Reading the Comics, November 13, 2019: I Could Have Posted This Wednesday Edition

Now let me discuss the comic strips from last week with some real meat to their subject matter. There weren’t many: after Wednesday of last week there were only casual mentions of any mathematics topic. But one of the strips got me quite excited. You’ll know which soon enough.

Mac King and Bill King’s Magic in a Minute for the 10th uses everyone’s favorite topological construct to do a magic trick. This one uses a neat quirk of the Möbius strip: that if sliced along the center of its continuous loop you get not two separate shapes but one Möbius strip of greater length. There are more astounding feats possible. If the strip were cut one-third of the way from an edge it would slice the strip into two shapes, one another Möbius strip and one a simple loop.

Or consider not starting with a Möbius strip. Make the strip of paper by taking one end and twisting it twice around, for a full loop, before taping it to the other end. Slice this down the center and what results are two interlinked rings. Or place three twists in the original strip of paper before taping the ends together. Then, the shape, cut down the center, unfolds into a trefoil knot. But this would take some expert hand work to conceal the loops from the audience while cutting. It’d be a neat stunt if you could stage it, though.

Presenting as a magic trick: cutting a loop of paper in half, along the loop's center, with the result being a single yet larger loop. The trick is to make the paper into a Moebius strip, and conceal the 'twist' so that your audience does not know what they see.
Mac King and Bill King’s Magic in a Minute for the 10th of November, 2019. This and other mathematics-based tricks featured in Magic In A Minute are at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 10th uses mathematics as obfuscation. We value mathematics for being able to make precise and definitely true statements. And for being able to describe the world with precision and clarity. But this has got the danger that people hear mathematical terms and tune out, trusting that the point will be along soon after some complicated talk.

Confession Tip: Use Statistics. Kid: 'Mom! Dad! Did you know that, in your immediate area, teen pregnancy may be as high as 100 percent?!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 10th of November, 2019. This and many other essays discussing Saturday Morning Breakfast Cereal are at this link.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 11th would be a Pi Day joke if it hadn’t run in November. But when this strip first ran, in 2010, Pi Day was not such a big event in the STEM/Internet community. The Boychuks couldn’t have known.

The formulas on the blackboard are nearly all legitimate, and correct, formulas for the value of π. The upper-left and the lower-right formulas are integrals, and ones that correspond to particular trigonometric formulas. The The middle-left and the upper-right formulas are series, the sums of infinitely many terms. The one in the upper right, \sum \frac{1}{n^2} = \frac{\pi^2}{6} , was roughly proven by Leonhard Euler. Euler developed a proof that’s convincing, but that assumed that infinitely-long polynomials behave just like finitely-long polynomials. In this context, he was correct, but this can’t be generally trusted to happen. We’ve got proofs that, to our eyes, seem rigorous enough now.

On the blackboard several calculus formulas for the value of pi. At the table a scientist type says to another, while eating a slice of pie, 'I don't know why, Haskins, but I've had a craving for this all day.'
Brian Boychuk and Ron Boychuk’s The Chuckle Brothers rerun for the 11th of November, 2019. It originally ran the 29th of November, 2010. This and other essays mentioning The Chuckle Brothers, which has gone into perpetual reruns, are at this link.

The center-left formula doesn’t look correct to me. To my eye, this looks like a mistaken representation of the formula

\pi = 2 \sum_{k = 0}^{\infty} \frac{2^k \cdot k!^2}{\left(2k + 1\right)!}

But it’s obscured by Haskins’s head. It may be that this formula’s written in a format that, in full, would be correct. There are many, many formulas for π (here’s Mathworld’s page of them and here’s Wikipedia’s page of π formulas); it’s impossible to list them all.

The center-right formula is interesting because, in part, it looks weird. It’s written out as

\pi = \frac{4}{6+}\frac{1^2}{6+}\frac{3^2}{6+}\frac{5^2}{6+}\frac{7^2}{6+} \cdots

That looks at first glance like something’s gone wrong with one of those infinite-product series for π. Not so; this is a notation used for continued fractions. A continued fraction has a string of denominators that are typically some whole number plus another fraction. Often the denominator of that fraction will itself be a whole number plus another fraction. This gets to be typographically challenging. So we have this notation instead. Its syntax is that

a + \frac{b}{c + \frac{d}{e + \frac{f}{g}}} = a + \frac{b}{c+} \frac{d}{e+} \frac{f}{g}

There are many attractive formulas for π. It’s temping to say this is because π is such a lovely number it naturally has beautiful formulas. But more likely humans are so interested in π we go looking for formulas with some appealing sequence to them. There are some awful-looking formulas out there too. I don’t know your tastes, but for me I feel my heart cool when I see that π is equal to four divided by this number:

\sum_{n = 0}^{\infty} \frac{(-1)^n (4n)! (21460n + 1123)}{(n!)^4 441^{2n + 1} 2^{10n + 1}}

however much I might admire the ingenuity which found that relationship, and however efficiently it may calculate digits of π.

Eno, on a coffee date: 'So you claim you're a teacher, huh?' Teacher: 'What do you mean, 'claim'?' Eno; 'What's 30 divided by 5?' Teacher: 'Six!' Eno: 'OK, you check out.'
Glenn McCoy and Gary McCoy’s The Duplex for the 13th of November, 2019. This surprises me by not being a new comic tag. Essays mentioning The Duplex are at this link.

Glenn McCoy and Gary McCoy’s The Duplex for the 13th uses skill at arithmetic as shorthand for proving someone’s a teacher. There’s clearly some implicit idea that this is a school teacher, probably for elementary schools, and doesn’t have a particular specialty. But it is only three panels; they have to get the joke done, after all.

And that’s all for the comic strips this week. Come Sunday I should have another Reading the Comics post. And the Fall 2019 A-to-Z draws closer to its conclusion with two more essays, trusting that I can indeed write them, for Tuesday and Thursday. I also have something disturbing to write about for Wednesday. Can’t wait.

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

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