Mathematical functions depicted as stick figure dance moves. (via @mulegirl)

Mathematical functions depicted as stick figure dance moves. (via @mulegirl)

In a lecture given in 1924, German mathematician David Hilbert introduced the idea of the paradox of the Grand Hotel, which might help you wrap your head around the concept of infinity. (Spoiler alert: it probably won’t help…that’s the paradox.) In his book One Two Three… Infinity, George Gamow describes Hilbert’s paradox:

Let us imagine a hotel with a finite number of rooms, and assume that all the rooms are occupied. A new guest arrives and asks for a room. “Sorry,” says the proprietor, “but all the rooms are occupied.” Now let us imagine a hotel with an infinite number of rooms, and all the rooms are occupied. To this hotel, too, comes a new guest and asks for a room.

“But of course!” exclaims the proprietor, and he moves the person previously occupying room N1 into room N2, the person from room N2 into room N3, the person from room N3 into room N4, and so on…. And the new customer receives room N1, which became free as the result of these transpositions.

Let us imagine now a hotel with an infinite number of rooms, all taken up, and an infinite number of new guests who come in and ask for rooms.

“Certainly, gentlemen,” says the proprietor, “just wait a minute.”

He moves the occupant of N1 into N2, the occupant of N2 into N4, and occupant of N3 into N6, and so on, and so on…

Now all odd-numbered rooms became free and the infinite of new guests can easily be accommodated in them.

This TED video created by Jeff Dekofsky explains that there are similar strategies for finding space in such a hotel for infinite numbers of infinite groups of people and even infinite amounts of infinite numbers of infinite groups of people (and so on, and so on…) and is very much worth watching:

(via brain pickings)

Given that there’s so much mathematicians don’t know about prime numbers, you might be surprised to learn that there’s a very simple regular expression for detecting prime numbers:

`/^1?$|^(11+?)\\1+$/`

If you’ve got access to Perl on the command line, try it out with some of these (just replace [number] with any integer):

`perl -wle 'print "Prime" if (1 x shift) !~ /^1?$|^(11+?)\\1+$/' [number]`

An explanation is here which I admit I did not quite follow. A commenter at Hacker News adds a bit more context:

However while cute, it is very slow. It tries every possible factorization as a pattern match. When it succeeds, on a string of length n that means that n times it tries to match a string of length n against a specific pattern. This is O(n^2). Try it on primes like 35509, 195341, 526049 and 1030793 and you can observe the slowdown.

Using only squares, triangles, and the condition that each shape wants to move if less than 1/3 of its neighbors are like it, watch how extreme segregation appears in even the most random mixing of shapes.

These little cuties are 50% Triangles, 50% Squares, and 100% slightly shapist. But only slightly! In fact, every polygon prefers being in a diverse crowd. You can only move them if they’re unhappy with their immediate neighborhood. Once they’re OK where they are, you can’t move them until they’re unhappy with their neighbors again. They’ve got one, simple rule: “I wanna move if less than 1/3 of my neighbors are like me.”

Harmless, right? Every polygon would be happy with a mixed neighborhood. Surely their small bias can’t affect the larger shape society that much? Well… And… our shape society becomes super segregated. Daaaaang. Sometimes a neighborhood just becomes square, and it’s not their fault if no triangles wanna stick around. And a triangular neighborhood would welcome a square, but they can’t help it if squares ain’t interested.

Super super fascinating. Take your time and go through and play with all the interactive widgets. (via @ftrain)

What’s a large number? A billion? A billion times a billion? A billion to the billionth power? A googol? A googolplex? A googolplex is 10^googol, BTW:

So a googol is 1 with just 100 zeros after it, which is a number 10 billion times bigger than the grains of sand that would fill the universe. Can you possibly imagine what kind of number is produced when you put a googol zeros after the 1?

That’s pretty big, right? Not. Even. It turns out you can construct numbers that are so much larger than a googolplex, that it’s gonna light your head on fire just to read about them. Put on your asbestos hat and feast your eyes on Graham’s Number.

Moving up another level, exponentiation is iterated multiplication. Instead of saying 3 x 3 x 3 x 3, exponentiation allows me to bundle that string into the more concise 3^4.

Now, the thing is, this is where most people stop. In the real world, exponentiation is the highest operation we tend to ever use in the hyperoperation sequence. And when I was envisioning my huge googolplex^googolplex number, I was doing the very best I could using the highest level I knew — exponentiation. On Level 3, the way to go as huge as possible is to make the base number massive and the exponent number massive. Once I had done that, I had maxed out.

The key to breaking through the ceiling to the really big numbers is understanding that you can go up more levels of operations — you can keep iterating up infinitely. That’s the way numbers get truly huge.

You might get lost around the “power tower feeding frenzy” bit or the “power tower feeding frenzies psycho festival” bit, but persist…the end result is really just beyond superlatives. (via @daveg)

**Update:** In this video, you can listen to the inventor of Graham’s number, Ron Graham, explain all about it.

(via @eightohnine)

Acclaimed science and math writer Simon Singh has written a book on the mathematics of The Simpsons, The Simpsons and Their Mathematical Secrets. Boing Boing has an excerpt.

The principles of rubber sheet geometry can be extended into three dimensions, which explains the quip that a topologist is someone who cannot tell the difference between a doughnut and a coffee cup. In other words, a coffee cup has just one hole, created by the handle, and a doughnut has just one hole, in its middle. Hence, a coffee cup made of a rubbery clay could be stretched and twisted into the shape of a doughnut. This makes them homeomorphic.

By contrast, a doughnut cannot be transformed into a sphere, because a sphere lacks any holes, and no amount of stretching, squeezing, and twisting can remove the hole that is integral to a doughnut. Indeed, it is a proven mathematical theorem that a doughnut is topologically distinct from a sphere. Nevertheless, Homer’s blackboard scribbling seems to achieve the impossible, because the diagrams show the successful transformation of a doughnut into a sphere. How?

Although cutting is forbidden in topology, Homer has decided that nibbling and biting are acceptable. After all, the initial object is a doughnut, so who could resist nibbling? Taking enough nibbles out of the doughnut turns it into a banana shape, which can then be reshaped into a sphere by standard stretching, squeezing, and twisting. Mainstream topologists might not be thrilled to see one of their cherished theorems going up in smoke, but a doughnut and a sphere are identical according to Homer’s personal rules of topology. Perhaps the correct term is not homeomorphic, but rather Homermorphic.

The Fields Medal is viewed as the greatest honor in mathematics; the Nobel of math. Today, Iranian mathematician Maryam Mirzakhani became the first woman (and Iranian) to win a Fields Medal.

Maryam Mirzakhani has made stunning advances in the theory of Riemann surfaces and their moduli spaces, and led the way to new frontiers in this area. Her insights have integrated methods from diverse fields, such as algebraic geometry, topology and probability theory.

In hyperbolic geometry, Mirzakhani established asymptotic formulas and statistics for the number of simple closed geodesics on a Riemann surface of genus g. She next used these results to give a new and completely unexpected proof of Witten’s conjecture, a formula for characteristic classes for the moduli spaces of Riemann surfaces with marked points.

In dynamics, she found a remarkable new construction that bridges the holomorphic and symplectic aspects of moduli space, and used it to show that Thurston’s earthquake flow is ergodic and mixing.

Most recently, in the complex realm, Mirzakhani and her coworkers produced the long sought-after proof of the conjecture that - while the closure of a real geodesic in moduli space can be a fractal cobweb, defying classification - the closure of a complex geodesic is always an algebraic subvariety.

Get all that? Adolescent math fans, you have a new role model. She does math like a girl. Here’s more on Mirzakhani from Quanta Magazine.

**Update:** Sad news…Mirzakhani died in July 2017 from cancer. She was 40 years old.

What if you wanted to cut a bagel in half not for toasting or sandwich purposes, but to explore its topology and mildly astonish your friends?

If you cut a bagel along a mÃ¶bius strip pattern, you end up with two separate halves that form interlocking rings, as shown below.

Geoge Hart, who cut this bagel and made this video, is an engineering professor at SUNY-Stony Brook and “mathematical sculptor. On his web site, he offers two bagel-derived math problems: *What is the ratio of the surface area of this linked cut to the surface area of the usual planar bagel slice?* and *Modify the cut so the cutting surface is a one-twist Mobius strip. *

Via @mark_e_evans and The Onion A/V Club.

The book cover for Naive Set Theory by Paul Halmos is so so good:

The cover is a riff on, I think, Russell’s Paradox, a problem with naive set theory described by Bertrand Russell in 1901 about whether sets can contain themselves.

Russell’s paradox is based on examples like this: Consider a group of barbers who shave only those men who do not shave themselves. Suppose there is a barber in this collection who does not shave himself; then by the definition of the collection, he must shave himself. But no barber in the collection can shave himself. (If so, he would be a man who does shave men who shave themselves.)

Reminds me of David Pearson’s genius cover for Benjamin’s The Work of Art in the Age of Mechanical Reproduction.

In France, pie charts are called “le camembert” after the cheese. Or sometimes “un diagramme en fromage” (cheese diagram). In Brazil, they are pizza charts. (via numberphile & reddit)

It’s possible to make a .zip file that contains itself infinitely many times. So a 440 byte file could conceivably be expanded into eleventy dickety two zootayunafliptobytes of data and beyond. Here’s the full explanation.

“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.

As Pi Day approaches, it time for a refresher course, courtesy of Steven Strogatz, on what pi actually means and how you can visualize calculating it. It’s all about rearranging the pieces of a circle in a calculus-ish sort of way:

Planet Money: always buy the bigger pizza because geometry.

The math of why bigger pizzas are such a good deal is simple. A pizza is a circle, and the area of a circle increases with the square of the radius.

So, for example, a 16-inch pizza is actually four times as big as an 8-inch pizza.

And when you look at thousands of pizza prices from around the U.S., you see that you almost always get a much, much better deal when you buy a bigger pizza.

What do you think you get if you add 1+2+3+4+5+… all the way on up to infinity? Probably a massively huge number, right? Nope. You get a small negative number:

This is, by a wide margin, the most noodle-bending counterintuitive thing I have ever seen. Mathematician Leonard Euler actually proved this result in 1735, but the result was only made rigorous later and now physicists have been seeing this result actually show up in nature. Amazing. (thx, chris)

**Update:** Of course (of course!) the actual truth seems more complicated, hinging on what “sum” means mathematically, etc. (via @cenedella)

**Update:** As usual, Phil Plait sorts things out on this complicated situation. (via @theory)

A short time before his death, Benoît B. Mandelbrot filmed an interview with Errol Morris. Morris charmingly starts off my asking Mandelbrot where “the fractal stuff” came from.

Note: as always, the “B.” in “Benoît B. Mandelbrot” stands for “Benoît B. Mandelbrot”. (via @sampotts)

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.

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

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)

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.”

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:

I also enjoyed this dicussion of what a distribution of actual cash from Yahoo to Tumblr would be like.

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).

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)

**Update:** Here’s a good profile of and interview with Zhang.

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.”

(via @dunstan)

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).

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

(via mlkshk)

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)

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

(via ★vuokko)

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

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.

(via @pomeranian99)

One of the many reasons to love the wooden water towers found on the tops of NYC buildings is that the structures themselves reveal the math behind how they work.

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.