## Neil deGrasse Tyson on pi and other constants —

Both the Bible and the Indiana State Legislature have tried to redefine pi to equal something much more simple than 3.14159265358979323846264338327950 ...

## 86.54% liked this —

Science blogger Matt Springer analyzes the surprisingly fascinating math behind Reddit upvotes.

# Probability theory for programmers

Jeremy Kun, a mathematics PhD student at the University of Illinois in Chicago, has posted a wonderful primer on probability theory for programmers on his blog. It's a subject vital to machine learning and data-mining, and it's at the heart of much of the stuff going on with Big Data. His primer is lucid and easy to follow, even for math ignoramuses like me.

For instance, suppose our probability space is $\Omega = \left \{ 1, 2, 3, 4, 5, 6 \right \}$ and $f$ is defined by setting $f(x) = 1/6$ for all $x \in \Omega$ (here the “experiment” is rolling a single die). Then we are likely interested in more exquisite kinds of outcomes; instead of asking the probability that the outcome is 4, we might ask what is the probability that the outcome is even? This event would be the subset $\left \{ 2, 4, 6 \right \}$, and if any of these are the outcome of the experiment, the event is said to occur. In this case we would expect the probability of the die roll being even to be 1/2 (but we have not yet formalized why this is the case).

As a quick exercise, the reader should formulate a two-dice experiment in terms of sets. What would the probability space consist of as a set? What would the probability mass function look like? What are some interesting events one might consider (if playing a game of craps)?

(Image: Dice, a Creative Commons Attribution (2.0) image from artbystevejohnson's photostream)

# Math Fleet: command a space squadron and defend planet Earth with the power of math

Kevin Kelly told me about Phil Scale's new iOS application to teach kids arithmetic. It's called Math Fleet and it sounds great. (Phil created Kevin's Asia Grace photobook app, which is also wonderful.)

Phil says:

I've been an independent iOS app developer for four years, and my wife, Jennifer, and I work together from our home in Austin creating games and educational apps. Our newest app is called Math Fleet, an action game set in space where players must use quick math skills to save Earth from invasion, all while dodging asteroids and battling enemy star fighters.

The inspiration for the game came from our sons, Jack, Luke and Dylan (ages 7, 6 and 3) for whom I've downloaded and tried many educational apps and games. At the beginning of the summer we were all home together and I was brainstorming the next game, which I knew I wanted to set in space. I had my sons Jack and Luke playing a math game I had downloaded to earn playing time for the other games they really wanted to play. We all agreed though that the math game I had them playing, which cost me $5, wasn't a very good game and there really wasn't very much math in it. I knew I could write a better game and we started talking about what we would do differently, and in that moment, we decided our mission was to create not just a better game, but the most awesome action math game out there, and that Jack and Luke would be in the driver's seat guiding how the game would take shape. It became the ultimate geek dad summer for me as I fully committed myself to making their ideas a reality, and some of their ideas were pretty challenging to implement. Such as Luke's idea that the user pilot multiple ships, customizable and upgradable -- the eventual foundation of the game; Or the Patrol Sector concept drawn up by my 7-year-old, Jack. Throughout the eight month journey, as I coded, they tested prototypes of controls, menus, action sequences, effects, space weapons, and ships. They told me what they liked and what they didn't like, what they'd do differently, and in many cases they would contribute key design concepts delivered as crayon drawings or Lego models. I learned more about game design by watching them, listening, and discussing ideas than I learned from writing my previous two games. Finally, after eight months and 300,000 lines of code, Math Fleet is officially released, and is hopefully everything we set out to create to create an exciting math game that kids actually want to play, while also challenging their minds by combining fast problem solving with the stress and distraction of piloting a space fleet. Math Fleet # Crypto and Bletchley Park podcast from BBC's Infinite Monkey Cage BBC Radio 4's great math and science show "The Infinite Monkey Cage" did a great (and very funny) episode on crypto and Bletchley Park, with Robin Ince, Brian Cox, Dave Gorman, Simon Singh and Dr Sue Black. (via Schneier) ## MoMath, more problems — Here's an awesome activity for anybody who happens to be in New York City. Next week, on December 15th, The National Museum of Mathematics (MoMath) will open at a location near the Flatiron Building. Opening weekend festivities (and the museum, itself) look really cool. # Amazing, invisible work that goes on when you click an HTTPS link Jeff Moser has a clear, fascinating enumeration of all the incredible math stuff that happens between a server and your browser when you click on an HTTPS link and open a secure connection to a remote end. It's one of the most important (and least understood) parts of the technical functioning of the Internet. People sometimes wonder if math has any relevance to programming. Certificates give a very practical example of applied math. Amazon's certificate tells us that we should use the RSA algorithm to check the signature. RSA was created in the 1970's by MIT professors Ron *R*ivest, Adi *S*hamir, and Len *A*dleman who found a clever way to combine ideas spanning 2000 years of math development to come up with a beautifully simple algorithm: You pick two huge prime numbers "p" and "q." Multiply them to get "n = p*q." Next, you pick a small public exponent "e" which is the "encryption exponent" and a specially crafted inverse of "e" called "d" as the "decryption exponent." You then make "n" and "e" public and keep "d" as secret as you possibly can and then throw away "p" and "q" (or keep them as secret as "d"). It's really important to remember that "e" and "d" are inverses of each other. Now, if you have some message, you just need to interpret its bytes as a number "M." If you want to "encrypt" a message to create a "ciphertext", you'd calculate: C ≡ Me (mod n) This means that you multiply "M" by itself "e" times. The "mod n" means that we only take the remainder (e.g. "modulus") when dividing by "n." For example, 11 AM + 3 hours ≡ 2 (PM) (mod 12 hours). The recipient knows "d" which allows them to invert the message to recover the original message: Cd ≡ (Me)d ≡ Me*d ≡ M1 ≡ M (mod n) # Free Coursera Calculus course with hand-drawn animated materials Robert Ghrist from University of Pennsylvania wrote in to tell us about his new, free Coursera course in single-variable Calculus, which starts on Jan 7. Calculus is one of those amazing, chewy, challenging branches of math, and Ghrist's hand-drawn teaching materials look really engaging. Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: the introduction and use of Taylor series and approximations from the beginning; * a novel synthesis of discrete and continuous forms of Calculus; * an emphasis on the conceptual over the computational; and * a clear, entertaining, unified approach. Calculus: Single Variable (Thanks, Robert!) # Tallest possible Lego tower height calculated The good folks on the most-excellent BBC Radio/Open University statistical literacy programme More or Less decided to answer a year-old Reddit argument about how many Lego bricks can be vertically stacked before the bottom one collapses. They got the OU's Dr Ian Johnston to stress-test a 2X2 Lego in a hydraulic testing machine, increasing the pressure to some 4,000 Newtons, at which point the brick basically melted. Based on this, they calculated the maximum weight a 2X2 brick could bear, and thus the maximum height of a Lego tower: The average maximum force the bricks can stand is 4,240N. That's equivalent to a mass of 432kg (950lbs). If you divide that by the mass of a single brick, which is 1.152g, then you get the grand total of bricks a single piece of Lego could support: 375,000. So, 375,000 bricks towering 3.5km (2.17 miles) high is what it would take to break a Lego brick. "That's taller than the highest mountain in Spain. It's significantly higher than Mount Olympus [tallest mountain in Greece], and it's the typical height at which people ski in the Alps," Ian Johnston says. "So if the Greek gods wanted to build a new temple on Mount Olympus, and Mount Olympus wasn't available, they could just - but no more - do it with Lego bricks. As long as they don't jump up and down too much." # Some planets are harder to leave than others At his Psychology Today blog, Michael Chorost delves into a question about exoplanets that I've not really thought much about before — how easy they would be to leave. Many of the potentially habitable exoplanets that we've found — the ones we call "Earth-like" — are actually a lot bigger than Earth. That fact has an effect — both on how actually habitable those planets would be for us humans and how easily any native civilizations that developed could slip the surly bonds of gravity and make it to outer space. The good news, says Chorost is that the change in surface gravity wouldn't be as large as you might guess, even for planets much bigger than Earth. The bad news: Even a relatively small increase in surface gravity can mean a big increase in how fast a rocket would have to be going in order to leave the planet. It starts with one equation — SG=M/R^2. Let’s try it with [exoplanet] HD 40307g, using data from the Habitable Exoplanet Catalog. Mass, 8.2 Earths. Radius, 2.4 times that of Earth. That gets you a surface gravity of 1.42 times Earth. ... it’s amazingly easy to imagine a super-Earth with a comfortable gravity. If a planet had eight Earth masses and 2.83 times the radius, its surface gravity would be exactly 1g. This is the “Fictional Planet” at the bottom of the table. Fictional Planet would be huge by Earth standards, with a circumference of 70,400 miles and an area eight times larger. Does that mean we could land and take off with exactly the same technology we use here, assuming the atmosphere is similar? Actually, no. Another blogger, who who goes by the moniker SpaceColonizer, pointed out that Fictional Planet has a higher escape velocity than Earth. Put simply, escape velocity is how fast you have to go away from a planet to ensure that gravity can never bring you back. For Earth, escape velocity is about 25,000 miles per hour. Fictional Planet has an escape velocity 68% higher. That’s 42,000 miles per hour. Read the full story at Psychology Today blogs Thanks to Apollo 18, who also helped with the math for Chorost's post. ## Gifts for the space fans in your life — If yesterday's BoingBoing Gift Guide didn't give you enough holiday ideas, Popular Science has a collection of gifts for aspiring rocket scientists. Includes meteorite jewelry, a scarf printed with a pattern inspired by measurement systems, and some natty blazers designed by NASA. # Make a green bean matherole! (And other math-based Thanksgiving treats) Vi Hart is Khan Academy's professional mathemusician. (Yeah, I KNOW, right?) And, this year, she's making the most delightfully nerdy Thanksgiving dinner ever. It begins with green bean matherole, topped with fried Borromean onion rings. But, besides the fact that it's finished with crispy, delicious hyperbolic geometry, what makes the matherole a matherole? Vectors. Like the rings, vectors are part of geometry. They've got a magnitude (think: size of the green bean) and they've got a direction (think: which way the green bean is pointing). Most importantly, a single vector can be part of a field of vectors. And that, my friends, is an excellent starting point for a 9 x 13 pan full of beans. # Klein bottle bottle opener Yes, it's$72. But this 3-D printed metal sculpture/bottle opener is fantastic. And so is its marketing copy.

The problem of beer That it is within a 'bottle', i.e. a boundaryless compact 2-manifold homeomorphic to the sphere. Since beer bottles are not (usually) pathological or "wild" spheres, but smooth manifolds, they separate 3-space into two non-communicating regions: inside, containing beer, and outside, containing you. This state must not remain.

Read the rest of the product description and, you know, maybe buy the bottle opener, too. If you're feeling spendy.

# Wall Street is not made up of "numbers guys"

Chad Orzel's post, "Financiers Still Aren’t Rocket Scientists" is a timely reminder that Mitt Romney and other Wall Street Types are not, by and large, superhero math geniuses with their fingers on the arcane numeric truths underpinning all reality. Some quants are genuinely impressive mathematicians, but the industry's reputation for "numbers guys," is just wrong-o.

You would think that the 2008 economic meltdown, in which the financial industry broke the entire world when they were blindsided by the fact that housing prices can go down as well as up, might have cut into the idea of Wall Street bankers as geniuses, but evidently not. The weird idea that the titans of investment banking are the smartest people on the planet continues to persist, even among people who ought to know better– another thing that bugged me about Chris Hayes’s Twilight of the Elites was the way he uncritically accepted the line that Wall Street was the very peak of the meritocracy. It’s not hard to see where it originates– Wall Street types can’t go twenty minutes without telling everybody how smart they are– but it’s hard to see why so many people accept such blatant propaganda without question.

Look, Romney was an investment banker and corporate raider at Bain Capital. This is admittedly vastly more quantitative work than, say, being a journalist, but it doesn’t make him a “numbers guy.” The work that they do relies almost as much on luck and personal connections as it does on math– they’re closer to being professional gamblers than mathematical scientists. This is especially true of Bain and Romney, as was documented earlier this year– Bain made some bad bets before Romney got there, and was deep in the hole, and he got them out in large part by exploiting government connections and a sort of hostage-taking brinksmanship, creating a situation in which their well-deserved bankruptcy would’ve created a nightmare for the people they owed money, which bought them enough time for some other bets to pay off.

Romney has no shortage of nerve, and while he creeps me out, he has the sort of faux charm that works well in the finance community. But he’s not a “numbers guy” in any sense that looks meaningful from over here in the land of science. He can do the math needed to add up his personal fortune, but the game that he made his money playing isn’t a rigorously mathematical one– people get rich in finance as much by playing hunches and cutting sharp deals as by crunching numbers. There are people who make their way in that business by taking a rigorously data-driven approach to investing– one of the many things I need to write up for the blog at some point is a review of a forthcoming book called The Physics of Wall Street– but they’re nowhere near a majority of the industry, and Romney’s not one of them.

Financiers Still Aren’t Rocket Scientists (via Making Light)

## Math + Too Much Free Time = —

Here is a detailed analysis of the amount of time it would take to ride a hypothetical elevator down through the Earth's core and back out the other side of the planet. Apparently, this has something to do with the remake of Total Recall. But it's interesting even if (like me) you have no intention of seeing that movie. (Via Rhett Allain)