Wed. Dec 7th, 2022
Geometry imposes some real design constraints on Darth Vader's desire for a spherical Death Star.
enlarge / Geometry imposes some real design constraints on Darth Vader’s desire for a spherical Death Star.

According to math teacher Ben Orlin, choosing to build the Death Star in the shape of a sphere may not have been the wisest move by classic Star Wars villain Darth Vader. He explores this burning question, and much more, in his fantastic new book, Math with bad drawingsto Orlin’s blog of the same name.

Orlin began to use his rough drawings as a teaching aid. One day he drew a dog on his board to illustrate a math problem, and it was so bad that the class burst out laughing. “Watching 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 photos would be crucial in helping readers visualize the mathematical abstractions. Not having any particular artistic talent, he chose to just do it ahead of time. And so the blog “Mathematics with Bad Drawings” was born.

The book is a more polished, comprehensive discussion of the concepts that crisscross Orlin’s blog, with its signature biting humor, refreshingly light-hearted conversational tone and, of course, lots and lots of very bad drawings. It’s a great, entertaining book for beginners and math fans alike, because Orlin excels at finding new ways to connect math with real-world problems — or in the case of the Death Star, with problems in fictional worlds.

Bulbs in space

Architecture and design are full of math, and “Geometry doesn’t reveal anyone, not even evil realms,” Orlin writes. The Star Wars movies never really explain the details behind the Death Star’s construction, giving Orlin the perfect opportunity to speculate about what could have happened behind the scenes. This takes the form of imagined dialogues between the team responsible for building it.

Lord Vader has swapped out design shapes like pyramids, cubes, or a cylindrical Death Pencil because he loves symmetry.

The Death Star is a near-perfect sphere, a hundred miles across, with a planet-evaporating laser. In Orlin’s version, Lord Vader has design shapes like pyramids, cubes, or a cylindrical Death Pencil, as he loves symmetry, and a sphere is the maximally symmetrical shape. But aerodynamics is hell, because how much force an object experiences from air molecules as it travels through an atmosphere depends on (a) whether those molecules hit the surface parallel or (b) perpendicular at a 90-degree angle.

As Grand Moff Tarkin points out, unlike the sensibly designed Star Destroyers, where air molecules usually look off the sides as the spacecraft travels through the atmosphere, the Death Star is just one massive surface. The air molecules always hit it at almost perfect right angles, so the craft has to bear the full impact of the impact. So the Death Star can’t visit planets directly, meaning it can’t get into an atmosphere and occasionally vaporize a continent while exploding The Imperial March through speakers. It must remain in the vacuum of space, where there is no air resistance.

That poses another problem. The Death Star is built 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 the shape of a sphere is about 400 kilometers in diameter, which is considerably larger than the ~160 km long Death Star.

“Ice bodies, it’s even bigger, because ice isn’t that dense, so you don’t get the same kind of gravity critical point,” he says. Therefore, the hypothetical physicist of Orlin’s Imperial team insists that the Death Star should be more lumpy, in the form of an asteroid. Orlin’s conclusion: “The Death Star isn’t nearly big enough. But maybe imperial steel is very, very dense and so it becomes 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.” The quartet, created by the late English statistician Frank Anscombe, consists of four datasets that look completely different when you draw them. In one, the data is a straight line with a single outlier. In another, the data is spread out in a straight line. For yet another, there is a horizontal line with one crazy outlier. And finally, the fourth data set makes a parabola. But summary statistics can obscure these crucial differences.

Finding the math in the everyday world.
enlarge / Finding the math in the everyday world.

Ben Orlin

To illustrate: Orlin imagines students from a hypothetical Anscombe Academy of Witchcraft and Wizardry. That’s where students have to take exams in four classes: Potions, Transfiguration, Charms, and Defense Against the Dark Arts. We are interested in comparing how long a student studies for a test with their actual score on that test because we want to find an interesting correlation. If we just look at the summaries of the data, the stats all come down to the same thing. On average, students 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 if you look at the actual data, they’re completely different,” Orlin says. Transfigurations follows a nice straight line: every extra hour of studying results in a 0.35 improvement in the test score. Potions shows the data points scattered along a line, with some random interference with noise: more research helps, but it’s not certain. Graphing the data for the Charms class creates a neat parabola where studying can improve your score to a point, but with diminishing marginal returns. Studying for more than 10 hours in a row with the addition of fatigue will have a negative effect on your performance. As for Defense Against the Dark Arts, that’s the horizontal line with one crazy outlier – one pesky overachiever studying for 19 hours and passing the test will artificially boost the average a lot.

“The point is, even if we use all of our summary stats, we can still miss things,” Orlin says. This is especially true when we examine very small data sets. The whole point of statistics is to eliminate complexity: to simplify and summarize. “But we must always remember that they are summaries, otherwise we will read very different things as if they were the same,” he warns.

Win the jackpot

One of Orlin’s favorite chapters is entitled ‘The Ten People You Meet in Line for the Lottery’. This chapter aims to break down conventional stereotypes about who is most likely to play. “The stereotype is that the lottery is a tax on the uneducated or that it is played almost entirely by lower-income people,” he says. “That’s just not true.” Massachusetts, for example, is a high-income, high-educated state, but it’s also the state with the most lottery spending.

So who could you meet in line? There’s the Gamer, who buys tickets for fun. The Educated Fool, “a rare witty creature who does with ‘expected value’ what the fools always do with education: confusing partial truth with total wisdom.” The Big Roller, who thinks he can exceed the risk by buying every possible combination of numbers. You have the dutiful taxpayer, the dreamer, the crab lover, and so on.

And then there’s the Kid Who Just Turned 18, aka Orlin himself. He admits that he once bought two lottery tickets, on his 18th birthday, despite fully understanding the futility of the odds. He didn’t win. Most of the time he regretted buying two tickets instead of one. “Either I live in the world where probability works, in which case my lottery ticket isn’t going to win, or I live in some kind of magical world that wants me to win the lottery, in which case I don’t have to buy two,” he explains. from. “In neither world does it make sense to buy a second ticket.”

That is thinking like a real mathematician.

By akfire1

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