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kottke.org posts about mathematics

The beauty of mathematics

posted by Jason Kottke   Dec 18, 2015

The “hidden” mathematics and order behind everyday objects & phenomenon like spinning tops, dice, magnifying glasses, and airplanes. (via @stevenstrogatz)

Einstein’s first proof

posted by Jason Kottke   Nov 19, 2015

Steven Strogatz walks us through the first mathematical proof Albert Einstein did when he was a boy: a proof of the Pythagorean theorem.

Einstein, unfortunately, left no such record of his childhood proof. In his Saturday Review essay, he described it in general terms, mentioning only that it relied on “the similarity of triangles.” The consensus among Einstein’s biographers is that he probably discovered, on his own, a standard textbook proof in which similar triangles (meaning triangles that are like photographic reductions or enlargements of one another) do indeed play a starring role. Walter Isaacson, Jeremy Bernstein, and Banesh Hoffman all come to this deflating conclusion, and each of them describes the steps that Einstein would have followed as he unwittingly reinvented a well-known proof.

Twenty-four years ago, however, an alternative contender for the lost proof emerged. In his book “Fractals, Chaos, Power Laws,” the physicist Manfred Schroeder presented a breathtakingly simple proof of the Pythagorean theorem whose provenance he traced to Einstein.

Of course, that breathtaking simplicity later became a hallmark of Einstein’s work in physics. See also this brilliant visualization of the Pythagorean theorem

P.S. I love that two of the top three most popular articles on the New Yorker’s web site right now are about Albert Einstein.

Brilliant visualization of the Pythagorean Theorem

posted by Jason Kottke   Oct 26, 2015

Everyone knows that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. What this video presupposes is, fuck yeah math!

PhotoMath iOS app can do your homework for you

posted by Jason Kottke   Oct 15, 2015

PhotoMath

Some iOS apps still seem like magic. Case in point: PhotoMath. Here’s how it works. You point your camera at a math problem and PhotoMath shows the answer. It’ll even give you a step-by-step explanation and solution.

The mathematical secrets of Pascal’s triangle

posted by Jason Kottke   Sep 16, 2015

Pascal’s triangle1 is a simple arrangement of numbers in a triangle…rows are formed by the successive addition of numbers in previous rows. But out of those simple rows comes deep and useful mathematical relationships related to probability, fractals, squares, and binomial expansions. (via digg)

  1. As the video says, Pascal was nowhere near the discoverer of this particular mathematical tool. By the time he came along in 1653, the triangle had already been described in India (possibly as early as the 2nd century B.C.) and later in Persia and China.

Cool furniture alert: the Fibonacci Shelf

posted by Jason Kottke   Jul 29, 2015

The Fibonacci Shelf by designer Peng Wang might not be the most functional piece of furniture, but I still want one.

Fibonacci Shelf

Fibonacci Shelf

The design of the shelf is based on the Fibonacci sequence of numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, …), which is related to the Golden Rectangle. When assembled, the Fibonacci Shelf resembles a series of Golden Rectangles partitioned into squares. (via ignant)

Fibonacci sequence hidden in ordinary division problem

posted by Jason Kottke   Jul 06, 2015

If you divide 1 by 999,999,999,999,999,999,999,998,999,999,999,999,999,999,999,999 (that’s 999 quattuordecillion btw), the Fibonacci sequence neatly pops out. MATH FTW!

Fibonacci division

At the end of Carl Sagan’s Contact (spoilers!), the aliens give Ellie a hint about something hidden deep in the digits of π. After a long search, a circle made from a sequence of 1s and 0s is found, providing evidence that intelligence was built into the fabric of the Universe. I don’t know if this Fibonacci division thing is on quite the same level, but it might bake your noodle if you think about it too hard. (via @stevenstrogatz)

Update: From svat at Hacker News, an explanation of the magic behind the math.

It’s actually easier to understand if you work backwards and arrive at the expression yourself, by asking yourself: “If I wanted the number that starts like 0.0…000 0…001 0…001 0…002 0…003 0…005 0…008 … (with each block being 24 digits long), how would I express that number?”

(thx, taylor)

Web Mandelbrot

posted by Jason Kottke   May 04, 2015

Mandelbrot

This web app allows you to explore the Mandelbrot set interactively…just click and zoom. I had an application like this on my computer in college, but it only went a few zooms deep before crashing though. There was nothing quite like zooming in a bunch of times on something that looked like a satellite photo of a river delta and seeing something that looks exactly like when you started. (via @stevenstrogatz)

Finding Zero

posted by Jason Kottke   Apr 23, 2015

Finding Zero

The latest book from Amir Aczel, who has written previously about the compass, the Large Hadron Collider, and Fermat’s Last Theorem, is Finding Zero: A Mathematician’s Odyssey to Uncover the Origins of Numbers…in particular, the number zero.

Finding Zero is an adventure filled saga of Amir Aczel’s lifelong obsession: to find the original sources of our numerals. Aczel has doggedly crisscrossed the ancient world, scouring dusty, moldy texts, cross examining so-called scholars who offered wildly differing sets of facts, and ultimately penetrating deep into a Cambodian jungle to find a definitive proof.

The NY Times has a review of the book, written by another Amir, Amir Alexander, who wrote a recent book on infinitesimals, aka very nearly zero. (via @pomeranian99)

The beauty of pi puts infinity within reach

posted by Jason Kottke   Mar 13, 2015

I’m dreading it. No hope of solving any equations that day, what with the pie-eating contests, the bickering over the merits of pi versus tau (pi times two), and the throwdowns over who can recite more digits of pi. Just stay off the streets at 9:26:53, when the time will approximate pi to ten places: 3.141592653.

The New Yorker’s Steven Strogatz on why pi matters.

Dancing mathematics

posted by Jason Kottke   Mar 02, 2015

Dancing Math

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

The Infinite Hotel Paradox

posted by Jason Kottke   Feb 19, 2015

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)

A regular expression for finding prime numbers

posted by Jason Kottke   Feb 12, 2015

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.

Parable of the Polygons

posted by Jason Kottke   Dec 08, 2014

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)

The “power tower feeding frenzy” and hyper-large numbers

posted by Jason Kottke   Dec 04, 2014

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)

The Simpsons and their mathematical secrets

posted by Jason Kottke   Nov 13, 2014

Simpsons Math

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.

Doing math like a girl

posted by Jason Kottke   Aug 12, 2014

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.

Möbius Bagel

posted by Tim Carmody   Jul 31, 2014

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?

bagel cutting pattern.jpg

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

bagel9.jpg

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.

Fantastic recursive book cover

posted by Jason Kottke   Jun 23, 2014

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

Naive Set Theory Cover

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.

Cheese charts

posted by Jason Kottke   Jun 18, 2014

Camembert chart

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)

The infinitely large .zip file

posted by Jason Kottke   Jun 16, 2014

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.

Louis C.K. seeks cure for the Common Core

posted by Jason Kottke   May 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?

posted by Jason Kottke   Mar 13, 2014

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:

Pi Calc

Always buy the bigger pizza

posted by Jason Kottke   Feb 26, 2014

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.

The sum of all positive integers

posted by Jason Kottke   Jan 16, 2014

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)

Short Errol Morris film on Benoit Mandelbrot

posted by Jason Kottke   Nov 18, 2013

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)

How to win at The Price is Right

posted by Jason Kottke   Nov 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 triangle

posted by Jason Kottke   Oct 09, 2013

Sierpinski Curved

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)

The unhappy traveling salesman problem

posted by Jason Kottke   Sep 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 calculations

posted by Jason Kottke   May 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:

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).