kottke.org posts about mathematics
The abacus counting device dates back thousands of years but has, in the past century, been replaced by calculators and computers. But studies show that abacus use can have an effect on how well people learn math. In this excerpt adapted from his new book Learn Better, education researcher Ulrich Boser writes about the abacus and how people learn.
Researchers from Harvard to China have studied the device, showing that abacus students often learn more than students who use more modern approaches.
UC San Diego psychologist David Barner led one of the studies, and he argues that abacus training can significantly boost math skills with effects potentially lasting for decades.
“Based on everything we know about early math education and its long-term effects, I’ll make the prediction that children who thrive with abacus will have higher math scores later in life, perhaps even on the SAT,” Barner told me.
Ignore the hyperbolic “and it changed my life” in the title…this piece is interesting throughout. For example, this passage on the strength of the mind-body connection and the benefits of learning by doing:
When first I watched high school abacus whiz Serena Stevenson, her hand gestures seemed like a pretentious affect, like people who wear polka-dot bow ties. But it turned out that her finger movements weren’t really all that dramatic, and on YouTube, I watched students with even more theatrical gesticulations. What’s more, the hand movements turned out to be at the heart of the practice, and without any arm or finger motions, accuracy can drop by more than half.
Part of the explanation for the power of the gestures goes to the mind-body connection. But just as important is the fact that abacus makes learning a matter of doing. It’s an active, engaging process. As one student told me, abacus is like “intellectual powerlifting.”
Psychologist Rich Mayer has written a lot about this idea, and in study after study he has shown that people gain expertise by actively producing what they know. As he told me: “Learning is a generative activity.”
I’d never heard of the concept of overlearning before:
Everybody from actors learning lines, to musicians learning new songs, to teachers trying to impart key facts to students has observed that learning has to “sink in” in the brain. Prior studies and also the new one, for example, show that when people learn a new task and then learn a similar one soon afterward, the second instance of learning often interferes with and undermines the mastery achieved on the first one.
The new study shows that overlearning prevents against such interference, cementing learning so well and quickly, in fact, that the opposite kind of interference happens instead. For a time, overlearning the first task prevents effective learning of the second task — as if learning becomes locked down for the sake of preserving master of the first task. The underlying mechanism, the researchers discovered, appears to be a temporary shift in the balance of two neurotransmitters that control neural flexibility, or “plasticity,” in the part of the brain where the learning occurred.
“These results suggest that just a short period of overlearning drastically changes a post-training plastic and unstable [learning state] to a hyperstabilized state that is resilient against, and even disrupts, new learning,” wrote the team led by corresponding author Takeo Watanabe, the Fred M. Seed Professor of Cognitive Linguistic and Psychological Sciences at Brown.
Yay! It’s Fractal Friday! (It’s not, I just made that up.) But anyway, courtesy of Christopher Night, you can play around with this Julia set fractal. It works in a desktop browser (by moving the mouse) or on your phone (by dragging your finger).
The Julia set, if you don’t remember, goes thusly: Let f(z) be a complex rational function from the plane into itself, that is, f(z)=p(z)/q(z) f(z)=p(z)/q(z), where p(z) and q(z) are complex polynomials. Then there is a finite number of open sets F1, …, Fr, that are left invariant by f(z) which, uh, is um… yay! Fractal Friday! The colors are so pretty!
If you spin these sculptures by artist John Edmark at a certain speed and light them with a strobe, they appear to animate in slowly trippy ways.
Blooms are 3-D printed sculptures designed to animate when spun under a strobe light. Unlike a 3D zoetrope, which animates a sequence of small changes to objects, a bloom animates as a single self-contained sculpture. The bloom’s animation effect is achieved by progressive rotations of the golden ratio, phi (ϕ), the same ratio that nature employs to generate the spiral patterns we see in pinecones and sunflowers. The rotational speed and strobe rate of the bloom are synchronized so that one flash occurs every time the bloom turns 137.5º (the angular version of phi).
The effect seems computer generated (but obviously isn’t) and is better than I anticipated. (via colossal)
Update: While not as visually smooth as his sculptures, Edmark’s rotation of an artichoke under strobe lighting deftly demonstrates the geometric rules followed by plants when they grow.
Here we see an artichoke spinning while being videotaped at 24 frames-per-second with a very fast shutter speed (1/4000 sec). The rotation speed is chosen to cause the artichoke to rotate 137.5º — the golden angle — each time a frame is captured, thus creating the illusion that the leaves are moving up or down the surface of the artichoke. The reason this works is that the artichoke grows by producing new leaf one at a time, with each new leaf positioned 137.5º around the center from the previous leaves. So, in a sense, this video reiterates the artichoke’s growth process.
Update: This similar sculpture by Takeshi Murata is quite impressive as well.
The Monty Hall Problem is one of those things that demonstrates just how powerful a pull common sense has on the human reasoning process. The problem itself is easily stated: there are three doors and behind one of them there is a prize and behind the other two, nothing. You choose a door in hopes of finding the prize and then one of the other two doors is opened to reveal nothing. You are offered the opportunity to switch your guess to the other door. Do you take it?
Common sense tells you that switching wouldn’t make any difference. There are two remaining doors, the prize is randomly behind one of them, why would switching yield any benefit? But as the video explains and this simulation shows, counterintuition prevails: you should switch every time.
America was introduced to the difficulty of the problem by Marilyn vos Savant in her column for Parade magazine in 1990.1 In a follow-up explanation of the question, vos Savant offered a quite simple “proof” of the always switch method (from Wikipedia). Let’s assume you pick door #1, here are the possible outcomes:
||Result if you stay
||Result if you switch
As you can see, staying yields success 33% of the time while if you switch, you win 2 times out of three (67%), a result verified by a properly written simulator. In his Straight Dope column, Cecil Adams explained it like so (after he had gotten it wrong in the first place):
A friend of mine did suggest another way of thinking about the problem that may help clarify things. Suppose we have the three doors again, one concealing the prize. You pick door #1. Now you’re offered this choice: open door #1, or open door #2 and door #3. In the latter case you keep the prize if it’s behind either door. You’d rather have a two-in-three shot at the prize than one-in-three, wouldn’t you? If you think about it, the original problem offers you basically the same choice. Monty is saying in effect: you can keep your one door or you can have the other two doors, one of which (a non-prize door) I’ll open for you.
See also the case of the plane and the conveyor belt (sorry not sorry, I couldn’t resist).
For a recent episode of Numberphile, David Eisenbud explains the Collatz Conjecture, a math problem that is very easy to understand but has an entire book devoted to it and led famous mathematician Paul Erdős to say “this is a problem for which mathematics is perhaps not ready”.
The problem is easily stated: start with any positive integer and if it is even, divide it by 2 and if odd multiply it by 3 and add 1. Repeat the process indefinitely. Where do the numbers end up? Infinity? 1? Loneliness? Somewhere in-between? My favorite moment of the video:
16. Whoa, a very even number.
I love math and I love this video. (via df)
When you look at some plants, you can just see the mathematics behind how the leaves, petals, and veins are organized.
As a child, Danica McKellar played Winnie Cooper on The Wonder Years. After the show was over, McKellar had difficulty breaking away from other people’s perceptions of her. But in college, she discovered an aptitude for mathematics, went on to have a theorem named after her — not because she was famous but because she’d helped prove it — and forged a new identity. (via @stevenstrogatz)
When Grade-A nerds get together and talk about programming and math, a popular topic is P vs NP complexity. There’s a lot to P vs NP, but boiled down to its essence, according to the video:
Does being able to quickly recognize correct answers [to problems] mean there’s also a quick way to find [correct answers]?
Most people suspect that the answer to that question is “no”, but it remains famously unproven.
In fact, one of the outstanding problems in computer science is determining whether questions exist whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure. Problems like the one listed above certainly seem to be of this kind, but so far no one has managed to prove that any of them really are so hard as they appear, i.e., that there really is no feasible way to generate an answer with the help of a computer.
According to this 2011 article on Mark Eaton and other 7-footers who have played in the NBA, there are only 70 American men between 20 and 40 who are 7 feet tall…and that more than 1 in 6 of them get to play in the NBA.
The curve shaped by the CDC’s available statistics, however, does allow one to estimate the number of American men between the ages of 20 and 40 who are 7 feet or taller: fewer than 70 in all. Which indicates, by further extrapolation, that while the probability of, say, an American between 6’6” and 6’8” being an NBA player today stands at a mere 0.07%, it’s a staggering 17% for someone 7 feet or taller.
Being seven feet tall is absurdly tall and comes with a whole host of challenges, from bumping one’s head on door frames to difficulty finding clothes to health issues. Some of these difficulties arise out of simple geometry: as height and width increase, volume increases more quickly.1
The possession of certain prime numbers is illegal in the US. For instance, one of these primes can be used to break a DVD’s copyright encryption.
Mathematicians have calculated pi out to more than 13 trillion decimal places, a calculation that took 208 days. NASA’s Marc Rayman explains that in order to send out probes and slingshot them accurately throughout the solar system, NASA needs to use only 15 decimal places, or 3.141592653589793. How precise are calculations with that number? This precise:
The most distant spacecraft from Earth is Voyager 1. It is about 12.5 billion miles away. Let’s say we have a circle with a radius of exactly that size (or 25 billion miles in diameter) and we want to calculate the circumference, which is pi times the radius times 2. Using pi rounded to the 15th decimal, as I gave above, that comes out to a little more than 78 billion miles. We don’t need to be concerned here with exactly what the value is (you can multiply it out if you like) but rather what the error in the value is by not using more digits of pi. In other words, by cutting pi off at the 15th decimal point, we would calculate a circumference for that circle that is very slightly off. It turns out that our calculated circumference of the 25 billion mile diameter circle would be wrong by 1.5 inches. Think about that. We have a circle more than 78 billion miles around, and our calculation of that distance would be off by perhaps less than the length of your little finger.
When was humanity’s calculation of pi accurate enough for NASA? In 1424, Persian astronomer and mathematician Jamshid al-Kashi calculated pi to 17 digits.
From 3Blue1Brown, a quick video showing some space-filling curves.
In the book In Pursuit of the Unknown, Ian Stewart discusses how equations from the likes of Pythagoras, Euler, Newton, Fourier, Maxwell, and Einstein have been used to build the modern world.
I love how as time progresses, the equations get more complicated and difficult for the layperson to read (much less understand) and then Boltzmann and Einstein are like, boom!, entropy is increasing and energy is proportional to mass, suckas!
The “hidden” mathematics and order behind everyday objects & phenomenon like spinning tops, dice, magnifying glasses, and airplanes. (via @stevenstrogatz)
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.
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!
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.
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)
The Fibonacci Shelf by designer Peng Wang might not be the most functional piece of furniture, but I still want one.
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)
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!
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?”
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)
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)
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.
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:
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.
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.