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.
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.
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)
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.↩
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)
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.
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.
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:
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]
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.
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.
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.
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.)
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.
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)
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)
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.
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