Trivia of Use to Fibonacci … or MacGyver

From Futility Closet

Since the ratio of kilometers to miles is very close to the Golden Ratio, consecutive terms later in the Fibonacci series (once you get to 3) are approximate conversions of those distances.
1, 1, 2, 3, 5, 8, 13, 21, …

3 miles is 5 km, 5 miles is 8 km, etc.

This knowledge, along with some bubble gum and a paper clip, should be sufficient to thwart the bad guys in at least three ways. Get on with it!

How Does Calculus Compare?

Your math teacher, Darth Vader. (Boy, is he strict!)

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The ability to calculate the length of the hypotenuse is insignificant next to the power of the force.

I have to add — I would have noted that Vader’s board skills leave something to be desired, but he might have found my lack of faith to be disturbing.

The Other Manhole Cover

Drilling Square Holes at Linearly Independent. All about the Reuleaux triangle and rotations thereof.

[I]t has the same maximum width regardless of how it is rotated. this property was thought to be only possessed by circles once and yet here’s a simple and apt counter example.

Constant width shapes can make great manhole covers, as no orientation of the cover will let it fall through the manhole. A few years ago Microsoft interviewers asked job applicants why manhole covers were round and this was thought to be one of the best answers.
Another more important property of constant width shapes is that they can rotate inside parallel lines.

Game Theory

A side comment by Matt about quizzes triggered a thought (so many of these interactions are induced rather than spontaneous)

I have all my old lecture notes and materials so the only real thing I have to do is make up new quizzes. Students are good at nothing if not gaming the system and they’d notice repeated quizzes pretty quickly.

When I TA’d I did labs, but the same idea applied. It was assumed that the students had access to old lab reports and exams (especially if they were in a fraternity or sorority) so the one thing we could make different was a question or two tacked on to the end of the calculations. And that did trip up a couple of students, who had obviously just copied from some old report to which they had access. Professors had various strategies about re-using questions, but I think the use of computers has made it far easier to keep a large database and mix-and-match questions that simple memorization of old exams prohibitive for introductory classes.

When I was teaching in the navy it wasn’t an issue. Quizzes didn’t count toward your grade, so there was no real incentive to cheat, other than trying to get out of some extra problems to be worked because the instructor might assign them to people who failed several quizzes. There was no master file of exam questions because they were treated as restricted material — the students did not keep them, and they were strictly accounted for. But to cut down on the possibility of some “oral tradition” information flow between the different classes in session, questions were not re-used until the class that had taken that exam had graduated.

We had one incident that occurred just before I had transferred into one division — an exam went missing. The most likely explanation is someone miscounted, but what was recorded was that there were 126 exams (and they were numbered) and after the exam was administered #126 was nowhere to be found. So the exam was assumed to have been compromised for future tests, and all of the questions on the exam had to be removed from the exam bank. As it turned out, I inherited the job of writing that particular exam, so it fell to me to repopulate the stockpile — two brand spankin’ new questions per exam for the next year, so I got a lot of practice coming up with new material. Which isn’t that hard, because an old question with new numbers and solving for a different variable is a “new” question. The use of old questions wasn’t laziness, though — we didn’t grade on a curve, and the goal was to test each class the same, so you kept statistics, and made tests that had a predicted result of between a 3.1 and 3.2 on a 4.0 scale. A venerable question was well-trusted, and a new or changed question could throw the result off. If a class got an unexpectedly high or low score (usually low), an audit was initiated to try and ensure that there was nothing hinky going on. This was especially odious for the early exams, before the class had a chance establish itself as being above- or below-average. If a class had underperformed on earlier exams, tanking a later exam didn’t raise eyebrows. But at least once the conclusion was that it was the Russian judge a new question or two were harder than had been predicted, and had shaved a few tenths off the score.

But even within that strict paradigm, an exam-writer could game the system a little. No matter how much you’d drill it into the students’ heads to skip a tough question and go back to it later, there were those who didn’t. They’d invariably leave an easy question or two blank because they took too much time on another question that they still got mostly wrong. So putting tougher questions toward the front would tend to lower scores a little bit.

The Necessity of Mathematics

Awesome megapost over at Science after Sunclipse, covering many overlapping issues on the topic.

To use mathematics in the natural sciences, we first decide how we wish to represent some aspect of the world in mathematical form. We then take the diagrams and equations we’ve written and manipulate them according to logical rules, and in so doing, we try to make predictions about Nature, to anticipate what we’ll see in places we have not yet looked. If additional observations corroborate our expectations, then we’re on the right track. (It’s rarely so clean-cut as that — the process can spread across thousands of people and multiple generations of activity — but that’s the gist of it.) Several skill sets are involved: one must know how to idealize the world, and then how to work with that idealization. Remarkably enough, our schools fail to teach either skill.

Don't Fall Behind

Ketchup

A whole lot about the king of condiments, without getting into shear thinning and thixotropic properties.

It explains why Barenaked Ladies can’t find the fancy dijon ketchup they want in “If I Had a $1,000,000”

What Heinz had done was come up with a condiment that pushed all five of these primal buttons. The taste of Heinz’s ketchup began at the tip of the tongue, where our receptors for sweet and salty first appear, moved along the sides, where sour notes seem the strongest, then hit the back of the tongue, for umami and bitter, in one long crescendo. How many things in the supermarket run the sensory spectrum like this?

The business decision of empowering kids

A typical five-year-old consumes about sixty per cent more ketchup than a typical forty-year-old, and the company realized that it needed to put ketchup in a bottle that a toddler could control. “If you are four—and I have a four-year-old—he doesn’t get to choose what he eats for dinner, in most cases,” Keller says. “But the one thing he can control is ketchup. It’s the one part of the food experience that he can customize and personalize.”

With some multivariable optimization thrown in.