Who Can Name the Bigger Number?Jun 08 2009

Regarding the game of Who Can Name the Bigger Number?, Scott Aaronson shows that while 9^9^9^9 might cut the mustard in the first couple of rounds, the numbers and the notation used to express them get much more complicated.

Exponentials are familiar, relevant, intimately connected to the physical world and to human hopes and fears. Using the notational systems I'll discuss next, we can concisely name numbers that make exponentials picayune by comparison, that subjectively speaking exceed 9^9^9^9 as much as the latter exceeds 9.

See also the Wikipedia entry for large numbers.

Math and the City (and the elephant)May 20 2009

This should provide a sufficient amount of "whoa" for the day: mathematically speaking, how are elephants and big cities the same? A: both cities and elephants have developed a similar level of efficiency in the distribution of resources and transportation.

Geoffrey West of the Santa Fe Institute and his colleagues Jim Brown and Brian Enquist have argued that a 3/4-power law is exactly what you'd expect if natural selection has evolved a transport system for conveying energy and nutrients as efficiently and rapidly as possible to all points of a three-dimensional body, using a fractal network built from a series of branching tubes -- precisely the architecture seen in the circulatory system and the airways of the lung, and not too different from the roads and cables and pipes that keep a city alive.

(thx, john)

Typesetting the biggest primeApr 13 2009




Perfect pancake recipeFeb 25 2009

Or so says a mathematics teacher from the UK. The formula is:


In the complex formula L represents the number of lumps in the batter and C equals its consistency. The letter F stands for the flipping score, k is the ideal consistency and T is the temperature of the pan. Ideal temp of pan is represented by m, S is the length of time the batter stands before cooking and E is the length of time the cooked pancake sits before being eaten. The closer to 100 the result is -- the better the pancake.

However, a commenter notes:

According to that formula, if you left the pancake batter standing for ten years, (s-e) would be large, and so the pancake would be near perfect. If you let it stand for the same time as you left the pancake to cool, (s-e) would be zero and the pancake would be infinitely bad.

The suggestion to serve with sugar and lemon is clearly wrong as well. See also the formula for how tall high heels can go. (via buzzfeed)

Archimedes developed calculus?Jan 23 2009

According to a Greek text that was overwritten with Christian prayers, Archimedes worked out some of the principles of calculus over 1900 years before Newton and Leibniz. He called it The Method.

In The Method, Archimedes was working out a way to compute the areas and volumes of objects with curved surfaces, which was also one of the problems that motivated Newton and Leibniz. Ancient mathematicians had long struggled to "square the circle" by calculating its exact area. That problem turned out to be impossible using only a straightedge and compass, the only tools the ancient Greeks allowed themselves. Nevertheless, Archimedes worked out ways of computing the areas of many other curved regions.

Much more is explained in the book The Archimedes Codex. The entire text is available for free on Google Books (you might need this). (via long now)

Feynman on school textbooksOct 22 2008

Richard Feynman on the "perpetual absurdity" of school textbooks.





"I wound it up."

"And how did you get moving?"

"From eating."


(via rw)

Power of noodlesOct 03 2008

I know I've posted this one before but I'm probably gonna post it each time I run across it.

That's chef Kin Jing Mark stretching and dividing dough into super-thin noodles. Seeing this when I was a kid made a great impression on me about the wonder of mathematics.

23 of the toughest math questionsOct 02 2008

DARPA is soliciting research proposals for people wishing to solve one of twenty-three mathematical challenges, many of which deal with attempting to find a mathematical basis underlying biology.

What are the Fundamental Laws of Biology?: This question will remain front and center for the next 100 years. DARPA places this challenge last as finding these laws will undoubtedly require the mathematics developed in answering several of the questions listed above.

(via rw)

GuesstimationsSep 17 2008

The results of some back-of-the-envelope calculations:

- The air in the Empire State Building weighs about 4 million pounds.
- The energy consumption of the world's population will be greater than the energy coming from the sun in less than 500 years. (Peak photons?)

What's surprising about such estimates is how often they are very close to the reality. This is especially true in a multi-step approximation, where over- and underestimates at various steps tend to cancel each other out, usually resulting in something not too far off from the truth.

Both Microsoft and Google use questions like these as part of their job interview process. We did a bunch of them in my freshman physics class; I loved them.

Update: The NY Times says that these are called Fermi Problems. (via gerry)

GuillochesSep 04 2008



Population powers of tenJul 24 2008

Population densities in the United States vary over nine orders of magnitude.

In case you're wondering, the most densely populated block group is one in New York County, New York -- 3,240 people in 0.0097 square miles, for about 330,000 per square mile. The least dense is in the North Slope Borough of Alaska -- 3 people in 3,246 square miles, or one per 1,082 square miles. The Manhattan block group I mention here is 360 million times more dense than the Alaska one; population densities vary over a huge range.

That's approximately the same range from the height of an iPod to the diameter of the Earth. (via fakeisthenewreal)

Benoit Mandelbrot and Paola Antonelli talk about,Apr 17 2008



Bruce Bukiet is back with his annualApr 03 2008



O'Reilly Media and Wolfram Research are goingFeb 20 2008



Hilbert should be available before the end of the year.

Math and science-themed tattoosFeb 12 2008


What do you get when you crossFeb 07 2008

What do you get when you cross an ouroboros with a Möbius strip?

M.C. Escher knew: The dreaded Mouroboröbius!

Feast your eyes on this bit of loveliness.

The mathematics of well-balanced stacks of blocks.Dec 31 2007

The mathematics of well-balanced stacks of blocks. When I was a kid, I would make stacks like these for hours on end...constructing buildings was dull in comparison.

Infinite JestDec 10 2007

Infinite Jest










Sierpinski Gasket


A Beddian Year is one in whichNov 21 2007

A Beddian Year is one in which your age matches the last two digits of the year you were born.

What's sort of great about it is that it will happen to everybody if you live long enough. If you were born in 2000, it happens instantaneously. The people who were born at the end of the century have to take care of themselves.

Chart of the possible shapes and formsNov 20 2007


Update: kottke.org reader Flip passes along this article about the wavy edges of flowers, leaves and even garbage bags, summarizing it thusly:


(thx, flip)

20-minute video about how to turn aNov 14 2007

20-minute video about how to turn a sphere inside out without creases or sharp corners. Way more interesting than it sounds...watch until about 1:45 to have your mind blown a little bit. (via 3quarksdaily)

Even though a Rubik's Cube has about 43Sep 12 2007

Even though a Rubik's Cube has about 43 quintillion (that's 43000000000000000000) possible configurations, it's been proven possible to solve a cube starting in any one of those configurations in 26 moves or less. "Most researchers believe that just 20 steps are enough to solve any Rubik's Cube, but no one has proved it yet."

Cadaeic Cadenza is a 3834-word story bySep 10 2007

Cadaeic Cadenza is a 3834-word story by Mike Keith where each word in sequence has the same number of letters as the corresponding digit in pi. (thx, mark, who has more info on constrained writing) Related: The Feynman point is the sequence of six 9s which begins 762 digits into pi. "[Feynman] once stated during a lecture he would like to memorize the digits of pi until that point, so he could recite them and quip 'nine nine nine nine nine nine and so on.'"

Before the 2007 season started, mathematician Bruce Bukiet'sSep 06 2007

Before the 2007 season started, mathematician Bruce Bukiet's mathematical model predicted the NY Yankees would win 110 games this year...they might win 90 based on their current pace. Related: check out how the team salary vs. performance graph is shaping up as we move into the last few weeks of the 2007 season.

I love YouTube. This is a videoApr 05 2007



Perl one-liner for checking if a numberMar 20 2007

Perl one-liner for checking if a number is prime. (via daringfireball)

Mathematician Terence Tao won both the FieldsMar 14 2007

Mathematician Terence Tao won both the Fields Medal and a MacArthur genius grant last year. To dumb it down for all you Fields Medal non-winners out there, that's like doing Miss America and Miss Universe at the same time.

kottke.org is 9 years old todayMar 14 2007




The Golden Ratio and its appearance inMar 07 2007

The Golden Ratio and its appearance in the music of Nintendo's Zelda games.

A list of distinctive numbers and whyFeb 21 2007

A list of distinctive numbers and why they're so special.

Update: Then again, maybe every number is interesting. (thx, edmund)

Rule of thumb to avoid photographing peopleFeb 18 2007

Rule of thumb to avoid photographing people with their eyes closed: divide the number of people by three (or by two if the light is bad). That means that if you're taking a photo of 12 people, you need to take at least 4 photos to have a good chance of getting a photo with everyone's eyes open. (via photojojo)


Formulas for writing reviews of music, restaurants,Jan 31 2007

Formulas for writing reviews of music, restaurants, and boutique clothing stores (???). "What the a lacked in x, the b made up for in y. Where a = a menu item, x = a characteristic often used in conjunction with fast cars, b = a menu item, and y = an adjective generally used by Victorian novelists to describe a young woman." (via airbag)

Tupper's Self-Referential Formula is an equation thatJan 25 2007

Tupper's Self-Referential Formula is an equation that when graphed, displays the formula itself.

Update: In computing, a quine is a program which "produces its complete source code as its only output". (thx, sam)

A look at Saks Fifth Avenue's newDec 20 2006

A look at Saks Fifth Avenue's new logo and identity. The identity system consists of cutting up the logo into patterns....98,137,610,226,945,526,221,323,127,451,938,506, 431,029,735,326,490,840,972,261,848,186,538, 906,070,058,088,365,083,852,800,000,000,000 possible patterns.

Here's the 3129 character code you need toDec 20 2006

Here's the 3129 character code you need to enter into a car's keyless entry pad in order to guess the 5-digit passcode. It'll take you 20 minutes or less to enter it. (via j-walk)

Hmm, perhaps Richard Taylor's fractal analysis ofDec 04 2006


Slashdot recently crossed 16,777,216 (that's 2^24) comments, after whichNov 10 2006

Slashdot recently crossed 16,777,216 (that's 2^24) comments, after which no more could be added because of a database misconfiguration. (via waxy)

What's so special about the number 1729? (TurnsOct 16 2006

What's so special about the number 1729? (Turns out, quite a lot.)

On the heels of the possible proofOct 05 2006

On the heels of the possible proof of the Poincare conjecture, another of the $1 million Clay Institute Millennium Prize problems has possibly been proven: the Navier-Stokes equations. The paper is available here.

Update: Oops, "This paper is being withdrawn by the author due a serious flaw." (thx, tj)

Poincare conjecture battle continuesSep 18 2006





A short interview with Grigory Perelman, theAug 28 2006

A short interview with Grigory Perelman, the Russian mathematician who proved the Poincare conjecture and turned down the Fields Medal. "Newspapers should be more discerning over who they write about. They should have more taste." (thx pedro)

Mathematical politics and the Poincare ConjectureAug 23 2006

As I mentioned yesterday, the New Yorker published an article by Sylvia Nasar1 and David Gruber about the recent proof of the Poincare Conjecture2. (Previous coverage in the NY Times and the Guardian.) The article, which is unavailable from the New Yorker's web site (they've now made it available), contains the only interview I've seen with Grigory Perelman, the Russian mathematician who published a potential proof of the conjecture in late 2002, gave a series of lectures in the US, and then went back to Russia. Since then, he hasn't communicated with anyone about the proof, has quit mathematics, and recently refused the Fields Medal, the most prestigious award that mathematics has to offer, saying:

It was completely irrelevent for me. Everybody understood that if the proof is correct then no other recognition is needed.

Meanwhile, a Chinese group of mathematicians, led by Shing-Tung Yau3, are claiming that Perelman's proof was too complicated and are offering a reworked proof instead of Perelman's. That is, they're claiming the first complete proof of the conjecture. Yau The active director of Yau's mathematics institute explained the relative contributions thusly:

Hamilton contributed over fifty per cent; the Russian, Perelman, about twenty five per cent; and the Chinese, Yau, Zhu, and Cao et al., about thirty per cent. (Evidently, simple addition can sometimes trip up even a mathematician.)

Clearly the Chinese gave more than 100% in solving this proof, but Yau is regarded by some mathematicians as attempting to grab glory that does not belong to him. John Morgan, a mathematician at Columbia University, says:

Perelman already did it and what he did was complete and correct. I don't seen anything that [Yau et al.] did different.

Yau wants to be associated with the proof of the Poincare Conjecture, to have China associated with it, and for his student, Zhu, to be elevated in status by it. The $1 million in prize money for the proof of the conjecture offered by the Clay Mathematics Institute can't be far from Yau's mind as well. For his part, Grigory Perelman won't say whether he'll accept the prize money until it is offered. Stay tuned, I guess.

[1] Nasar wrote A Beautiful Mind, a book about mathematician John Nash.

[2] Poincare (properly written as Poincaré) is pronounced Pwan-cah-RAY, not Poyn-care as I said it up until a few weeks ago.

[3] Yau proved a conjecture by Eugenio Calabi which gave birth to a highly useful mathematical structure called a Calabi-Yau manifold; Yau won the Fields Medal for it. The C-Y manifold is important in string theory and Andrew Wiles used it as part of his proof of Fermat's Last Theorem. In short, Yau is a mathematical stud, no question.

Grigory Perelman, who I posted about lastAug 22 2006

Grigory Perelman, who I posted about last week, has indeed won the Fields Medal for his possible proof of the Poincare Conjecture but declined the award. The current New Yorker has an article (not online) about the whole deal which I have yet to read.

David from Ironic Sans is staying inAug 18 2006

David from Ironic Sans is staying in a New Hampshire residence once owned by Norbert Wiener, a mathematician and the founder of cybernetics and reports back about what's on Wiener's bookshelves.

With math immortality, the Fields Medal, and $1Aug 17 2006

With math immortality, the Fields Medal, and $1 million on the line, an eccentric Russian mathematician comes from out of nowhere, proves the Poincare conjecture, and then disappears again. A whodunnit with thousands of pages of mathematical formulas.

The Wolfram Integrator uses a web versionAug 15 2006

The Wolfram Integrator uses a web version of Mathematica to find integrals of functions. We used Mathematica a lot in college to help visualize examples from math and physics classes. (via rw)

The Biology of B-Movie Monsters, or whyAug 07 2006

The Biology of B-Movie Monsters, or why you just can't scale living things up (a la King Kong) or down (like in Fantastic Voyage) without consequence. One key problem: with a theoretical 20 foot tall human, mass increases much faster than bone strength and at some point, his skeleton wouldn't be able to support the weight.

Works of fiction (books mosly) which deal significantly with mathematics.Aug 07 2006

Works of fiction (books mosly) which deal significantly with mathematics.

Using the sequential serial numbers of capturedJul 21 2006

Using the sequential serial numbers of captured German tanks, Allied statisticians accurately determined the number of tanks the Nazis were producing each month.

A Manchester scientist has come up withApr 19 2006

A Manchester scientist has come up with a mathematical formula to assess the perfection of the female derriere. "Dr Holmes said that Kylie Minogue, whose celebrated bottom relaunched her career with the help of a pair of hotpants, would almost certainly score a perfect 80."

Modern mathematical proofs are so complex thatFeb 21 2006

Modern mathematical proofs are so complex that it's becoming impossible to prove them with absolute certainty. About a 1980 proof, an expert says "twenty-five years later we're still not sure if it's correct or not. We sort of think it is, but no one's ever written down the complete proof". I don't think I heard my math teachers ever say "sort of".

Catching cheaters with Benford's LawFeb 21 2006

Benford's Law describes a curious phenomenon about the counterintuitive distribution of numbers in sets of non-random data:

A phenomenological law also called the first digit law, first digit phenomenon, or leading digit phenomenon. Benford's law states that in listings, tables of statistics, etc., the digit 1 tends to occur with probability ~30%, much greater than the expected 11.1% (i.e., one digit out of 9). Benford's law can be observed, for instance, by examining tables of logarithms and noting that the first pages are much more worn and smudged than later pages (Newcomb 1881). While Benford's law unquestionably applies to many situations in the real world, a satisfactory explanation has been given only recently through the work of Hill (1996).

I first heard of Benford's Law in connection with the IRS using it to detect tax fraud. If you're cheating on your taxes, you might fill in amounts of money somewhat at random, the distribution of which would not match that of actual financial data. So if the digit "1" shows up on Al Capone's tax return about 15% of the time (as opposed to the expected 30%), the IRS can reasonably assume they should take a closer look at Mr. Capone's return.

Since I installed Movable Type 3.15 back in March 2005, I have been using its "post to the future" option pretty regularly to post my remaindered links...and have been using it almost exclusively for the last few months[1]. That means I'm saving the entries in draft, manually changing the dates and times, and then setting the entries to post at some point in the future. For example, an entry with a timestamp like "2006-02-20 22:19:09" when I wrote the draft might get changed to something like "2006-02-21 08:41:09" for future posting at around 8:41 am the next morning. The point is, I'm choosing basically random numbers for the timestamps of my remaindered links, particularly for the hours and minutes digits. I'm "cheating"...committing post timestamp fraud.

That got me thinking...can I use the distribution of numbers in these post timestamps to detect my cheating? Hoping that I could (or this would be a lot of work wasted), I whipped up a MT template that produced two long strings of numbers: 1) one of all the hours and minutes digits from the post timestamps from May 2005 to the present (i.e. the cheating period), 2) and one of all the hours and minutes digits from Dec 2002 - Jan 2005 (i.e. the control group). Then I used a PHP script to count the numbers in each string, dumped the results into Excel, and graphed the two distributions together. And here's what they look like, followed by a table of the values used to produce the chart:

Catching cheaters

Digit   5/05-now   12/02-1/05   Difference

As expected, 1 & 2 show up less than they should during the cheating period, but not overly so[2]. The real fingerprint of the crime lies with the 8s. The number 8 shows up during the cheating period ~64% more than expected. After thinking about it for awhile, I came up with an explanation for the abundance of 8s. I often schedule posts between 8am-9am so that there's stuff on the site for the early-morning browse and I usually finish off the day with something between 6pm-7pm (18:00 - 19:00). Not exactly the glaring evidence I was expecting, but you can still tell.

The obvious next question is, can this technqiue be utilized for anything useful? How about detecting comment, trackback. or ping spam? I imagine IPs and timestamps from these types of spam are forged to at least some extent. The difficulties are getting enough data to be statistically significant (one forged timestamp isn't enough to tell anything) and having "clean" data to compare it against. In my case, I knew when and where to look for the cheating...it's unclear if someone who didn't know about the timestamp tampering would have been able to detect it. I bet companies with services that deal with huge amounts of spam (Gmail, Yahoo Mail, Hotmail, TypePad, Technorati) could use this technique to filter out the unwanted emails, comments, trackbacks, or pings...although there's probably better methods for doing so.

[1] I've been doing this to achieve a more regular publishing schedule for kottke.org. I typically do a lot of work in the evening and at night and instead of posting all the links in a bunch from 10pm to 1am, I space them out over the course of the next day. Not a big deal because increasing few of the links I feature are time-sensitive and it's better for readers who check back several times a day for updates...they've always got a little something new to read.

[2] You'll also notice that the distributions don't quite follow Benford's Law either. Because of the constraints on which digits can appear in timestamps (e.g. you can never have a timestamp of 71:95), some digits appear proportionally more or less than they would in statistical data. Here's the distribution of digits of every possible time from 00:00 to 23:59:

1 - 25.33
2 - 17.49
3 - 12.27
4 - 10.97
5 - 10.97
6 - 5.74
7 - 5.74
8 - 5.74
9 - 5.74

Three economists share a cab, getting offDec 09 2005

Three economists share a cab, getting off at three different destinations. How do they split the fare? For answers, you might look to John Nash or the Talmud.

An investigation into the properties of numberDec 07 2005

An investigation into the properties of number spirals (and prime numbers).

Brian Greene on Einstein's most famous equation,Oct 05 2005

Brian Greene on Einstein's most famous equation, E =mc^2. When he finally gets around to it in the middle of the article, Greene's got a pretty good layman's explanation of what the formula actually means.

Freeman Dyson on his friend and colleagueOct 04 2005

Freeman Dyson on his friend and colleague Richard Feynman for The New York Review of Books.

Neat applet that displays orbit patterns for various particle arrangements.Sep 30 2005

Neat applet that displays orbit patterns for various particle arrangements.

When Teen Talk Barbie came out in 1989Sep 30 2005

When Teen Talk Barbie came out in 1989 saying things like "math is hard", could you imagine if blogs had existed at the time? The whole internet would have exploded with rage.

Birthday hookySep 27 2005

Today is my birthday -- I'm 2^5!** -- so I'm taking the day off. No posts or links, aside from this one.

** That's ! as in exclamation point, not ! as in factorial. I'm not 1.33 x 10^36 years old today.




    1. Design is the easy part.
    3. Content is king.
    4. Read. Read. Read.
    5. Think first, then design.
    6. Never forget how lucky you are. Enjoy yourself.



The list of the 100 greatest theorems inSep 19 2005

The list of the 100 greatest theorems in mathematics is topped by The Irrationality of the Square Root of 2 from that nutball Pythagoras. Jesus, who does Godel have to sleep with to get higher on this list...I mean, all the man did was destroy math! (I know, I know, oversimplification, please don't send me any email....) (via cyn-c)

WolframTones lets you generate and download ringtonesSep 12 2005

WolframTones lets you generate and download ringtones based on patterns created by cellular automata systems. Anything's better than the Crazy Frog, yeah?

Fun little quiz on eight grade math...Aug 29 2005

Fun little quiz on eight grade math...can you pass? I got 9/10 (got tripped up on what I thought was a trick question but wasn't...erroneous! erroneous!).

The competitive Scrabble world is starting toAug 24 2005

The competitive Scrabble world is starting to see some top-notch players for whom English is not their native language. At he highest level of competition, "Scrabble's secret is that it's a math game: board geometry, strategic decision making, probability and chance." And sometimes it's better not knowing English so the player can focus solely on the memorization of patterns and gameplay. Interesting stuff.

xThink Calculator is a math calculation programAug 23 2005

xThink Calculator is a math calculation program that recognizes handwritten input from a Tablet PC (check out the screenshots). Pretty darn nifty and reminiscent of Denim, a tool for UI design. (thx nick)

Spirals on nanoparticles show order, specifically ourAug 16 2005

Spirals on nanoparticles show order, specifically our friend the Fibonacci sequence, which can be seen in places like seashells and plants. In the case of the nanoparticles, the Fibonacci pattern results from minimizing the stress energy in the system.

"The hairy ball theorem of algebraic topologyAug 05 2005

"The hairy ball theorem of algebraic topology states that, in layman's terms, 'one cannot comb the hair on a ball in a smooth manner'". Heh. Looks like Wikipedia has some new measures in placeto deal with spam/trolls: "This page has been protected from editing to deal with vandalism."

Pi, God, and apartment supercomputersJul 18 2005

The New Yorker recently ran a feature on how a couple of mathematicians helped The Met photograph a part of The Hunt of the Unicorn tapestries. That same week, they ran from their extensive archives a 1992 profile of the same mathematicians, brothers David and Gregory Chudnovsky. The Chudnovskys were then engaged in calculating as many digits of pi as they could using a homemade supercomputer housed in their Manhattan apartment. There's some speculation that director Darren Aronfsky based his 1998 film, Pi, on the Chudnovskys and after reading the above article, there's little doubt that's exactly what he did:




(Oh, and as for patterns hidden in pi, we've already found one. It's called the circle. Just because humans discovered circles first and pi later shouldn't mean that the latter is derived from the former.)

Strange attractors and fashion photography meet atMay 13 2005


How a couple of mathematicians helped theApr 28 2005

How a couple of mathematicians helped the Met accurately photograph some priceless tapestries. The difficulty in piecing together the different photographs was because when the tapestries were taken off the wall, they "began to breathe, expanding, contracting, shifting"...that is, they were changing between photos.

Everything and MoreNov 30 2003

Everything and More




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