Reading the Comics, December 11, 2015: So, That Didn’t Work Edition

I’d hoped that running a slightly-too-soon edition of Reading the Comics would let me have a better-sized edition for later in this week. Then everybody did comics for the 11th of December. I can have a series of awkward-sized essays or just run what I have. I wonder which I’ll do.

Aaron McGruder’s The Boondocks for the 6th of December is a student-resisting-the-problem joke. It ran originally the 24th of September, 2000, if the copyright information is right. The original problem — “what is 24 divided by 4 minus 2” — is a reasonable one for at least some level of elementary school. (I’m vague on just what grade Caesar is supposed to be in. It’s a problem for any strip with wise-beyond-their-years children. Peanuts plays with this by having the kids give book reports on Peter Rabbit and Tess of the d’Urbervilles.) What makes it a challenge is that you know to know the order of operations. Should you divide 24 by 4 first, and subtract 2 from that, or should you take 4 minus 2 and then divide 24 by whatever that number is?

Absent any confounding information, you should always do multiplication and division before you do addition and subtraction. So this suggests 24 divided by 4, giving us 6, and then subtract 2, giving us 4. The only relevant confounding information, though, would be the direction to do something else first. That’s indicated by putting something in parentheses. (Or brackets, if you have so many parentheses the symbols are getting confusing.) A thing in parentheses has higher priority and should be calculated first. But there’s no way to tell parentheses in dialogue. The best the teacher could do is say something like “24 divided by the quantity four minus two”, or even, “24 divided by parenthesis four minus two close parenthesis”. That’s awkward but it is what we resort to even in the mathematics department.

Eric the Circle for the 6th of December, this one by “Scooterpiggy”, is the anthropomorphic-numerals joke this essay. You might fuss that there’s a difference between a circle and zero. The earliest examples of zero seem to have been simple dots. But the circle, or at least elliptical, shape of zero grew pretty fast. Maybe in a couple of centuries. Maybe there’s something in the empty loop that suggests what it stands for.

Tom Thaves’s Frank and Ernest for the 6th of December tosses in a statistics pun for the final panel. The statistics use of “median” is the number that half the data is less than and half the data is greater than. It’s one of several quantities that get called an “average”. In this case it’s average because if you picked a data point at random you’d be as likely to be above as below the median. In data sets that aren’t too weird, that will usually be pretty close to the arithmetic mean. The arithmetic mean is the thing normal people mean by “average”. It’ll also typically be near the most common value. That most common value mathematicians and statisticians call the “mode”.

I don’t know if the use of “median” for the middle strip of a divided road shares an etymology with the statistics use of the word. It might be one use might have inspired the other, perhaps as metaphor. But the similarity between “being in the middle of the data” and “being in the middle of the street” is straightforward for English.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th of December pinpoints a common failure mode of experts. (The strip almost surely ran before, sometime. The only method I have to find out when, though, is to post an incorrect date and make someone correct me. So let me say it originally ran on Singapore National Day, 2009.) Mathematics is especially prone to it. It’s so seductive to teach something the way an expert sees it. This is usually in a rigorously thought-out, open-ended, flexible method. After all, why would you ever teach something that wasn’t exactly right, with “right” being “the ways experts see things”? A teacher knows the answer: the expert understanding of a thing is hard to get to. That’s why having it takes expertise. The comic strip’s explanation of fractions is correct and reasonable. But it brings up why Bertrand Russell and Alfred North Whitehead needed over four hundred pages to establish 1 + 1 equals 2. That’s a lot of intellectual scaffolding for the quality of paint job required. Sometimes it’s easier to start with a quick and dirty explanation, and then go back later and rebuild the understanding if a student needs it.

Rick Stromoski’s Soup to Nutz for the 7th of December puts forth a kind of Zeno’s paradox problem in the guise of compound interest. If doing something increases life expectancy by a certain percentage, then, how much of the extra time one gets do you need to be immortal? I’m amused by this although I can’t imagine modest alcohol consumption increasing lifespan by 20 percent. (I assume 20 percent of the average expected lifespan.) If the effect were anything near that big the actuaries would have noticed and ordered people to drink long ago.

On looking at all this, I think I’ll save the December 11th strips for later. This is enough text for this early in the morning.

The Set Tour, Part 10: Lots of Spheres

The next exhibit on the Set Tour here builds on a couple of the previous ones. First is the set Sⁿ, that is, the surface of a hypersphere in n+1 dimensions. Second is Bⁿ, the ball — the interior — of a hypersphere in n dimensions. Yeah, it bugs me too that Sⁿ isn’t the surface of Bⁿ. But it’d be too much work to change things now. The third has lurked implicitly since all the way back to Rⁿ, a set of n real numbers for which the ordering of the numbers matters. (That is, that the set of numbers 2, 3 probably means something different than the set 3, 2.) And fourth is a bit of writing we picked up with matrices. The selection is also dubiously relevant to my own thesis from back in the day.

S^{n x m} and B^{n x m}

Here ‘n’ and ‘m’ are whole numbers, and I’m not saying which ones because I don’t need to tie myself down. Just as with Rⁿ and with matrices this is a whole family of sets. Each different pair of n and m gives us a different set S^{n x m} or B^{n x m}, but they’ll all look quite similar.

The multiplication symbol here is a kind of multiplication, just as it was in matrices. That kind is called a “direct product”. What we mean by S^{n x m} is that we have a collection of items. We have the number m of them. Each one of those items is in Sⁿ. That’s the surface of the hypersphere in n+1 dimensions. And we want to keep track of the order of things; we can’t swap items around and suppose they mean the same thing.

So suppose I write S^{2 x 7}. This is an ordered collection of seven items, every one of which is on the surface of a three-dimensional sphere. That is, it’s the location of seven spots on the surface of the Earth. S^{2 x 8} offers similar prospects for talking about the location of eight spots.

With that written out, you should have a guess what B^{n x m} means. Your guess is correct. It’s a collection of m things, each of them within the interior of the n-dimensional ball.

Now the dubious relevance to my thesis. My problem was modeling a specific layer of planetary atmospheres. The model used for this was to pretend the atmosphere was made up of some large number of vortices, of whirlpools. Just like you see in the water when you slide your hand through the water and watch the little whirlpools behind you. The winds could be worked out as the sum of the winds produced by all these little vortices.

In the model, each of these vortices was confined to a single distance from the center of the planet. That’s close enough to true for planetary atmospheres. A layer in the atmosphere is not thick at all, compared to the planet. So every one of these vortices could be represented as a point in S², the surface of a three-dimensional sphere. There would be some large number of these points. Most of my work used a nice round 256 points. So my model of a planetary atmosphere represented the system as a point in the domain S^{2 x 256}. I was particularly interested in the energy of this set of 256 vortices. That was a function which had, as its domain, S^{2 x 256}, and as range, the real numbers R.

But the connection to my actual work is dubious. I was doing numerical work, for the most part. I don’t think my advisor or I ever wrote S^{2 x 256} or anything like that when working out what I ought to do, much less what I actually did. Had I done a more analytic thesis I’d surely have needed to name this set. But I didn’t. It was lurking there behind my work nevertheless.

The energy of this system of vortices looked a lot like the potential energy for a bunch of planets attracting each other gravitationally, or like point charges repelling each other electrically. We work it out by looking at each pair of vortices. Work out the potential energy of those two vortices being that strong and that far apart. We call that a pairwise interaction. Then add up all the pairwise interactions. That’s it. [1] The pairwise interaction is stronger as each vortex is stronger; it gets weaker as the vortices get farther apart.

In gravity or electricity problems the strength falls off as the reciprocal of the distance between points. In vortices, the strength falls off as minus one times the logarithm of the distance between points. That’s a difference, and it meant that a lot of analytical results known for electric charges didn’t apply to my problem exactly. That was all right. I didn’t need many. But it does mean that I was fibbing up above, when I said I was working with S^{2 x 256}. Pause a moment. Do you see what the fib was?

I’ll put what would otherwise be a footnote here so folks have a harder time reading right through to the answer.

[1] Physics majors may be saying something like: “wait, I see how this would be the potential energy of these 256 vortices, but where’s the kinetic energy?” The answer is, there is none. It’s all potential energy. The dynamics of point vortices are weird. I didn’t have enough grounding in mechanics when I went into them.

That’s all to the footnote.

Here’s where the fib comes in. If I’m really picking sets of vortices from all of the set S^{2 x 256}, then, can two of them be in the exact same place? Sure they can. Why couldn’t they? For precedent, consider R³. In the three-dimensional vectors I can have the first and third numbers “overlap” and have the same value: (1, 2, 1) is a perfectly good vector. Why would that be different for an ordered set of points on the surface of the sphere? Why can’t vortex 1 and vortex 3 happen to have the same value in S²?

The problem is if two vortices were in the exact same position then the energy would be infinitely large. That’s not unique to vortices. It would be true for masses and gravity, or electric charges, if they were brought perfectly on top of each other. Infinitely large energies are a problem. We really don’t want to deal with them.

We could deal with this by pretending it doesn’t happen. Imagine if you dropped 256 poker chips across the whole surface of the Earth. Would you expect any two to be on top of each other? Would you expect two to be exactly and perfectly on top of each other, neither one even slightly overhanging the other? That’s so unlikely you could safely ignore it, for the same reason you could ignore the chance you’ll toss a coin and have it come up tails 56 times in a row.

And if you were interested in modeling the vortices moving it would be incredibly unlikely to have one vortex collide with another. They’d circle around each other, very fast, almost certainly. So ignoring the problem is defensible in this case.

Or we could be proper and responsible and say, “no overlaps” and “no collisions”. We would define some set that represents “all the possible overlaps and arrangements that give us a collision”. Then we’d say we’re looking at S^{2 x 256} except for those. I don’t think there’s a standard convention for “all the possible overlaps and collisions”, but Ω is a reasonable choice. Then our domain would be S^{2 x 256} \ Ω. The backslash means “except for the stuff after this”. This might seem unsatisfying. We don’t explicitly say what combinations we’re excluding. But go ahead and try listing all the combinations that would produce trouble. Try something simple, like S^{2 x 4}. This is why we hide all the complicated stuff under a couple ordinary sentences.

It’s not hard to describe “no overlaps” mathematically. (You would say something like “vortex number j and vortex number k are not at the same position”, with maybe a rider of “unless j and k are the same number”. Or you’d put it in symbols that mean the same thing.) “No collisions” is harder. For gravity or electric charge problems we can describe at least some of them. And I realize now I’m not sure if there is an easy way to describe vortices that collide. I have difficulty imagining how they might, since vortices that are close to one another are pushing each other sideways quite intently. I don’t think that I can say they can’t, though. Not without more thought.

Reading the Comics, December 5, 2015: Awkward Break Edition

I confess I’m dissatisfied with this batch of Reading the Comics posts. I like having something like six to eight comics for one of these roundups. But there was this small flood of mathematically-themed comics on the 6th of December. I could either make do with a slightly short edition, or have an overstuffed edition. I suppose it’s possible to split one day’s comics across two Reading the Comics posts, but that’s crazy talk. So, a short edition today.

Jef Mallett’s Frazz for the 4th of December was part of a series in which Caulfield resists learning about reciprocals. The 4th offers a fair example of the story. At heart the joke is just the student-resisting-class, or student-resisting-story-problems. It certainly reflects a lack of motivation to learn what they are.

We use reciprocals most often to write division problems as multiplication. “a ÷ b” is the same as “a times the reciprocal of b”. But where do we get the reciprocal of b from? … Well, we can say it’s the multiplicative inverse of b. That is, it’s whatever number you have to multiply ‘b’ by in order to get ‘1’. But we’re almost surely going to find that taking 1 and dividing it by b. So we’ve swapped out one division problem for a slightly different one. This doesn’t seem to be getting us anywhere.

But we have gotten a new idea. If we can define the multiplication of things, we might be able to get division for almost free. Could we divide one matrix by another? We can certainly multiply a matrix by the inverse of another. (There are complications at work here. We’ll save them for another time.) A lot of sets allow us to define things that make sense as addition and multiplication. And if we can define a complicated operation in terms of addition and multiplication … If we follow this path, we get to do things like define the cosine of a matrix. Then we just have to figure out why we’d want have a cosine of a matrix.

There’s a simpler practical use of reciprocals. This relates to numerical mathematics, computer work. Computer chips do addition (and subtraction) really fast. They do multiplication a little slower. They do division a lot slower. Division is harder than multiplication, as anyone who’s done both knows. However, dividing by (say) 4 is the same thing as multiplying by 0.25. So if you know you need to divide by a number a lot, then it might make for a faster program to change division into multiplication by a reciprocal. You have to work out the reciprocal, but if you only have to do that once instead of many times over, this might make for faster code. Reciprocals are one of the tools we can use to change a mathematical process into something faster.

(In practice, you should never do this. You have a compiler that does this, and you should let it do its work. But it’s enlightening to know these are the sorts of things your compiler is looking for when it turns your code into something the computer does. And looking for ways to do the same work in less time is a noble side of mathematics.)

Charles Schulz’s Peanuts for the 4th of December (originally from 1968, on the same day) sees Peppermint Patty’s education crash against a word problem. It’s another problem in motivating a student to do a word problem. I admit when I was a kid I’d have been enchanted by this puzzle. But I was a weird one.

Dave Coverly’s Speed Bump for the 4th of December is a mathematics-symbols joke as applied to toast. I think you could probably actually sell those. At least the greater-than and the less-than signs. The approximately-equal-to signs would be hard to use. And people would think they were for bacon anyway.

Ruben Bolling’s Super-Fun-Pak Comix for the 4th of December showcases Young Albert Einstein. That counts as mathematical content, doesn’t it? The strip does make me wonder if they’re still selling music CDs and other stuff for infant or even prenatal development. I’m skeptical that they ever did any good, but it isn’t a field I’ve studied.

Bill Whitehead’s Free Range for the 5th of December uses a blackboard full of mathematical and semi-mathematical symbols to denote “stuff too complicated to understand”. The symbols don’t parse as anything. It is authentic to mathematical work to sometimes skip writing all the details of a thing and write in instead a few words describing it. Or to put in an abbreviation for the thing. That often gets circled or boxed or in some way marked off. That keeps us from later on mistaking, say, “MUB” as the product of M and U and B, whatever that would mean. Then we just have to remember we meant “minimum upper bound” by that.

Exam Grades And Ramsey Theory

While I don’t have any topics overwhelming my search-engine profile, I do see the rise in people looking up what they need to pass their class, or to get a desired minimum grade. The sorry answer is, they needed to start work sooner. But here’s the formula for working it out, for whatever your course average to date is, whatever score you want, whatever extra credit is available, all that. And here’s tables for some of the common cases, if you’re afraid of the formulas.

Ramsey theory in the dining room… https://t.co/wXqXEG869S pic.twitter.com/1psi3UTd9B

— Brian Hayes (@bit_player) November 30, 2015

And for pleasantly recreational mathematics … I forget which mathematics twitter account that I follow posted the above. But it links to “Ramsey Theory in the Dining Room”. I’d mentioned the field last month because its question about organizing dinner-party guests somehow got a Dear Abby correspondent all angry in the late 70s.

Brian Hayes there ran across an application of the theory that gets away from dinner-party invites and into table place settings. It’s worth a read, particularly for the challenge posed. Hayes thought, briefly, he had solved a question in Ramsey Theory, one that’s easy to understand — you’ll understand the question — but that everyone else in the field has found too hard to answer. He doubted his result, but didn’t think until the next day of why he was wrong. Can you spot where he went wrong? (It’s a subtle flaw in the reasoning, but one an eight-year-old would understand, so I recommend trying to think like one.)

Reading the Comics, December 2, 2015: The Art Of Maths Edition

Bill Amend’s FoxTrot Classics for the 28th of November (originally run in 2004) depicts a “Christmas Card For Smart People”. It uses the familiar motif of “ability to do arithmetic” as denoting smartness. The key to the first word is remembering that mathematicians use the symbol ‘e’ to represent a number that’s just a little over 2.71828. We call the number ‘e’, or something ‘the base of the natural logarithm’. It turns up all over the place. If you have almost any quantity that grows or that shrinks at a speed proportional to how much there is, and describe how much of stuff there is over time, you’ll find an ‘e’. Leonhard Euler, who’s renowned for major advances in every field of mathematics, is also renowned for major advances in notation in physics, and he gave us ‘e’ for that number.

The key to the second word there is remembering from physics that force equals mass times acceleration. Therefore the force divided by the acceleration is …

And so that inspires this essay’s edition title. There are several comics in this selection that are about the symbols or the representations of mathematics, and that touch on the subject as a visual art.

Matt Janz’s Out of the Gene Pool for the 28th of November first ran the 26th of October, 2002. It would make for a good word problem, too, with a couple of levels: given the constraints of (a slightly looser) budget, how do they get the greatest number of cookies? Or if some cookies are better than others, how do they get the most enjoyment from their cookie purchase? Working out the greatest amount of enjoyment within a given cookie budget, with different qualities of cookies, can be a good introduction to optimization problems and how subtle they can be.

Bill Holbrook’s On The Fastrack for the 29th of November speaks in support of accounting. It’s a worthwhile message. It doesn’t get much respect, not from the general public, and not from typical mathematics department. The general public maybe thinks of accounting as not much more than a way companies nickel-and-dime them. If the mathematics departments I’ve associated with are fair representatives, accounting isn’t even thought of except by the assistant professor doing a seminar on financial mathematics. (And I’m not sure accounting gets mentioned there, since there’s exciting stuff about the Black-Scholes Equation and options markets to think about instead.) This despite that accounting is probably, by volume, the most used part of mathematics. Anyway, Holbrook’s strip probably won’t get the field a better reputation. But it has got some great illustrations of doing things with numbers. The folks in mathematics departments certainly have had days feeling like they’ve done each of these things.

Bill Holbrook’s On The Fastrack for the 29th of November, 2015. While the strip’s focus is accountants, it is true that most mathematicians will spend hours overhauling their 6’s.

Dave Coverly’s Speed Bump for the 30th of November is a compound interest joke. I admit I’ve told this sort of joke myself, proposing that the hour cut out of the day in spring when Daylight Saving Time starts comes back as a healthy hour and three minutes in autumn when it’s taken out of saving. If I can get the delivery right I might have someone going for that three minutes.

Mikael Wulff and Anders Morgenthaler’s Truth Facts for the 30th of November is a Venn diagram joke for breakfast. I would bet they’re kicking themselves for not making the intersection be the holes in the center.

Mark Anderson’s Andertoons for this week interests me. It uses a figure to try explaining how to relate gallon and quart an pint and other units relate to each other. I like it, but I’m embarrassed to say how long it took in my life to work out the relations between pints, quarts, gallons, and particularly whether the quart or the pint was the larger unit. I blame part of that on my never really having to mix a pint of something with a quart of something else, which ought to have sorted that out. Anyway, let’s always cherish good representations of information. Good representations organize information and relationships in ways that are easy to remember, or easy to reconstruct or extend.

John Graziano’s Ripley’s Believe It or Not for the 2nd of December tries to visualize how many ways there are to arrange a Rubik’s Cube. Counting off permutations of things by how many seconds it’d take to get through them all is a common game. The key to producing a staggering length of time is that it one billion seconds are nearly 32 years, and the number of combinations of things adds up really really fast. There’s over eight billion ways to draw seven letters in a row, after all, if every letter is equally likely and if you don’t limit yourself to real or even imaginable words. Rubik’s Cubes have a lot of potential arrangements. Graziano misspells Rubik, but I have to double-check and make sure I’ve got it right every time myself. I didn’t know that about the pigeons.

Charles Schulz’s Peanuts for the 2nd of December (originally run in 1968) has Peppermint Patty reflecting on the beauty of numbers. I don’t think it’s unusual to find some numbers particularly pleasant and others not. Some numbers are easy to work with; if I’m trying to add up a set of numbers and I have a 3, I look instinctively for a 7 because of how nice 10 is. If I’m trying to multiply numbers, I’d so like to multiply by a 5 or a 25 than by a 7 or an 18. Typically, people find they do better on addition and multiplication with lower numbers like two and three, and get shaky with sevens and eights and such. It may be quirky. My love is a wizard with 7’s, but can’t do a thing with 8. But it’s no more irrational than the way a person might a pyramid attractive but a sphere boring and a stellated icosahedron ugly.

I’ve seen some comments suggesting that Peppermint Patty is talking about numerals, that is, the way we represent numbers. That she might find the shape of the 2 gentle, while 5 looks hostile. (I can imagine turning a 5 into a drawing of a shouting person with a few pencil strokes.) But she doesn’t seem to say one way or another. She might see a page of numbers as visual art; she might see them as wonderful things with which to play.

How November 2015 Treated My Mathematics Blog

So after a couple dismal months my ratings appear to be up. The number of page views and of visitors, in fact, seem to be at all-time highs. At least they’re at highs for the past twelve months. I would like to think that the depressed readings of September and October — 708 page views and 381 visitors; 733 page views and 405 visitors, respectively — are behind me. November saw 1,215 page views and 519 visitors.

Some of this is an accident. My humor blog got a tidal wave of readers courtesy The Onion AV Club. The AV Club wrote up a piece about the sad end of the comic strip Apartment 3-G, and I’ve written a shocking amount about the soap strip. They mentioned me. And as I’ve used my comic strip posts there to mention my Reading the Comics series here, some curious people followed along.

That said, I’m not sure how many of those readers were AV Club curiosity-seekers. A crude estimate suggests somewhere a little over two hundred were. So even discounting that something near a thousand regular-style reders came in and looked around, and that’s nice to see. It’s back up to about where the readership was before the mysterious dropoff, in July, that many suspect results from mobile devices being incorrectly read.

For the roster of countries, well, the top was the United States as always, with some 837 page views. The United Kingdom came in with 62. The Canada appears third at 50 views, and the Philippines next at 20. The Singapore and the Australia tie at 19.

Single-reader countries this past month were Algeria, Argentina, Belgium, Egypt, Finland, Israel, Jamaica, Malaysia, Mexico, Nigeria, Puerto Rico, Romania, Thailand, Turkey, and Vietnam. Belgium, Nigeria, and Thailand are repeats from October. No country’s on a three-month streak.

The Reading the Comics posts are as ever the most popular group and I’ve bundled them under the one category tag. But my Ramsey Theory question turned out to be slightly more popular than any of them in November. After grouping together all the comics posts, the most popular articles look like:

1. Why Was Someone Upset With Ramsey Theory In 1979? a question about a dimly remembered Dear Abby-class question.
2. Reading the Comics, an ongoing series.
3. How October Treated My Mathematics Blog, and yes, I risk an endless loop by mentioning this here.
4. How Many Trapezoids I Can Draw and goodness it’s nice to see the trapezoids turning up again.
5. How Antifreeze Works, one of my little pointers to someone else’s interesting writing.

Nothing really dominated my search term queries this month. Some of the things that turned up were:

• illustration of electromagnetic wave theory scientist comics strip
• james clerk maxwell comics (I’m not sure I have any of these; this suggests I ought to be finding some.)
• origin is the gateway to your entire gaming universe. (I’ve had this explained to me, but I forget what it means.)
• places 1975 miles from charlotte nc (I know of none specifically 1,975 miles away.)
• if i got 70 percent in all exams what grade do i need on final to pass course? (This I can help with.)

December starts with my blog here at 30,298 page views, and with 543 WordPress followers. I expect it’ll be overtaken in page views by my humor blog sometime soon.

A Timeline Of Mathematics Education

this is interesting: US timeline of maths education: https://t.co/hdJA66iUac pic.twitter.com/oUqR18aU4M

— danny brown (@dannytybrown) November 27, 2015

As Danny Brown’s tweet above promises, this is an interesting timeline. It’s a “work in progress” presentation by one David Allen that tries to summarize the major changes in the teaching of mathematics in the United States.

It’s a presentation made on Prezi, and it appears to require Flash (and at one point it breaks, at least on my computer, and I have to move around rather than use the forward/backward buttons). And the compilation is cryptic. It reads better as a series of things for further research than anything else. Still, it’s got fascinating data points, such as when algebra became a prerequisite for college, and when it and geometry moved from being college-level mathematics to high school-level mathematics.

Reading the Comics, November 27, 2015: 30,000 Edition

By rights, if this installment has any title it should be “confident ignorance”. That state appears in many of the strips I want to talk about. But according to WordPress, my little mathematics blog here reached its 30,000th page view at long last. This is thanks largely to spillover from The Onion AV Club discovering my humor blog and its talk about the late comic strip Apartment 3-G. But a reader is a reader. And I want to celebrate reaching that big, round number. As I write this I’m at 30,162 page views, because there were a lot of AV Club-related readers.

Bob Weber Jr’s Slylock Fox for the 23rd of November maybe shouldn’t really be here. It’s just a puzzle game that depends on the reader remembering that two rectangles put against the other can be a rectangle again. It also requires deciding whether the frame of the artwork counts as one of the rectangles. The commenters at Comics Kingdom seem unsure whether to count squares as rectangles too. I don’t see any shapes that look more clearly like squares to me. But it’s late in the month and I haven’t had anything with visual appeal in these Reading the Comics installments in a while. Later we can wonder if “counting rectangles in a painting” is the most reasonable way a secret agent has to pass on a number. It reminds me of many, many puzzle mysteries Isaac Asimov wrote that were all about complicated ways secret agents could pass one bit of information on.

Bob Weber Jr’s Slylock Fox for the 23rd of November, 2015. I suppose the artist is lucky they weren’t hiding out at number 38, or she wouldn’t have been able to make such a compellingly symmetric diagram.

Ryan North’s Dinosaur Comics for the 23rd of November is a rerun from goodness knows when it first ran on Quantz.com. It features T Rex thinking about the Turing Test. The test, named for Alan Turing, says that while we may not know what exactly makes up an artificial intelligence, we will know it when we see it. That is the sort of confident ignorance that earned Socrates a living. (I joke. Actually, Socrates was a stonecutter. Who knew, besides the entire philosophy department?) But the idea seems hard to dispute. If we can converse with an entity in such a way that we can’t tell it isn’t human, then, what grounds do we have for saying it isn’t human?

T Rex has an idea that the philosophy department had long ago, of course. That’s to simply “be ready for any possible opening with a reasonable conclusion”. He calls this a matter of brute force. That is, sometimes, a reasonable way to solve problems. It’s got a long and honorable history of use in mathematics. The name suggests some disapproval; it sounds like the way you get a new washing machine through a too-small set of doors. But sometimes the easiest way to find an answer is to just try all the possible outcomes until you find the ones that work, or show that nothing can. If I want to know whether 319 is a prime number, I can try reasoning my way through it. Or I can divide it by all the prime numbers from 2 up to 17. (The square root of 319 is a bit under 18.) Or I could look it up in a table someone already made of the prime numbers less than 400. I know what’s easier, if I have a table already.

The problem with brute force — well, one problem — is that it can be longwinded. We have to break the problem down into each possible different case. Even if each case is easily disposed of, the number of different cases can grow far too fast to be manageable. The amount of working time required, and the amount of storage required, can easily become too much to deal with. Mathematicians, and computer scientists, have a couple approaches for this. One is getting bigger computers with more memory. We might consider this the brute force method to solving the limits of brute force methods.

Or we might try to reduce the number of possible cases, so that less work is needed. Perhaps we can find a line of reasoning that covers many cases. Working out specific cases, as brute force requires, can often give us a hint to what a general proof would look like. Or we can at least get a bunch of cases dealt with, even if we can’t get them all done.

Jim Unger’s Herman rerun for the 23rd of November turns confident ignorance into a running theme for this essay’s comic strips.

Eric Teitelbaum and Bill Teitelbaum’s Bottomliners for the 24th of November has a similar confient ignorance. This time it’s of the orders of magnitude that separate billions from trillions. I wanted to try passing off some line about how there can be contexts where it doesn’t much matter whether a billion or a trillion is at stake. But I can’t think of one that makes sense for the Man At The Business Company Office setting.

Reza Farazmand’s Poorly Drawn Lines for the 25th of November is built on the same confusion about the orders of magnitude that Bottomliners is. In this case it’s ants that aren’t sure about how big millions are, so their confusion seems more natural.

The ants are also engaged in a fun sort of recreational mathematics: can you estimate something from little information? You’ve done that right, typically, if you get the size of the number about right. That it should be millions rather than thousands or hundreds of millions; that there should be something like ten rather than ten thousand. These kinds of problems are often called Fermi Problems, after Enrico Fermi. This is the same person the Fermi Paradox is named after, but that’s a different problem. The Fermi Paradox asks if there are extraterrestrial aliens, why we don’t see evidence of them. A Fermi Problem is simpler. Its the iconic example is, “how many professional piano tuners are there in New York?” It’s easy to look up how big is the population of New York. It’s possible to estimate how many pianos there should be for a population that size. Then you can guess how often a piano needs tuning, and therefore, how many full-time piano tuners would be supported by that much piano-tuning demand. And there’s probably not many more professional piano tuners than there’s demand for. (Wikipedia uses Chicago as the example city for this, and asserts the population of Chicago to be nine million people. I will suppose this to be the Chicago metropolitan region, but that still seems high. Wikipedia says that is the rough population of the Chicago metropolitan area, but it’s got a vested interest in saying so.)

Mark Anderson’s Andertoons finally appears on the 27th. Here we combine the rational division of labor with resisting mathematics problems.

The Set Tour, Part 9: Balls, Only The Insides

Last week in the tour of often-used domains I talked about Sⁿ, the surfaces of spheres. These correspond naturally to stuff like the surfaces of planets, or the edges of surfaces. They are also natural fits if you have a quantity that’s made up of a couple of components, and some total amount of the quantity is fixed. More physical systems do that than you might have guessed.

But this is all the surfaces. The great interior of a planet is by definition left out of Sⁿ. This gives away the heart of what this week’s entry in the set tour is.

Bⁿ

Bⁿ is the domain that’s the interior of a sphere. That is, B³ would be all the points in a three-dimensional space that are less than a particular radius from the origin, from the center of space. If we don’t say what the particular radius is, then we mean “1”. That’s just as with the Sⁿ we meant the radius to be “1” unless someone specifically says otherwise. In practice, I don’t remember anyone ever saying otherwise when I was in grad school. I suppose they might if we were doing a numerical simulation of something like the interior of a planet. You know, something where it could make a difference what the radius is.

It may have struck you that B³ is just the points that are inside S². Alternatively, it might have struck you that S² is the points that are on the edge of B³. Either way is right. Bⁿ and S^n-1, for any positive whole number n, are tied together, one the edge and the other the interior.

Bⁿ we tend to call the “ball” or the “n-ball”. Probably we hope that suggests bouncing balls and baseballs and other objects that are solid throughout. Sⁿ we tend to call the “sphere” or the “n-sphere”, though I admit that doesn’t make a strong case for ruling out the inside of the sphere. Maybe we should think of it as the surface. We don’t even have to change the letter representing it.

As the “n” suggests, there are balls for as many dimensions of space as you like. B² is a circle, filled in. B¹ is just a line segment, stretching out from -1 to 1. B³ is what’s inside a planet or an orange or an amusement park’s glass light fixture. B⁴ is more work than I want to do today.

So here’s a natural question: does Bⁿ include S^n-1? That is, when we talk about a ball in three dimensions, do we mean the surface and everything inside it? Or do we just mean the interior, stopping ever so short of the surface? This is a division very much like dividing the real numbers into negative and positive; do you include zero among other set?

Typically, I think, mathematicians don’t. If a mathematician speaks of B³ without saying otherwise, she probably means the interior of a three-dimensional ball. She’s not saying anything one way or the other about the surface. This we name the “open ball”, and if she wants to avoid any ambiguity she will say “the open ball Bⁿ”.

“Open” here means the same thing it does when speaking of an “open set”. That may not communicate well to people who don’t remember their set theory. It means that the edges aren’t included. (Warning! Not actual set theory! Do not attempt to use that at your thesis defense. That description was only a reference to what’s important about this property in this particular context.)

If a mathematician wants to talk about the ball and the surface, she might say “the closed ball Bⁿ”. This means to take the surface and the interior together. “Closed”, again, here means what it does in set theory. It pretty much means “include the edges”. (Warning! See above warning.)

Balls work well as domains for functions that have to describe the interiors of things. They also work if we want to talk about a constraint that’s made up of a couple of components, and that can be up to some size but not larger. For example, suppose you may put up to a certain budget cap into (say) six different projects, but you aren’t required to use the entire budget. We could model your budgeting as finding the point in B⁶ that gets the best result. How you measure the best is a problem for your operations research people. All I’m telling you is how we might represent the study of the thing you’re doing.

Reading the Comics, November 21, 2015: Communication Edition

And then three days pass and I have enough comic strips for another essay. That’s fine by me, really. I picked this edition’s name because there’s a comic strip that actually touches on information theory, and another that’s about a much-needed mathematical symbol, and another about the ways we represent numbers. That’s enough grounds for me to use the title.

Samson’s Dark Side Of The Horse for the 19th of November looks like this week’s bid for an anthropomorphic numerals joke. I suppose it’s actually numeral cosplay instead. I’m amused, anyway.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 19th of November makes a patent-law joke out of the invention of zero. It’s also an amusing joke. It may be misplaced, though. The origins of zero as a concept is hard enough to trace. We can at least trace the symbol zero. In Finding Zero: A Mathematician’s Odyssey to Uncover the Origins of Numbers, Amir D Aczel traces out not just the (currently understood) history of Arabic numerals, but some of how the history of that history has evolved, and finally traces down the oldest known example of a written (well, carved) zero.

Tony Cochrane’s Agnes for the 20th of November is at heart just a joke about a student’s apocalyptically bad grades. It contains an interesting punch line, though, in Agnes’s statement that “math people are dreadful spellers”. I haven’t heard that before. It might be a joke about algebra introducing letters into numbers. But it does seem to me there’s a supposition that mathematics people aren’t very good writers or speakers. I do remember back as an undergraduate other people on the student newspaper being surprised I could write despite majoring in physics and mathematics. That may reflect people remembering bad experiences of sitting in class with no idea what the instructor was going on about. It’s easy to go from “I don’t understand this mathematics class” to “I don’t understand mathematics people”.

Steve Sicula’s Home and Away for the 20th of November is about using gambling as a way to teach mathematics. So it would be a late entry for the recent Gambling Edition of the Reading The Comics posts. Although this strip is a rerun from the 15th of August, 2008, so it’s actually an extremely early entry.

Ruben Bolling’s Tom The Dancing Bug for the 20th of November is a Super-Fun-Pak Comix installment. And for a wonder it hasn’t got a Chaos Butterfly sequence. Under the Guy Walks Into A Bar label is a joke about a horse doing arithmetic that itself swings into a base-ten joke. In this case it’s suggested the horse would count in base four, and I suppose that’s plausible enough. The joke depends on the horse pronouncing a base four “10” as “ten”, when the number is actually “four”. But the lure of the digits is very hard to resist, and saying “four” suggests the numeral “4” whatever the base is supposed to be.

Mark Leiknes’s Cow and Boy for the 21st of November is a rerun from the 9th of August, 2008. It mentions the holographic principle, which is a neat concept. The principle’s explained all right in the comic. The idea was first developed in the late 1970s, following the study of black hole thermodynamics. Black holes are fascinating because the mathematics of them suggest they have a temperature, and an entropy, and even information which can pass into and out of them. This study implied that information about the three-dimensional volume of the black hole was contained entirely in the two-dimensional surface, though. From here things get complicated, though, and I’m going to shy away from describing the whole thing because I’m not sure I can do it competently. It is an amazing thing that information about a volume can be encoded in the surface, though, and vice-versa. And it is astounding that we can imagine a logically consistent organization of the universe that has a structure completely unlike the one our senses suggest. It’s a lasting and hard-to-dismiss philosophical question. How much of the way the world appears to be structured is the result of our minds, our senses, imposing that structure on it? How much of it is because the world is ‘really’ like that? (And does ‘really’ mean anything that isn’t trivial, then?)

I should make clear that while we can imagine it, we haven’t been able to prove that this holographic universe is a valid organization. Explaining gravity in quantum mechanics terms is a difficult point, as it often is.

Dave Blazek’s Loose Parts for the 21st of November is a two- versus three-dimensions joke. The three-dimension figure on the right is a standard way of drawing x-, y-, and z-axes, organized in an ‘isometric’ view. That’s one of the common ways of drawing three-dimensional figures on a two-dimensional surface. The two-dimension figure on the left is a quirky representation, but it’s probably unavoidable as a way to make the whole panel read cleanly. Usually when the axes are drawn isometrically, the x- and y-axes are the lower ones, with the z-axis the one pointing vertically upward. That is, they’re the ones in the floor of the room. So the typical two-dimensional figure would be the lower axes.