Opting to build the Death Star in the shape of a sphere may not have been classic Star Wars villain Darth Vader's wisest move, according to math teacher Ben Orlin. He investigates this burning question, and so much more, in his fabulous new book, Math with Bad Drawings, after Orlin's blog of the same name.
Orlin started using his crude drawings as a teaching tool. He drew a figure of a dog one day on his chalkboard to illustrate a math problem, and it was so bad the class broke out in laughter. "To see the alleged expert reveal himself as the worst in the room at something—anything—can humanize him and, perhaps, by extension, the subject," he writes. When he started his blog, he knew that pictures would be crucial to helping readers visualize the mathematical abstractions. Since he had no particular artistic talent, he opted to just cop to it up front. And thus, the "Math with Bad Drawings" blog was born.
The book is a more polished, extensive discussion of the concepts that pepper Orlin's blog, featuring his trademark caustic wit, a refreshingly breezy conversational tone, and of course, lots and lots of very bad drawings. It's a great, entertaining read for neophytes and math fans alike, because Orlin excels at finding novel ways to connect the math to real-world problems—or in the case of the Death Star, to problems in fictional worlds.
Spheres in space
Architecture and design are filled with math, and "Geometry yields to no one, not even evil empires," Orlin writes. The Star Wars movies never really explain the specifics behind the construction of the Death Star, giving Orlin the perfect opportunity to indulge in speculation about what could have happened behind the scenes. This takes the form of imagined dialogues among the team responsible for constructing it.
Lord Vader nixed design shapes like pyramids, cubes, or a cylindrical Death Pencil, because he adores symmetry.
The Death Star is a near-perfect sphere, a hundred miles across, with a planet-vaporizing laser. In Orlin's version, Lord Vader nixed design shapes like pyramids, cubes, or a cylindrical Death Pencil, because he adores symmetry, and a sphere is the maximally symmetric shape. But it's hell on the aerodynamics, since how much force an object experiences from air molecules as it travels through an atmosphere depends on (a) whether those molecules are hitting its surface in parallel or (b) perpendicularly at 90-degree angles.
As Grand Moff Tarkin points out, unlike the sensibly designed Star Destroyers, where air molecules mostly glance off the sides as the spacecraft travel through the atmosphere, the Death Star is just one huge surface area. The air molecules are always hitting it at near-perfect right angles, so the vessel must bear the full brunt of the impact. So the Death Star cannot visit planets directly, meaning it could not enter an atmosphere and vaporize the occasional continent while blasting The Imperial March through loudspeakers. It has to remain in the vacuum of space, where there is no air resistance.
That brings up another issue. The Death Star was constructed in space, a realm where massive things (moons, planets) tend to take on a spherical shape due to gravity. But when Orlin did the calculations, he found that the size at which objects take on the shape of a sphere is about 400 kilometers in diameter, which is significantly larger than the ~160km Death Star.
"Bodies of ice, it's even larger, because ice is not as dense, so you don't get the same sort of gravitational critical point," he says. That's why Orlin's hypothetical Imperial team physicist keeps insisting the Death Star should be more lumpy, shaped like an asteroid. Orlin's conclusion: "The Death Star is not nearly big enough. But maybe Imperial steel is very, very dense and thus goes spherical much faster."
Damn lies and statistics
Statistics is another discipline ripe for Orlin's razor-sharp wit: "A statistic is an imperfect witness. It tells the truth but never the whole truth." For example, when discussing the correlation coefficient, he cites "Anscombe's Quartet." Devised by the late English statistician Frank Anscombe, the Quartet consists of four datasets that look completely different when you graph them out. In one, the data is a straight line with a single outlier. In another, the data is scattered around a straight line. For yet another, there is a horizontal line with one crazy outlier. And finally, the fourth dataset makes a parabola. But summary statistics can obscure these crucial differences.
To illustrate, Orlin envisions students at a hypothetical Anscombe Academy of Witchcraft and Wizardry. That's where students must take exams in four classes: Potions, Transfiguration, Charms, and Defense Against the Dark Arts. We're interested in comparing how long a student studies for a test with their actual score on said test, because we want to find an interesting correlation. If we just look at summaries of the data, the statistics all come up the same. Students on average spend the same amount of time studying. They get the same average score. The variance in the scores is the same. And the correlation between study time and final score is the same.
"But when you look at the actual data, they're totally different," says Orlin. Transfigurations follows a neat straight line: every extra hour of study results in a 0.35 improvement in the test score. Potions shows the data points scattered around a line, with some random noisy interference: more study helps, but it's not a sure thing. Graphing the data for the Charms class produces a tidy parabola where studying may improve your score to a point but with diminishing marginal returns. Studying for more than 10 hours straight with the addition of fatigue will adversely affect your performance. As for Defense Against the Dark Arts, that's the horizontal line with one crazy outlier—one annoying overachiever who studies for 19 hours and aces the test will artificially strengthen the average by a lot.
"The point is, even when we're using all our summary statistics, we might still be missing things," says Orlin. This is particularly the case when we examine very small datasets. The whole point of statistics is to eliminate complexity: to simplify and summarize. "But we've always got to remember that they're summaries or else we'll wind up reading very different things as though they're the same," he cautions.
Hitting the jackpot
One of Orlin's favorite chapters is entitled "The Ten People You Meet in Line for the Lottery." This chapter is designed to shatter conventional stereotypes about who is most likely to play. "The stereotype is that the lottery is a tax on the uneducated or it's played almost entirely by lower-income people," he says. "That's just not true." Massachusetts, for example, is a high-income, highly educated state, yet it's also the state with the most lottery spending.
So who might you meet in line? There's the Gamer, who buys tickets for pleasure. The Educated Fool, "a rare sap-brained creature who does with 'expected value' what the foolish always do with education: mistake partial truth for total wisdom." The Big Roller, who thinks he can transcend risk by buying every possible combination of numbers. There's the Dutiful Taxpayer, The Dreamer, The Enthusiast for Scratching Things, and so on.
And then there's the Kid Who Just Turned 18, aka, Orlin himself. He admits to once buying two lottery tickets, on his 18th birthday, despite understanding full well the futility of the odds. He didn't win. Mostly he regretted buying two tickets instead of one. "Either I live in the world where probability works, in which case my ticket is not gonna win, or I live in some kind of magical world that wants me to win the lottery, in which case I don't need to buy two," he explains. "In neither world does it make sense to buy a second ticket."
That's thinking like a true mathematician.
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