Randall Munroe’s xkcd has a table of slightly wrong equations and identities.

The table gives that the number of seconds in a year is 75⁴ = 31640625 or (using the “RENT method”) 525,600 × 60 = 31536000. (This refers to the song Seasons of Love, which features the number of minutes in a year prominently in its lyrics.) I’ve always been partial to “pi seconds is a nanocentury” (attributed to Tom Duff), which gives π × 10⁷ for the number of seconds in a year, but you have to know π for this to be useful.

Also, while I’m on the subject: it’s easy to remember that the circumference of the Earth is 4 × 10⁷ meters, or 40,000 kilometers; the meter was originally defined to be one ten-millionth of the distance from the North Pole to the equator through Paris. Call this C, or 2πr. The surface area of the Earth is therefore 4πr² = C²/π, which works out to (1.6/π) × 10¹⁴ square meters; may as well let π = 3.2 for the moment and call that 5 × 10¹³. Munroe gives 69⁸, which I suppose has some amusement value.

game theory used on a British game show

Using R to solve a simple numerical puzzle

A calculator that requires an approximate answer before it will give you an exact answer.

The Traveling Salesman movie, premiering in Philadelphia on June 16.

The kaleidoscopic patterns of cathedral ceilings.

From Language Log: longer reviews of wines are correlated with higher ratings (they are!) and whether baboons can tell the difference between English words and non-words (they can, but it looks like that’s because the “non-words” have different letter frequencies).

What does the average of all fonts (on one person’s computer) look like?

It turns out that overlaying letters from different fonts on each other doesn’t work too well – you get blurry results, like composite faces. But if you just take evenly-spaced points along the boundary of each letter in each font, and average them together, you get something quite readable.

I’m reminded of Metafont, which is used by TeX to specify fonts; each font is specified by a bunch of parameters, so averaging fonts becomes averaging numbers. There are some nice illustrations of this in the chapter on Metafont in Douglas Hofstadter’s book Metamagical Themas.

(via Hacker News)

Coursera is offering free online classes from Princeton, Stanford, Michigan, and Penn, in a variety of field.

Andrew Ng is offering a course in Machine Learning, which appears to be the same class as the free Machine Learning class that was offered over the winter. I’m not sure exactly how this compares to the Stanford CS229 class but it looks interesting. It starts today and runs 10 weeks.

The pilot classes were Machine Learning, Introduction to Artificial Intelligence and Introduction to Databases (that link goes to a Stanford press release from August 2011), but Coursera is now offering classes fairly widely spread across the curriculum.

Jordan Ellenberg has written a couple interesting blog posts on the future of higher education: What, if anything, is the future of the university? and Several attacks on the previous post; see also this Crooked Timber comment thread. Cathy O’Neill suggests that online learning promotes passivity.

At Berkeley student evaluations of courses and instructors are still done on paper forms; we’re supposed to do them in class on a day when attendance is good. This is why I’m not doing them today – it’s Friday, and beautiful weather, and both of those always lower attendance. Although in practice, I tend to do the evaluations on a day when having a slightly shortened class makes sense, as opposed to introducing new ideas at the end of a normal-length class.

But let’s say an instructor is only interested in having the average of their evaluations be as large as possible. Wouldn’t it make sense to do the evaluations on a day when attendance is comparatively low? On a day with high attendance you’re likely to have the marginally interested students there, who would give lower evaluations. I would assume that the students who come every day like the class more.

(This is actually testable, if you could get everyone to fill out the evaluation. You could ask students “how often did you come to class?” and compare their self-reported attendance with their evaluations of the class.)

From Cross Validated (stats.stackexchange.com, a web site which I think deserves to be better known): What is the best introductory Bayesian statistics textbook?

Some of the recommendations from this thread that I’ve seen before:

• John Kruschke,Doing Bayesian Data Analysis with R and BUGS. This book could be improved by having kittens on the cover instead of puppies.
• At a somewhat more advanced level is Gelman, Carlin, Stern, and Rubin, Bayesian Data Analysis. I own a copy of this book but have not had the time to sit down and work through it carefully.
• Not entirely Bayesian, but entirely fun, is David MacKay’s Information Theory, Inference, and Learning Algorithms. (You can download it for free from the author’s web page; this is happily true of a lot of books published by Cambridge.)

A few weeks ago I mentioned that the propagation of errors is a bit tricky. Say we want to predict the acceleration in an Atwood machine. The machine consists of a string extended over a pulley with masses at either end, of masses M and m, with M > m. The acceleration is given by

where g is the acceleration due to gravity, which we assume is known exactly. Let’s set g = 1, so we’ll have

We previously found by analytic methods that if and , then . But it’s instructive to do a simulation.

Specifically, fix some large . For , let be normally distributed with mean 100 and standard deviation 1; let be normally distributed with mean 50 and standard deviation 1; and let . Then the mean and standard deviation of the are estimates of the expected acceleration and its error.

This is very easy in R:


n = 10^4;
M = rnorm(n, 100, 1);
m = rnorm(n, 50, 1);
a = (M-m)/(M+m);


When I ran this code I got mean(a) = 0.3333875, sd(a) = 0.00993982. Furthermore, the computed values of a are roughly normally distributed, as shown by this histogram and Q-Q plot. (The line on the Q-Q plot passes through the point (0, mean(a)) and has slope sd(a).)

This works even if the errors are not normally distributed. For example, we can draw the simulated data from a uniform distribution with the given mean and standard deviation:


Mu = runif(n, 100-sqrt(3), 100+sqrt(3))
mu = runif(n, 50-sqrt(3), 50+sqrt(3))
au = (Mu-mu)/(Mu+mu)


I got mean(au) = 0.3332557 and sd(au) = 0.009936519. The distribution of the simulated results is a bit unusual-looking:

There’s also a way to compute an approximation to the error of the result using calculus, but simulation is cheap.

World population density visualizer, by Derek Watkins, via Metafilter and gizmodo. The original idea goes back to William Bunge‘s “Continents and Islands of Mankind”, redrawn at Making Maps. There we have a map of the areas where population density is greater than 30 per square kilometer, roughly “where people live”; Watkins adds a slider so you can change that number “30” to anything from 5 to 500.  Here’s a static map of the same data.

You should in theory be able to determine the population of the world from something like this, but the slider only goes up to 500, so you can’t tell how many people live at densities greater than 500 per square kilometer; these are “urban” densities (roughly) and so that’s a lot of people. Robert Talbert mentioned something similar on Twitter a few days ago: can you estimate the population of Colorado from a population density map? Not really, since the population of Colorado is very concentrated.

Physicist uses math to beat traffic ticket, via Physics Central and Hacker News.

The paper is dated 1 April 2012, so it may be a joke, but the idea is at least theoretically reasonable. The ticket in question was for not stopping at a stop sign. The idea is that to a police officer, a driver might appear to not stop, despite having actually reached zero speed for a moment, if another car happens to obstruct the officer’s view at the critical moment.

But is that a “stop” anyway? Is there some minimum amount of time one must be stopped at a stop sign? Still, it’s a nice little piece of mathematical modeling.

Andrew Gelman asks: do statisticians practice what we preach in teaching? (His conclusion: no.)

Samuel Arbesman, Probability and game theory in The Hunger Games and Brett Keller, Hunger Games survival analysis. Andrew Gelman writes: “I think it’s always good to get practice. Analyzing a book/movie is like doing sports statistics; it can keep you in shape.”

Here’s that annual list of the year’s best jobs. Mathematician is #10.

MagicTile: Geometrical and topological analogues of Rubik’s Cube.

Luis Apiolaza has some ideas about what an introductory book on Bayesian statistics should be like; commenters there have listed some of their favorite such books. He also has a post on first impressions of Kruschke’s book “Doing Bayesian Data Analysis” (which has puppies on the cover!) and a link to a free-for-noncommercial-purposes PDF version of Joseph Kadane’s Principles of Uncertainty.