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 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 And now … if the center of a circle is the point which two-dimensional vector describes, and the radius of the circle is the number r, then the set of points at the end of two-dimensional vectors which describe a circle with center and radius r are the points for which the norm of the difference between and equals r; that is, for which is true.

But if we’re interested in three dimensions, in a sphere, then we use a three-dimensional vector and the three-dimensional norm, and find the three-dimensional vectors which describe a circle of radius r are the ones for which the equation 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 and the four-dimensional norm and we would find the hypersphere to be the points which make the equation 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.

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 2B^{2} 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.

i have a question. if feeling impish describes how you would feel by proving that two points on a line can be shown to be a sphere … which i THINK you are implying you can do … then .,..

if by some formula or definition (i’m not sure which) D, you can describe a forest of trees by the density of trees within a certain area … AND … by another formula or definition B, you can, determine the number of board feet contained within the density of trees within a certain area …

… can it be said that, since you already know the number of board feet needed to build a two bedroom home, you can use the relationship between D and B to determine the number of two bedroom homes that could be built within any area populated by the same density of trees?

wull … assuming this is true then, is proving that two points on a line can be a sphere (you little devil) the same thing as not being able to see the forest for the number of two family homes that could be built?

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Well, now, indeed, I am putting forth that two points on a line can be, in the right context, interpreted as a sphere, or at least the way a sphere happens to exist in a one-dimensional space.

And, now, I agree with you almost all the way about describing the forest and the number of two-family homes it could be made into, if you have the density of trees and the number of board feet needed to build a home. But I think the setup falls a little short of what’s needed because it doesn’t make clear that the density of trees is really the density of usable board-feet in that area, and I’m not sure that we have a clear idea of how much area there is in the forest. If we take those as given, though, then yes: we’ve got missing forests for two-family homes.

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well … first of all thanks for reading my earlier posts … the ones about rain, ice, the moon, universe, infinity, black holes, etc.

i have a fairly long comment about your comment.

well … first of all my comment was attempt to draw a comic analogy between my interpretation of YOUR interpretation that two points on a straight line could be interpreted as a sphere, at least in one dimensional space.

the really humorous aspect of the statement for me being that, you could make the statement with a straight face knowing that most of the people who read your blog would accept and understand exactly what you meant by being impish (you little devil).

The mere thought that making this interpretation would be ‘’impish’’ is so far from my lexicon that, despite the fact that the people in your milieu probably completely understood your meaning … the statement took on a massive amount (like … the amount of energy it would take to completely fill up a black hole … which i realize is a really stupid thing to say but maybe funny?) of absurdist humor, to me!

I mean, i had never in my life heard … or indeed THOUGHT that i’d ever hear … such a statement made about such a subject. The absurdity of the statement … the sheer unexpected aspect of it … (please interpret ‘’absurdist’’ as an example of writing something good) caused me laugh last night, so hard that i woke my wife up who had been sleeping for at least an hour.

In fact, the statement was ALMOST as funny as a comment you made recently about pulling some really interesting artwork … the ‘medium’ being i believe, ice … ) out from beneath your swimming pool heater, which also sent me into paroxysms of laughter.

For me … humor … is the result of seeing or hearing something totally unexpected.

Both of your statements reached the highest level of humor since … how could i (a total and complete layman) EVER consider a person ‘impish’ because they could interpret two points on a straight line as a sphere but ONLY in one dimensional space? (hahahahahah

AND that you had found some interesting art work beneath your swimming pool heater?!!! (i’ve got a smile on my face as i write this … but hope i don’t start laughing because i want to finish this comment soon …

so anyway … as a result of my interpretation of your statement, about spheres while being impish, i decided to make up a situation that was absurd but, that i thought was fairly logical … except that, at the end i gave it an ‘absurd’ twist based on the statement … “you can’t see the forest for the trees” … which i thought was absurd enough to be pretty funny … wull … it made me laugh anyway …

i don’t think you got the joke … but that’s ok since … the fact that you proceeded to tell me the falsities of my ‘’suppositions’’ and why my conclusions were not necessarily valid … this caused even greater paroxysms of laughter.

as to your conclusion about the variation of the density of wood invalidating my conclusion … i would disagree with you completely since given a large enough sample, perhaps in a specific geographic area (which i KNOW i didn’t mention … ) the variations would even out so that the number of two bedroom homes COULD be determined.

so there … ! This is just another example of you NOT being able to see the forest for the trees … !! i enjoy your blog and will return … ks

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Aw, well, now, I did expect that you were being whimsical with writing about forests and trees, but I also didn’t want you to think I was just passing over your comment, and when I thought about it I could think of something interesting to say about it, and maybe useful in thinking about how to think about problems.

I like to think one of my good traits is being unafraid to look like I didn’t get the joke. It gives other people something amusing to respond to, if nothing else, and makes me look like a better sport than I worry I am.

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on jeeze … i didn’t see the forest for the trees!! ” … yes: we’ve got missing forest for two family homes.” ha ha ha ha … i guess if you assume that the ‘exercise’ applies to the real world (what is ‘real world’?) then there WOULD be missing forests … pardon my uppityness … .

Anyway, not being part of the mathematics/physics demographic, would you say that overall, this group is obsessed with definition? i wonder if i’d get along better with groups of “scientists” than people outside of this field … (i know there are other fields concerned with definitions … ) since i tend to zero in on words that people use, within statements, that i don’t think apply to the situation which eventually tends to piss people off if taken too far … .

i have learned to overlook the use of what i think could be a more specific word, but the word used, still bangs around inside my mind.

what happens is, my brain starts thinking of absurd usages for the word while i’m stifling myself from saying something.

this tendency to think of absurd uses for words is a great asset when it comes to writing. the result is, i love being a hermit (from time to time) when i want to do a lot of writing since i can focus in on word usage and not bug the shit out of people. i wonder if mathematicians think/feel the same way about numbers?

these kind of ridiculous”theories” that i pass on to people usually leave them scratching their heads wondering what the fuck i’m talking about. do mathematicians overall, tend to hang out in cliques of like minded people for the same reason?

i’m being serious and speaking the truth about such matters, from my own perspective … but i think it’s absurdly hilarious at the same time. please forgive me if it seems like i’m being patronizing because my intent is NOT to be patronizing.

my latest ‘obsession’ revolves around my thoughts on a much broader scope of POLARITY in the world around us.

my attempts to explain these thoughts to my wife the other day, resulted in her getting pissed off at me which lead me to try to use that circumstance as an example of how polarity can ratchet up in the real world which … made matters worse.

the last comment or question i would like to make is … is there any reason why the Great Salt Lake couldn’t be made into a ‘super dynamo’ since salt water is a necessary ingrediant for making electricity?

i figure that, since plus and minus are mathematic terms maybe you’d have an opinion.

or, you could table this discussion since i’m sure you have much more important issues to deal with. (another example of polarity)

in any case … don’t waste your time if you have limited amounts of it .,.. and thank you so much for reading my blog … ks

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I’ve been doing a fair bit of thinking about this because I think it’s an interesting question: are mathematicians (and people in related fields) obsessed with definitions? I think I’ve come to conclude that while mathematicians (and that demographic) tend to think more, and maybe more critically, about definitions than average folks do, it’s not exactly because they’re obsessed with definitions, but because they’re interested in what they can do with definitions.

The big interest that mathematicians have, I think, is in finding interesting things which are true to say about something. But what makes something interesting to say? It’s probably got to be something which is implied by the system you’ve set up, but which isn’t obvious from the original setup. But if it’s not obvious, then it’s got to be something that can be deduced from the setup, and it has to be something which can be judged against the rules of the setup and said to be either true or false.

And that’s where definitions come in: if you don’t have a fairly good idea of, say, what a Therblig Number is, you can’t really say whether “2,038” is a Therblig Number, or whether it isn’t. And if nobody knows what a Therblig Number is (certainly I don’t, and I made up the name), it’s not going to be interesting to say whether it is or isn’t. Your idea of what the definition is might not be precise, and it might need revision as you find it implies things you don’t want it to, but you have an idea there is this thing called a Therblig Number and that it has some traits you find interesting enough to label, and that’s why definitions — or, more generally, working out what the properties of a well-defined problem are — end up being of interest to mathematician types.

As for the Great Salt Lake, I’m afraid I don’t actually know that salt water is necessary for making electricity, or how using it for electricity would affect the lake’s other uses, so I don’t think I know enough about the problem to venture an opinion.

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you bring up some really interesting points which i am going to LOVE to delve into with, having to do with art and science … (and i don’t mean art V.S. science) i don’t have time to delve at the moment though since it’s going on 1:00 a.m. here. but i WILL and i’m looking forward to it. i have reservations though, … you might say i am an over analytical person … so for every statement you make i’ll come up with a couple of conclusions which can branch off and soon i am lost in space … whoops .. i’m going off on a tangent right now … and i don’t want to … maybe one way to overcome this problem is to look at how i think … in terms of vectors … now …. i always think of vectors as straight lines … and i think if vectors are straight lines then they will intersect with other vectors … but, intersecting vectors are not ”tangents’ ‘ (which is the way i think … ) (jeeze, this gets complicated since i don’t know your lexicon so my use of ‘tangent’ and your usage is probably completely different than mine) … so while two vectors … after they intersect … even though they go off in two separate directions … this is not the same as a tangent … which ….. TO ME … a tangent would be a vector that suddenly splits … so that the vector goes off in two directions … … maybe this ‘species’ of vector has a definition i’m not aware of …. the point i was tying to make is … my mind … thinks in terms of split vectors … let’s call it a Therblig Vector, so discussions can get pretty tedious … i think the beauty of science is that you are looking for Truth … but by scientific definition, (i’m going out on a real long limb here … ) TRUTH is a single statement … E = mc2 …

there’s great beauty i n the language you use i order to find this ‘truth” because it’s so ‘precise’ … but based on the way i think … everything is ‘tangental” .. thoughts splitting and resplitting … to me, discoveries can be made by this constant re-splitting … re-splitting … going deeper and deeper into these splits …. so that there are many many discoveries made along the way as opposed to having an idea that there’s something then trying to find the language to confirm that the thing exists … vectors continually splitting gives an infinite places to go and i suppose an infinite amount of discoveries to make along the way … ………. i think there’s some truth to this since structurally, the brain is comprised of branching ‘vectors’ … as are … if you look .. at the structure of trees … their limbs … basically there are splits and splits and splits … reaching out collecting information … while at the same time the roots of the tree … as a reflection of the tree’s limbs .. are doing the same thing underground …

so i started out saying i had to go to bed and it’s now getting close to two o’clock and it’s all because i was being over analytical looking at the application of vectors and ”tangents” (by my definition) as they apply to how we think and whether it is best to journey through time seeking TRUTHS as opposed to seeking TRUTH … hey … please forgive me … i think i’m going off half cocked here … making all these statements … my final comment is …. that, maybe the definition of ART is … a random discovery made while searching for the right word or the right color … a search that’s almost random in nature waiting for something to ‘fall into place ” by following the right ‘split’ … or tangent …

it’s the idea that maybe this is the difference between pure art and pure science … DON’T GET ME WRONG … I’M NOT SAYING THAT WITHIN MATH. OR SCIENE IN GENERAL there isn’t creativity … i guess i’m writing more about the different languages and different approaches … even though i know there is plenty of overlap … i’m really sorry if i’ve confused you … i’m not going to proof read this … so i know it’s probably confusing but maybe there are a couple of grains of corn that we can harvest into a nicely organized corn cob that we both can look at and think is beautiful for the same reasons and for different reasons … jeeze … i had no idea where this whole thing was going … and didn’t even think i’d continue and … here i am … i hope you slept well … thanks … for even considering my words … ks

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I liked that the equation on the blackboard EQUALED z, an often used variable. Dark Side of the Horse often seems to have good math comics, doesn’t it?

I can think of times when that Frank and Ernest strip would be quite good to show in a classroom.

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Yes, letting the variable be z is one of those little touches of craftsmanship that makes Dark Side of the Horse stand out in these mathematics roundups. I don’t know Samson’s biography. It’s easy to suppose she or he might have a mathematics-inclined background, although it’s just as easy to suppose she agrees with the notion that having the irrelevant details check out makes the overall joke stronger.

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