Reading the Comics, March 10, 2015: Shapes Of Things Edition

If there’s a theme running through today’s collection of mathematics-themed comic strips it’s shapes: I have good reason to talk about a way of viewing circles and spheres and even squares and boxes; and then both Euclid and men’s ties get some attention.

Eric the Circle (March 5), this one by “regina342”, does a bit of shape-name-calling. I trust that it’s not controversial that a rectangle is also a parallelogram, but people might be a bit put off by describing a circle as a sphere, what with circles being two-dimensional figures and spheres three-dimensional ones. For ordinary purposes of geometry that’s a fair enough distinction. Let me now make this complicated.

When you want to work in a multidimensional space it’s almost irresistible to start working with vectors; we can think of a vector $\vec{x}$ as representing how far and in what direction the point we’re interested in is from some reference point. (A little arrow on top is the popular notation to denote something is a vector, although if it’s too much of a bother to typeset an arrow, a bold typeface is next-best. The arrow is probably easier to write out on paper.) The reference point is dubbed the origin, to give it that sense of maybe being from a dystopian-yet-sluggish early-70s science fiction movie, although You can add and subtract vectors just the same way you can ordinary numbers. We also have an operation called “norm” which works rather like the absolute value of ordinary numbers does: it describes how far away a point is, without saying anything about in what direction it is. [1] We normally write that $\|\vec{x}\|$ And now … if the center of a circle is the point which two-dimensional vector $\vec{a}$ describes, and the radius of the circle is the number r, then the set of points at the end of two-dimensional vectors $\vec{x}$ which describe a circle with center $\vec{a}$ and radius r are the points for which the norm of the difference between $\vec{x}$ and $\vec{a}$ equals r; that is, for which $\|\vec{x} - \vec{a} \| = r$ is true.

But if we’re interested in three dimensions, in a sphere, then we use a three-dimensional vector $\vec{a}$ and the three-dimensional norm, and find the three-dimensional vectors $\vec{x}$ which describe a circle of radius r are the ones for which the equation $\|\vec{x} - \vec{a}\| = r$ is true. If we wanted the four-dimensional equivalent of a sphere (called a hypersphere if we want to be fussy), we would use a four-dimensional vectors $\vec{a}$ and the four-dimensional norm and we would find the hypersphere to be the points which make the equation $\|\vec{x} - \vec{a}\| = r$ true. For five dimensions (also called the hypersphere, because we start running out of stuff to call it) the shape is described by the same equation again. Being able to describe lots of very similar things by the same equation, albeit with different meanings behind the symbols, is a great advantage.

And so this is why we might not bother distinguishing between a circle and a sphere, however obviously relevant that difference might seem to be. And it’s why we might, if we were feeling a little impish, describe two points on a line to be a sphere.

[1] Warning! There are actually infinitely many different norms. I mean here the Euclidean norm, which matches the sort of straight-line distance you expect out of normal two- or three-dimensional space, and is what most mathematicians would assume you mean if you say “the norm” without qualifier. But there are different norms where, for example, this same equation would describe a square or a box instead of a sphere, and there are times you’d want that instead.

Samson’s Dark Side Of The Horse (March 6) gives me a little break since it just uses a blackboard full of equations to let the counting-sheep do a presumably more efficient job of getting Horace to sleep.

Dave Whamond’s Reality Check (March 6) is also an easy one to talk about since it just uses arithmetic as synecdoche for all the kinds of problems someone might have. Granted, addition is easier to illustrate as a problem than, say, a history essay would be. Yes, I’m bothered by the redundant equals sign at the last line of the problem too.

Bunny Hoest and John Reiner’s The Lockhorns (March 10) name-drops KenKen, which is a Sudoku-like puzzle in which not only do the digits have to be used exactly once in each row, each column, but the digits in each subdivision — not necessarily a rectangle — have to satisfy some arithmetic operation, such as that they add up to eight, or they multiply to 48. This brings an element of arithmetic into the pure logic puzzle of Sudoku, so I don’t blame Leroy for doing a bit of calculating on the side to see that he gets a consistent answer together.

Bunny Hoest and John Reiner’s The Lockhorns for the 10th of March, 2015. I have never completed a KenKen puzzle myself.

Tom Thaves’s Frank and Ernest (March 10) dubs Euclid of Alexandria as “Mister Obvious”, as Euclid shows off one of his less-controversial axioms. Euclid certainly, famously, organized Greek mathematics as it was understood into a form — axioms, propositions, definitions, proofs, corollaries, which employ diagrams for clarity and comprehension but do not depend on them for logical soundness — which is not just recognizable to us today as proper mathematics, but even defines what mathematics ought to look like to us. The axioms that Euclid starts the Elements — describing geometry and number theory — are largely ones that sound obvious, but that’s almost required by the nature of an axiom. Since an axiom is a statement you stipulate without proof to be true, it should either be something too obvious to prove or else something you just can’t do any useful work without. (There are many good versions of Euclid’s Elements online; if you use Java, here’s a nice one that shows off the propositions along with interactive figures.)

Tim Lachowski’s Get A Life (March 10, rerun) is one I thought I had mentioned before, and yeah, there it is, back from November of 2011. Anyway, I’m still bothered by the blackboard and its declaration that math “is nothing 2B2 squared of”.

John Graziano’s Ripley’s Believe It Or Not (March 10) claims there are 177,147 ways to tie a tie, which qualifies as mathematics content because every so often a mathematician gets a bit of press by revealing on a slow news day that there’s a new way to tie a tie. These are usually mathematicians who study knot theory, which is almost exactly what the name suggests: the study of the ways a perfectly flexible strand can be looped back and folded against itself. For a field that grew out of topology, the study of how to classify shapes, it’s one remarkably fruitful in real-world applications, since complicated problems like how proteins turn from chains of base molecules into shapes that do useful things, or bewildering ones like why Christmas tree lights get tangled up, fall within its domain. Tie-tying can almost be a sideline of all this heady work.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (March 10) does mention the Turing Test — the popular idea that while we may be able to say what intelligence is, we know it when we see it, which makes me sad we’ll never know how Plato would have reported Socrates’s reaction to that claim — as well as a “Robot Test”, using a tediously long piece of arithmetic to show mental power. I appreciate that for its nagging reminder that mathematics, even arithmetic, is not a culturally neutral way to test for intelligence, however temptingly impersonal a calculation might seem to be.

Of this set, hm. I think I’d give the edge to Frank and Ernest as the funniest, though nobody really manages to hit that sweet spot of being mathematical and solidly funny. I suppose everyone at Comic Strip Master Command was still slogging through the dull days in February when they were drawing these.