My Little 2021 Mathematics A-to-Z: Inverse

I owe Iva Sallay thanks for the suggestion of today’s topic. Sallay is a longtime friend of my blog here. And runs the Find the Factors recreational mathematics puzzle site. If you haven’t been following, or haven’t visited before, this is a fun week to step in again. The puzzles this week include (American) Thanksgiving-themed pictures.

Inverse.

When we visit the museum made of a visual artist’s studio we often admire the tools. The surviving pencils and crayons, pens, brushes and such. We don’t often notice the eraser, the correction tape, the unused white-out, or the pages cut into scraps to cover up errors. To do something is to want to undo it. This is as true for the mathematics of a circle as it is for the drawing of one.

If not to undo something, we do often want to know where something comes from. A classic paper asks can one hear the shape of a drum? You hear a sound. Can you say what made that sound? Fine, dismiss the drum shape as idle curiosity. The same question applies to any sensory data. If our hand feels cooler here, where is the insulation of the building damaged? If we have this electrocardiogram reading, what can we say about the action of the heart producing that? If we see the banks of a river, what can we know about how the river floods?

And this is the point, and purpose, of inverses. We can understand them as finding the causes of what we observe.

The first inverse we meet is usually the inverse function. It’s introduced as a way to undo what a function does. That’s an odd introduction, if you’re comfortable with what a function is. A function is a mathematical construct. It’s two sets — a domain and a range — and a rule that links elements in the domain to the range. To “undo” a function is like “undoing” a rectangle. But a function has a compelling “physical” interpretation. It’s routine to introduce functions as machines that take some numbers in and give numbers out. We think of them as ways to transform the domain into the range. In functional analysis get to thinking of domains as the most perfect putty. We expect functions to stretch and rotate and compress and slide along as though they were drawing a Betty Boop cartoon.

So we’re trained to speak of a function as a verb, acting on pieces of the domain. An element or point, or a region, or the whole domain. We think the function “maps”, or “takes”, or “transforms” this into its image in the range. And if we can turn one thing into another, surely we can turn it back.

Some things it’s obvious we can turn back. Suppose our function adds 2 to whatever we give it. We can get the original back by subtracting 2. If the function subtracts 32 and divides by 1.8, we can reverse it by multiplying by 1.8 and adding 32. If the function takes the reciprocal, we can take the reciprocal again. We have a bit of a problem if we started out taking the reciprocal of 0, but who would want to do such a thing anyway? If the function squares a number, we can undo that by taking the square root. Unless we started from a negative number. Then we have trouble.

The trouble is not every function has an inverse. Which we could have realized by thinking how to undo “multiply by zero”. To be a well-defined function, the rule part has to match elements in the domain to exactly one element in the range. This makes the function, in the impenetrable jargon of the mathematician, a “one-to-one function”. Or you can describe it with the more intuitive label of “bijective”.

But there’s no reason more than one thing in the domain can’t match to the same thing in the range. If I know the cosine of my angle is $\frac{1}{2}$, my angle might be 30 degrees. Or -30 degrees. Or 390 degrees. Or 330 degrees. You may protest there’s no difference between a 30 degree and a 390 degree angle. I agree those angles point in the same direction. But a gear rotated 390 degrees has done something that a gear rotated 30 degrees hasn’t. If all I know is where the dot I’ve put on the gear is, how can I know how much it’s rotated?

So what we do is shift from the actual cosine into one branch of the cosine. By restricting the domain we can create a function that has the same rule as the one we want, but that’s also one-to-one and so has an inverse. What restriction to use? That depends on what you want. But mathematicians have some that come up so often they might as well be defaults. So the square root is the inverse of the square of nonnegative numbers. The inverse Cosine is the inverse of the cosine of angles from 0 to 180 degrees. The inverse Sine is the inverse of the sine of angles from -90 to 90 degrees. The capital letters are convention to say we’re doing this. If we want a different range, we write out that we’re looking for an inverse cosine from -180 to 0 degrees or whatever. (Yes, the mathematician will default to using radians, rather than degrees, for angles. That’s a different essay.) It’s an imperfect solution, but it often works well enough.

The trouble we had with cosines, and functions, continues through all inverses. There are almost always alternate causes. Many shapes of drums sound alike. Take two metal bars. Heat both with a blowtorch, one on the end and one in the center. Not to the point of melting, only to the point of being too hot to touch. Let them cool in insulated boxes for a couple weeks. There’ll be no measurement you can do on the remaining heat that tells you which one was heated on the end and which the center. That’s not because your thermometers are no good or the flow of heat is not deterministic or anything. It’s that both starting cases settle to the same end. So here there is no usable inverse.

This is not to call inverses futile. We can look for what we expect to find useful. We are inclined to find inverses of the cosine between 0 and 180 degrees, even though 4140 through 4320 degrees is as legitimate. We may not know what is wrong with a heart, but have some idea what a heart could do and still beat. And there’s a famous example in 19th-century astronomy. After the discovery of Uranus came the discovery it did not move right. For a while it moved across the sky too fast for its distance from the sun. Then it started moving too slow. The obvious supposition was that there was another, not-yet-seen, planet, affecting its orbit.

The trouble is finding it. Calculating the orbit from what data they had required solving equations with 13 unknown quantities. John Couch Adams and Urbain Le Verrier attempted this anyway, making suppositions about what they could not measure. They made great suppositions. Le Verrier made the better calculations, and persuaded an astronomer (Johann Gottfried Galle, assisted by Heinrich Louis d’Arrest) to go look. Took about an hour of looking. They also made lucky suppositions. Both, for example, supposed the trans-Uranian planet would obey “Bode’s Law”, a seeming pattern in the size of planetary radiuses. The actual Neptune does not. It was near enough in the sky to where the calculated planet would be, though. The world is vaster than our imaginations.

That there are many ways to draw Betty Boop does not mean there’s nothing to learn about how this drawing was done. And so we keep having inverses as a vibrant field of mathematics.

Next week I hope to cover the letter ‘C’ and don’t think I’m not worried about what that ‘C’ will be. This week’s essay, and all the essays for the Little Mathematics A-to-Z, should be at this link. And all of this year’s essays, and all the A-to-Z essays from past years, should be at this link. Thank you for reading.

Reading the Comics, September 24, 2017: September 24, 2017 Edition

Comic Strip Master Command sent a nice little flood of comics this week, probably to make sure that I transitioned from the A To Z project to normal activity without feeling too lost. I’m going to cut the strips not quite in half because I’m always delighted when I can make a post that’s just a single day’s mathematically-themed comics. Last Sunday, the 24th of September, was such a busy day. I’m cheating a little on what counts as noteworthy enough to talk about here. But people like comic strips, and good on them for liking them.

Norm Feuti’s Gil for the 24th sees Gil discover and try to apply some higher mathematics. There’s probably a good discussion about what we mean by division to explain why Gil’s experiment didn’t pan out. I would pin it down to eliding the difference between “dividing in half” and “dividing by a half”, which is a hard one. Terms that seem almost alike but mean such different things are probably the hardest part of mathematics.

Russell Myers’s Broom Hilda looks like my padding. But the last panel of the middle row gets my eye. The squirrels talk about how on the equinox night and day “can never be of identical length, due to the angular size of the sun and atmospheric refraction”. This is true enough for the equinox. While any spot on the Earth might see twelve hours facing the sun and twelve hours facing away, the fact the sun isn’t a point, and that the atmosphere carries light around to the “dark” side of the planet, means daylight lasts a little longer than night.

Ah, but. This gets my mathematical modelling interest going. Because it is true that, at least away from the equator, there’s times of year that day is way shorter than night. And there’s times of year that day is way longer than night. Shouldn’t there be some time in the middle when day is exactly equal to night?

The easy argument for is built on the Intermediate Value Theorem. Let me define a function, with domain each of the days of the year. The range is real numbers. It’s defined to be the length of day minus the length of night. Let me say it’s in minutes, but it doesn’t change things if you argue that it’s seconds, or milliseconds, or hours, if you keep parts of hours in also. So, like, 12.015 hours or something. At the height of winter, this function is definitely negative; night is longer than day. At the height of summer, this function is definitely positive; night is shorter than day. So therefore there must be some time, between the height of winter and the height of summer, when the function is zero. And therefore there must be some day, even if it isn’t the equinox, when night and day are the same length

There’s a flaw here and I leave that to classroom discussions to work out. I’m also surprised to learn that my onetime colleague Dr Helmer Aslaksen’s grand page of mathematical astronomy and calendar essays doesn’t seem to have anything about length of day calculations. But go read that anyway; you’re sure to find something fascinating.

Mike Baldwin’s Cornered features an old-fashioned adding machine being used to drown an audience in calculations. Which makes for a curious pairing with …

Bill Amend’s FoxTrot, and its representation of “math hipsters”. I hate to encourage Jason or Marcus in being deliberately difficult. But there are arguments to make for avoiding digital calculators in favor of old-fashioned — let’s call them analog — calculators. One is that people understand tactile operations better, or at least sooner, than they do digital ones. The slide rule changes multiplication and division into combining or removing lengths of things, and we probably have an instinctive understanding of lengths. So this should train people into anticipating what a result is likely to be. This encourages sanity checks, verifying that an answer could plausibly be right. And since a calculation takes effort, it encourages people to think out how to arrange the calculation to require less work. This should make it less vulnerable to accidents.

I suspect that many of these benefits are what you get in the ideal case, though. Slide rules, and abacuses, are no less vulnerable to accidents than anything else is. And if you are skilled enough with the abacus you have no trouble multiplying 18 by 7, you probably would not find multiplying 17 by 8 any harder, and wouldn’t notice if you mistook one for the other.

Jef Mallett’s Frazz asserts that numbers are cool but the real insight is comparisons. And we can argue that comparisons are more basic than numbers. We can talk about one thing being bigger than another even if we don’t have a precise idea of numbers, or how to measure them. See every mathematics blog introducing the idea of different sizes of infinity.

Bill Whitehead’s Free Range features Albert Einstein, universal symbol for really deep thinking about mathematics and physics and stuff. And even a blackboard full of equations for the title panel. I’m not sure whether the joke is a simple absent-minded-professor joke, or whether it’s a relabelled joke about Werner Heisenberg. Absent-minded-professor jokes are not mathematical enough for me, so let me point once again to American Cornball. They’re the first subject in Christopher Miller’s encyclopedia of comic topics. So I’ll carry on as if the Werner Heisenberg joke were the one meant.

Heisenberg is famous, outside World War II history, for the Uncertainty Principle. This is one of the core parts of quantum mechanics, under which there’s a limit to how precisely one can know both the position and momentum of a thing. To identify, with absolutely zero error, where something is requires losing all information about what its momentum might be, and vice-versa. You see the application of this to a traffic cop’s question about knowing how fast someone was going. This makes some neat mathematics because all the information about something is bundled up in a quantity called the Psi function. To make a measurement is to modify the Psi function by having an “operator” work on it. An operator is what we call a function that has domains and ranges of other functions. To measure both position and momentum is equivalent to working on Psi with one operator and then another. But these operators don’t commute. You get different results in measuring momentum and then position than you do measuring position and then momentum. And so we can’t know both of these with infinite precision.

There are pairs of operators that do commute. They’re not necessarily ones we care about, though. Like, the total energy commutes with the square of the angular momentum. So, you know, if you need to measure with infinite precision the energy and the angular momentum of something you can do it. If you had measuring tools that were perfect. You don’t, but you could imagine having them, and in that case, good. Underlying physics wouldn’t spoil your work.

Probably the panel was an absent-minded professor joke.

How To Work Out The Length Of Time Between Two Dates

September 1999 was a heck of a month you maybe remember. There that all that excitement of the Moon being blasted out of orbit thanks to the nuclear waste pile up there getting tipped over or something. And that was just as we were getting over the final new episode of Mystery Science Theater 3000‘s first airing. That episode was number 1003, Merlin’s Shop of Mystical Wonders, which aired a month after the season finale because of one of those broadcast rights tangles that the show always suffered through.

Time moves on, and strange things happen, and show co-creator and first host Joel Hodgson got together a Kickstarter and a Netflix deal. The show’s Season Eleven is supposed to air starting the 14th of April, this year. The natural question: how long will we go, then, between new episodes of Mystery Science Theater 3000? Or more generally, how do you work out how long it is between two dates?

The answer is dear Lord under no circumstances try to work this out yourself. I’m sorry to be so firm. But the Gregorian calendar grew out of a bunch of different weird influences. It’s just hard to keep track of all the different 31- and 30-day months between two events. And then February is all sorts of extra complications. It’s especially tricky if the interval spans a century year, like 2000, since the majority of those are not leap years, except that the one century year I’m likely to experience was. And then if your interval happens to cross the time the local region switched from the Julian to the Gregorian calendar —

So my answer is don’t ever try to work this out yourself. Never. Just refuse the problem if you’re given it. If you’re a consultant charge an extra hundred dollars for even hearing the problem.

All right, but what if you really absolutely must know for some reason? I only know one good answer. Convert the start and the end dates of your interval into Julian Dates and subtract one from the other. I mean subtract the smaller number from the larger. Julian Dates are one of those extremely minor points of calendar use. They track the number of days elapsed since noon, Universal Time, on the Julian-calendar date we call the 1st of January, 4713 BC. The scheme, for years, was set up in 1583 by Joseph Justus Scalinger, calendar reformer, who wanted for convenience an index year so far back that every historically known event would have a positive number. In the 19th century the astronomer John Herschel expanded it to date-counting.

Scalinger picked the year from the convergence of a couple of convenient calendar cycles about how the sun and moon move as well as the 15-year indiction cycle that the Roman Empire used for tax matters (and that left an impression on European nations). His reasons don’t much matter to us. The specific choice means we’re not quite three-fifths of the way through the days in the 2,400,000’s. So it’s not rare to modify the Julian Date by subtracting 2,400,000 from it. The date starts from noon because astronomers used to start their new day at noon, which was more convenient for logging a whole night’s observations. Since astronomers started taking pictures of stuff and looking at them later they’ve switched to the new day starting at midnight like everybody else, but you know what it’s like changing an old system.

This summons the problem: so how do I know many dates passed between whatever day I’m interested in and the Julian Calendar 1st of January, 4713 BC? Yes, there’s a formula. No, don’t try to use it. Let the fine people at the United States Naval Observatory do the work for you. They know what they’re doing and they’ve had this calculator up for a very long time without any appreciable scandal accruing to it. The system asks you for a time of day, because the Julian Date increases as the day goes on. You can just make something up if the time doesn’t matter. I normally leave it on midnight myself.

So. The last episode of Mystery Science Theater 3000 to debut, on the 12th of September, 1999, did so on Julian Date 2,451,433. (Well, at 9 am Eastern that day, but nobody cares about that fine grain a detail.) The new season’s to debut on Netflix the 14th of April, 2017, which will be Julian Date 2,457,857. (I have no idea if there’s a set hour or if it’ll just become available at 12:01 am in whatever time zone Netflix Master Command’s servers are in.) That’s a difference of 6,424 days. You’re on your own in arguing about whether that means it was 6,424 or 6,423 days between new episodes.

If you do take anything away from this, though, please let it be the warning: never try to work out the interval between dates yourself.

Reading the Comics, July 13, 2016: Catching Up On Vacation Week Edition

I confess I spent the last week on vacation, away from home and without the time to write about the comics. And it was another of those curiously busy weeks that happens when it’s inconvenient. I’ll try to get caught up ahead of the weekend. No promises.

Art and Chip Samson’s The Born Loser for the 10th talks about the statistics of body measurements. Measuring bodies is one of the foundations of modern statistics. Adolphe Quetelet, in the mid-19th century, found a rough relationship between body mass and the square of a person’s height, used today as the base for the body mass index.Francis Galton spent much of the late 19th century developing the tools of statistics and how they might be used to understand human populations with work I will describe as “problematic” because I don’t have the time to get into how much trouble the right mind at the right idea can be.

No attempt to measure people’s health with a few simple measurements and derived quantities can be fully successful. Health is too complicated a thing for one or two or even ten quantities to describe. Measures like height-to-waist ratios and body mass indices and the like should be understood as filters, the way temperature and blood pressure are. If one or more of these measurements are in dangerous ranges there’s reason to think there’s a health problem worth investigating here. It doesn’t mean there is; it means there’s reason to think it’s worth spending resources on tests that are more expensive in time and money and energy. And similarly just because all the simple numbers are fine doesn’t mean someone is perfectly healthy. But it suggests that the person is more likely all right than not. They’re guides to setting priorities, easy to understand and requiring no training to use. They’re not a replacement for thought; no guides are.

Jeff Harris’s Shortcuts educational panel for the 10th is about zero. It’s got a mix of facts and trivia and puzzles with a few jokes on the side.

I don’t have a strong reason to discuss Ashleigh Brilliant’s Pot-Shots rerun for the 11th. It only mentions odds in a way that doesn’t open up to discussing probability. But I do like Brilliant’s “Embrace-the-Doom” tone and I want to share that when I can.

John Hambrock’s The Brilliant Mind of Edison Lee for the 13th of July riffs on the world’s leading exporter of statistics, baseball. Organized baseball has always been a statistics-keeping game. The Olympic Ball Club of Philadelphia’s 1837 rules set out what statistics to keep. I’m not sure why the game is so statistics-friendly. It must be in part that the game lends itself to representation as a series of identical events — pitcher throws ball at batter, while runners wait on up to three bases — with so many different outcomes.

Alan Schwarz’s book The Numbers Game: Baseball’s Lifelong Fascination With Statistics describes much of the sport’s statistics and record-keeping history. The things recorded have varied over time, with the list of things mostly growing. The number of statistics kept have also tended to grow. Sometimes they get dropped. Runs Batted In were first calculated in 1880, then dropped as an inherently unfair statistic to keep; leadoff hitters were necessarily cheated of chances to get someone else home. How people’s idea of what is worth measuring changes is interesting. It speaks to how we change the ways we look at the same event.

Dana Summers’s Bound And Gagged for the 13th uses the old joke about computers being abacuses and the like. I suppose it’s properly true that anything you could do on a real computer could be done on the abacus, just, with a lot ore time and manual labor involved. At some point it’s not worth it, though.

Nate Fakes’s Break of Day for the 13th uses the whiteboard full of mathematics to denote intelligence. Cute birds, though. But any animal in eyeglasses looks good. Lab coats are almost as good as eyeglasses.

David L Hoyt and Jeff Knurek’s Jumble for the 13th is about one of geometry’s great applications, measuring how large the Earth is. It’s something that can be worked out through ingenuity and a bit of luck. Once you have that, some clever argument lets you work out the distance to the Moon, and its size. And that will let you work out the distance to the Sun, and its size. The Ancient Greeks had worked out all of this reasoning. But they had to make observations with the unaided eye, without good timekeeping — time and position are conjoined ideas — and without photographs or other instantly-made permanent records. So their numbers are, to our eyes, lousy. No matter. The reasoning is brilliant and deserves respect.

Reading the Comics, May 6, 2016: Mistakes Edition

I knew my readership would drop off after I fell back from daily posting. Apparently it was worse than I imagined and nobody read my little blog here over the weekend. That’s fair enough; I had to tend other things myself. Still, for the purpose of maximizing the number of page views around here, taking two whole days off in a row was a mistake. There’s some more discussed in this Reading The Comics installment.

Word problems are dull. At least at the primary-school level. There’s all these questions about trains going in different directions or ropes sweeping out areas or water filling troughs. So Aaron McGruder’s Boondocks rerun from the 5th of May (originally run the 22nd of February, 2001) is a cute change. It’s at least the start of a legitimate word problem, based on the ways the recording industry took advantage of artists in the dismal days of fifteen years ago. I’m sure that’s all been fixed by now. Fill in some numbers and the question might interest people.

Glenn McCoy and Gary McCoy’s The Duplex for the 5th of May is a misunderstanding-fractions joke. I’m amused by the idea of messing up quarter-pound burgers. But it also brings to mind a summer when I worked for the Great Adventure amusement park and got assigned one day as cashier at the Great American Hamburger Stand. Thing is, I didn’t know anything about the stand besides the data point that they probably sold hamburgers. So customers would order stuff I didn’t know, and I couldn’t find how to enter it on the register, and all told it was a horrible mess. If you were stuck in that impossibly slow-moving line, I am sorry, but it was management’s fault; I told them I didn’t know what I was even selling. Also I didn’t know the drink cup sizes so I just charged you for whatever you said and if I gave you the wrong size I hope it was more soda than you needed.

On a less personal note, I have heard the claim about why one-third-pound burgers failed in United States fast-food places. Several chains tried them out in the past decade and they didn’t last, allegedly because too many customers thought a third of a pound was less than a quarter pound and weren’t going to pay more for less beef. It’s … plausible enough, I suppose, because people have never been good with fractions. But I suspect the problem is more linguistic. A quarter-pounder has a nice rhythm to it. A half-pound burger is a nice strong order to say. A third-pound burger? The words don’t even sound right. You have to say “third-of-a-pound burger” to make it seem like English, and it’s a terribly weak phrase. The fast food places should’ve put their money into naming it something that suggested big-ness but not too-big-to-eat.

Mark Tatulli’s Heart of the City for the 5th is about Heart’s dread of mathematics. Her expressed fear, that making one little mistake means the entire answer is wrong, is true enough. But how how much is that “enough”? If you add together someting that should be (say) 18, and you make it out to be 20 instead, that is an error. But that’s a different sort of error from adding them together and getting 56 instead.

And errors propagate. At least they do in real problems, in which you are calculating something because you want to use it for something else. An arithmetic error on one step might grow, possibly quite large, with further steps. That’s trouble. This is known as an “unstable” numerical calculation, in much the way a tin of picric acid dropped from a great height onto a fire is an “unstable” chemical. The error might stay about as large as it started out being, though. And that’s less troublesome. A mistake might stay predictable. The calculation is “stable” In a few blessed cases an error might be minimized by further calculations. You have to arrange the calculations cleverly to make that possible, though. That’s an extremely stable calculation.

And this is important because we always make errors. At least in any real calculation we do. When we want to turn, say, a formula like πr2 into a number we have to make a mistake. π is not 3.14, nor is it 3.141592, nor is it 3.14159265358979311599796346854418516. Does the error we make by turning π into some numerical approximation matter? It depends what we’re calculating, and how. There’s no escaping error and it might be a comfort to Heart, or any student, to know that much of mathematics is about understanding and managing error.

Joe Martin’s Boffo for the 6th of May is in its way about the wonder of very large numbers. On some reasonable assumptions — that our experience is typical, that nothing is causing traits to be concentrated one way or another — we can realize that we probably will not see any extreme condition. In this case, it’s about the most handsome men in the universe probably not even being in our galaxy. If the universe is large enough and people common enough in it, that’s probably right. But we likely haven’t got the least handsome either. Lacking reason to suppose otherwise we can guess that we’re in the vast middle.

David L Hoyt and Jeff Knurek’s Jumble for the 6th of May mentions mathematicians and that’s enough, isn’t it? Without spoiling the puzzle for anyone, I will say that “inocci” certainly ought to be a word meaning something. So get on that, word-makers.

Dave Blazek’s Loose Parts for the 6th brings some good Venn Diagram humor back to my pages. Good. It’s been too long.

Reading the Comics, December 13, 2015: More Nearly Like It Edition

This has got me closer to the number of comics I like for a Reading the Comics post. There’s two comics already in my file, for the 14th of December, but those can wait until later in the week.

David L Hoyt and Jeff Knurek’s Jumble for the 11th of December has a mathematics topic. The quotes in the final answer are the hint that it’s a bit of wordplay. The mention of “subtraction” is a hint.

Brian Kliban’s cartoon for the 11th of December (a rerun from who knows when) promises an Illegal Cube Den, and delivers. I’m just delighted by the silliness of it all.

Greg Evans’s Luann Againn for the 11th of December reprints the 1987 Luann. “Geometric principles of equitorial [sic] astronomical coordinate systems” gets mentioned as a math-or-physics-sounding complicated thing to do. The basic idea is to tell where things are in the sky, as we see them from the surface of the Earth. In an equatorial coordinate system we imagine — we project — where the plane of the equator is, and we can measure things as north or south of that plane. (North is on the same side that the Earth’s north pole is.) That celestial equator is functionally equivalent to longitude, although it’s called declination.

We also need something functionally equivalent to longitude; that’s called the right ascension. To define that, we need something that works like the prime meridian. Projecting the actual prime meridian out to the stars doesn’t work. The prime meridian is spinning every 24 hours and we can’t publish updated star charts that quickly. What we use as a reference meridian instead is spring. That is, it’s where the path of the sun in the sky crosses the celestial equator in March and the (northern hemisphere) spring.

There are catches and subtleties, which is why this makes for a good research project. The biggest one is that this crossing point changes over time. This is because the Earth’s orbit around the sun changes. So right ascensions of points change a little every year. So when we give coordinates, we have to say in which system, and which reference year. 2000 is a popular one these days, but its time will pass. 1950 and 1900 were popular in their generations. It’s boring but not hard to convert between these reference dates. And if you need this much precision, it’s not hard to convert between the reference year of 2000 and the present year. I understand many telescopes will do that automatically. I don’t know directly because I have little telescope experience, and I couldn’t even swear I had seen a meteor until 2013. In fairness, I grew up in New Jersey, so with the light pollution I was lucky to see night sky.

Peter Maresca’s Origins of the Sunday Comics for the 11th of December showcases a strip from 1914. That, Clare Victor Dwiggins’s District School for the 12th of April, 1914, is just a bunch of silly vignettes. It’s worth zooming in to look at. It’s got a student going “figger juggling” and that gives me an excuse to point out the strip to anyone who’ll listen.

Samson’s Dark Side of the Horse for the 13th of December enters another counting-sheep joke into the ranks. Tying it into angles is cute. It’s tricky to estimate angles by sight. I think people tend to over-estimate how big an angle is when it’s around fifteen or twenty degrees. 45 degrees is easy enough to tell by sight. But for angles smaller than that, I tend to estimate angles by taking the number I think it is and cutting it in half, and I get closer to correct. I’m sure other people use a similar trick.

Brian Anderson’s Dog Eat Doug for the 13th of December has the dog, Sophie, deploy a lot of fraction talk to confuse a cookie out of Doug. A lot of new fields of mathematics are like that the first time you encounter them. I am curious where Sophie’s reasoning would have led, if not interrupted. How much cookie might she have cadged by the judicious splitting of halves and quarters and, perhaps, eighths and such? I’m not sure where her patter was going.

Shannon Wheeler’s Too Much Coffee Man for the 13th of December uses the traditional blackboard full of symbols to denote a lot of deeply considered thinking. Did you spot the error?

Reading the Comics, November 20, 2014: Ancient Events Edition

I’ve got enough mathematics comics for another roundup, and this time, the subjects give me reason to dip into ancient days: one to the most famous, among mathematicians and astronomers anyway, of Greek shipwrecks, and another to some point in the midst of winter nearly seven thousand years ago.

Eric the Circle (November 15) returns “Griffinetsabine” to the writer’s role and gives another “Shape Single’s Bar” scene. I’m amused by Eric appearing with his ex: x is practically the icon denoting “this is an algebraic expression”, while geometry … well, circles are good for denoting that, although I suspect that triangles or maybe parallelograms are the ways to denote “this is a geometric expression”. Maybe it’s the little symbol for a right angle.

Jim Meddick’s Monty (November 17) presents Monty trying to work out just how many days there are to Christmas. This is a problem fraught with difficulties, starting with the obvious: does “today” count as a shopping day until Christmas? That is, if it were the 24th, would you say there are zero or one shopping days left? Also, is there even a difference between a “shopping day” and a “day” anymore now that nobody shops downtown so it’s only the stores nobody cares about that close on Sundays? Sort all that out and there’s the perpetual problem in working out intervals between dates on the Gregorian calendar, which is that you have to be daft to try working out intervals between dates on the Gregorian calendar. The only worse thing is trying to work out the intervals between Easters on it. My own habit for this kind of problem is to use the United States Navy’s Julian Date conversion page. The Julian date is a straight serial number, counting the number of days that have elapsed since noon Universal Time at what’s called the 1st of January, 4713 BCE, on the proleptic Julian calendar (“proleptic” because nobody around at the time was using, or even imagined, the calendar, but we can project back to what date that would have been), a year picked because it’s the start of several astronomical cycles, and it’s way before any specific recordable dates in human history, so any day you might have to particularly deal with has a positive number. Of course, to do this, we’re transforming the problem of “counting the number of days between two dates” to “counting the number of days between a date and January 1, 4713 BCE, twice”, but the advantage of that is, the United States Navy (and other people) have worked out how to do that and we can use their work.

Bill Hind’s kids-sports comic Cleats (November 19, rerun) presents Michael offering basketball advice that verges into logic and set theory problems: making the ball not go to a place outside the net is equivalent to making the ball go inside the net (if we decide that the edge of the net counts as either inside or outside the net, at least), and depending on the problem we want to solve, it might be more convenient to think about putting the ball into the net, or not putting the ball outside the net. We see this, in logic, in a set of relations called De Morgan’s Laws (named for Augustus De Morgan, who put these ideas in modern mathematical form), which describe what kinds of descriptions — “something is outside both sets A and B at one” or “something is not inside set A or set B”, or so on — represent the same relationship between the thing and the sets.

Tom Thaves’s Frank and Ernest (November 19) is set in the classic caveman era, with prehistoric Frank and Ernest and someone else discovering mathematics and working out whether a negative number times a negative number might be positive. It’s not obvious right away that they should, as you realize when you try teaching someone the multiplication rules including negative numbers, and it’s worth pointing out, a negative times a negative equals a positive because that’s the way we, the users of mathematics, have chosen to define negative numbers and multiplication. We could, in principle, have decided that a negative times a negative should give us a negative number. This would be a different “multiplication” (or a different “negative”) than we use, but as long as we had logically self-consistent rules we could do that. We don’t, because it turns out negative-times-negative-is-positive is convenient for problems we like to do. Mathematics may be universal — something following the same rules we do has to get the same results we do — but it’s also something of a construct, and the multiplication of negative numbers is a signal of that.

Mickey Mouse (November 20, rerun) — I don’t know who wrote or draw this, but Walt Disney’s name was plastered onto it — sees messages appearing in alphabet soup. In one sense, such messages are inevitable: jumble and swirl letters around and eventually, surely, any message there are enough letters for will appear. This is very similar to the problem of infinite monkeys at typewriters, although with the special constraint that if, say, the bowl has only two letters “L”, it’s impossible to get the word “parallel”, unless one of the I’s is doing an impersonation. Here, Goofy has the message “buried treasure in back yard” appear in his soup; assuming those are all the letters in his soup then there’s something like 44,881,973,505,008,615,424 different arrangements of letters that could come up. There are several legitimate messages you could make out of that (“treasure buried in back yard”, “in back yard buried treasure”), not to mention shorter messages that don’t use all those letters (“run back”), but I think it’s safe to say the number of possible sentences that make sense are pretty few and it’s remarkable to get something like that. Maybe the cook was trying to tell Goofy something after all.

Mark Anderson’s Andertoons (November 20) is a cute gag about the dangers of having too many axes on your plot.

Gary Delainey and Gerry Rasmussen’s Betty (November 20) mentions the Antikythera Mechanism, one of the most famous analog computers out there, and that’s close enough to pure mathematics for me to feel comfortable including it here. The machine was found in April 1900, in ancient shipwreck, and at first seemed to be just a strange lump of bronze and wood. By 1902 the archeologist Valerios Stais noticed a gear in the mechanism, but since it was believed the wreck far, far predated any gear mechanisms, the machine languished in that strange obscurity that a thing which can’t be explained sometimes suffers. The mechanism appears to be designed to be an astronomical computer, tracking the positions of the Sun and the Moon — tracking the actual moon rather than an approximate mean lunar motion — the rising and etting of some constellations, solar eclipses, several astronomical cycles, and even the Olympic Games. It’s an astounding mechanism, it’s mysterious: who made it? How? Are there others? What happened to them? How was the mechanical engineering needed for this developed, and what other projects did the people who created this also do? Any answers to these questions, if we ever know them, seem sure to be at least as amazing as the questions are.

When I saw the Maths History tweet about Edmond Halley’s birthday I wondered if the November 8th date given was the relevant one since, after all, in 1656 England was still on the Julian calendar. The MacTutor biography of him makes clear that the 8th of November is his Gregorian-date birthday, and he was born on the 29th of October by the calendar his parents were using, although it’s apparently not clear he was actually born in 1656. Halley claimed it was 1656, at least, and he probably heard from people who knew.

Halley is famous for working out the orbit of the comet that’s gotten his name attached, and correctly so: working out the orbits of comets was one of the first great accomplishments of Newtonian mechanics, and Halley’s work took into account how Jupiter’s gravitation distorts the orbit of a comet. It’s great work. And he’s also famous within mathematical and physics circles because it’s fair to wonder whether, without his nagging and his financial support, Isaac Newton would have published his Principia Mathematica. Astronomers note him as the first Western European astronomer to set up shop in the southern hemisphere and produce a map of that part of the sky, as well.

That hardly exhausts what’s interesting about him: for example, he joined in the late-17th-century fad for diving bell companies (for a while, you couldn’t lose money excavating wrecked ships, until finally everyone did) and even explored the bed of the English Channel in a diving bell of his own design. This is to me the most terrifying thing he did, and that’s even with my awareness he led two scientific sailing expeditions, one of which was cut short after among other things irreconcilable differences with the ship’s other commissioned officer, Lieutenant Edward Harrison (who blamed Halley for the oblivion which Harrison’s book on longitude received), and the second of which included a pause in Recife when Halley was put under guard by a man claiming to be the English consul, and who was actually an agent of the Royal African Company considering whether to seize Halley’s ship[1] as a prize.

After his second expedition Halley published charts showing the magnetic declination, how far a magnetic compass’s “north” is from true north, and introduced one of those great conceptual breakthroughs that charts can give us: he connected the lines showing the points where the declination was equal. These isolines are a magnificent way to diagram three-dimensional information on a two-dimensional chart; we see them in topographic maps, as the contour curves showing where a hill rises or a valley sinks. We see them in weather maps, the lines where the temperature is 70 or 80 Fahrenheit (or 20 or 25 Celsius, if you rather) or where the wind speed is some sufficiently alarming figure. We see them (in three-dimensional form) in medical imaging, where a region of constant density gets the same color and this is used to understand a complicated shape within. Not all these uses derive directly from Halley; as with all really good, widely usable concepts many people discovered the concept, but Halley was among the first to put them to obvious, prominent use.

And something that might serve as comfort to anyone who’s taking a birthday hard: at age 65, Halley began a study of the moon’s saros, the cycle patterns of different relative positions the Sun and Moon have in the sky which describe when eclipses happen. One cycle takes a bit over eighteen years to complete. Halley lived long enough to complete this work.

[1] The Paramore, which — I note because this is just the kind of world it was back then — was constructed in 1694 at the Royal Dockyard at Deptford on the River Thames for a scientific circumnavigation of the globe, and first sailed in April 1698 under Tsar Peter the Great, then busy travelling western Europe under ineffective cover to learn things which might modernize Russia. Halley had hoped to sail in 1696, but he was waylaid by his appointment to the Mint at Chester, courtesy of Newton.

Reading the Comics, July 18, 2014: Summer Doldrums Edition

Now, there, see? The school year (in the United States) has let out for summer and the rush of mathematics-themed comic strips has subsided; it’s been over two weeks since the last bunch was big enough. Given enough time, though, a handful of comics will assemble that I can do something with, anything, and now’s that time. I hate to admit also that they’re clearly not trying very hard with these mathematics comics as they’re not about very juicy topics. Call it the summer doldroms, as I did.

Mason Mastroianni and Mick Mastroianni’s B.C. (July 6) spends most of its text talking about learning cursive, as part of a joke built around the punch line that gadgets are spoiling students who learn to depend on them instead of their own minds. So it would naturally get around to using calculators (or calculator apps, which is a fair enough substitute) in place of mathematics lessons. I confess I come down on the side that wonders why it’s necessary to do more than rough, approximate arithmetic calculations without a tool, and isn’t sure exactly what’s gained by learning cursive handwriting, but these are subjects that inspire heated and ongoing debates so you’ll never catch me admitting either position in public.

Eric the Circle (July 7), here by “andel”, shows what one commenter correctly identifies as a “pi fight”, which might have made a better caption for the strip, at least for me, because Eric’s string of digits wasn’t one of the approximations to pi that I was familiar with. I still can’t find it, actually, and wonder if andel didn’t just get a digit wrong. (I might just not have found a good web page that lists the digits of various approximations to pi, I admit.) Erica’s approximation is the rather famous 22/7.

Richard Thompson’s Richard’s Poor Almanac (July 7, rerun) brings back our favorite set of infinite monkeys, here, to discuss their ambitious book set at the Museum of Natural History.

Tom Thaves’s Frank and Ernest (July 16) builds on the (true) point that the ancient Greeks had no symbol for zero, and would probably have had a fair number of objections to the concept.

Joe Martin’s Mr Boffo (July 18, sorry that I can’t find a truly permanent link) plays with one of Martin’s favorite themes, putting deep domesticity to great inventors and great minds. I suspect but do not know that Martin was aware that Einstein’s first wife, Mileva Maric, was a fellow student with him at the Swiss Federal Polytechnic. She studied mathematics and physics. The extent to which she helped Einstein develop his theories is debatable; as far as I’m aware the evidence only goes so far as to prove she was a bright, outside mind who could intelligently discuss whatever he might be wrangling over. This shouldn’t be minimized: describing a problem is often a key step in working through it, and a person who can ask good follow-up questions about a problem is invaluable even if that person doesn’t do anything further.

Charles Schulz’s Peanuts (July 18) — a rerun, of course, from the 21st of July, 1967 — mentions Sally going to Summer School and learning all about the astronomical details of summertime. Astronomy has always been one of the things driving mathematical discovery, but I admit, thinking mostly this would be a good chance to point out Dr Helmer Aslaksen’s page describing the relationship between the solstices and the times of earliest and latest sunrise (and sunset). It’s not quite as easy as finding when the days are longest and shortest. Dr Aslaksen has a number of fascinating astronomy- and calendar-based pages which I think worth reading, so, I hope you enjoy.

Reading the Comics, June 4, 2014: Intro Algebra Edition

I’m not sure that there is a theme to the most recent mathematically-themed comic strips that I’ve seen, all from GoComics in the past week, but they put me in mind of the stuff encountered in learning algebra, so let’s run with that. It’s either that or I start making these “edition” titles into something absolutely and utterly meaningless, which could be.

Marc Anderson’s Andertoons (May 30) uses the classic setup of a board full of equation to indicate some serious, advanced thinking going on, and then puts in a cute animal twist on things. I don’t believe that the equation signifies anything, but I have to admit I’m not sure. It looks quite plausibly like something which might turn up in quantum mechanics (the “h” and “c” and lambda are awfully suggestive), so if Anderson made it up out of whole cloth he did an admirable job. If he didn’t make it up and someone recognizes it, please, let me know; I’m curious what it might be.

Marc Anderson reappears on the second of June has the classic reluctant student upset with the teacher who knew all along what x was. Knowledge of what x is is probably the source of most jokes about learning algebra, or maybe mathematics overall, and it’s amusing to me anyway that what we really care about is not what x is particularly — we don’t even do ourselves any harm if we call it some other letter, or for that matter an empty box — but learning how to figure out what values in the place of x would make the relationship true.

Jonathan Lemon’s Rabbits Against Magic (May 31) has the one-eyed rabbit Weenus doing miscellaneous arithmetic on the way to punning about things working out. I suppose to get to that punch line you have to either have mathematics or gym class as the topic, and I wouldn’t be surprised if Lemon’s done a version with those weight-lifting machines on screen. That’s not because I doubt his creativity, just that it’s the logical setup.

Eric Scott’s Back In The Day (June 2) has a pair of dinosaurs wondering about how many stars there are. Astronomy has always inspired mathematics. After one counts the number of stars one gets to wondering, how big the universe could be — Archimedes, classically, estimated the universe was about big enough to hold 1063 grains of sand — or how far away the sun might be — which the Ancient Greeks were able to estimate to the right order of magnitude on geometric grounds — and I imagine that looking deep into the sky can inspire the idea that the infinitely large and the infinitely small are at least things we can try to understand. Trying to count stars is a good start.

Steve Boreman’s Little Dog Lost (June 2) has a stick insect provide the excuse for some geometry puns.

Brian and Ron Boychuk’s The Chuckle Brothers (June 4) has a pie shop gag that I bet the Boychuks are kicking themselves for not having published back in mid-March.

Reading the Comics, April 21, 2014: Bill Amend In Name Only Edition

Recently the National Council of Teachers of Mathematics met in New Orleans. Among the panelists was Bill Amend, the cartoonist for FoxTrot, who gave a talk about the writing of mathematics comic strips. Among the items he pointed out as challenges for mathematics comics — and partly applicable to any kind of teaching of mathematics — were:

• Accessibility
• Stereotypes
• What is “easy” and “hard”?
• I’m not exactly getting smarter as I age
• Newspaper editors might not like them

Besides the talk (and I haven’t found a copy of the PowerPoint slides of his whole talk) he also offered a collection of FoxTrot comics with mathematical themes, good for download and use (with credit given) for people who need to stock up on them. The link might be expire at any point, note, so if you want them, go now.

While that makes a fine lead-in to a collection of mathematics-themed comic strips around here I have to admit the ones I’ve seen the last couple weeks haven’t been particularly inspiring, and none of them are by Bill Amend. They’ve covered a fair slate of the things you can write mathematics comics about — physics, astronomy, word problems, insult humor — but there’s still interesting things to talk about. For example:

Reading the Comics, March 26, 2014: Kitchen Science Department

It turns out that three of the comic strips to be included in this roundup of mathematics-themed strips mentioned things that could reasonably be found in kitchens, so that’s why I’ve added that as a subtitle. I can’t figure a way to contort the other entries to being things that might be in kitchens, but, given that I don’t get to decide what cartoonists write about I think I’m doing well to find any running themes.

Ralph Hagen’s The Barn (March 19) is built around a possibly accurate bit of trivia which tries to stagger the mind by considering the numinous: how many stars are there? This evokes, to me at least, one of the famous bits of ancient Greek calculations (for which they get much less attention than the geometers and logicians did), as Archimedes made an effort to estimate how many grains of sand could fit inside the universe. Archimedes had apparently little fear of enormous numbers, and had to strain the Greek system for representing numbers to get at such enormous quantities. But he was an ingenious reasoner: he was able to estimate, for example, the sizes and distances to the Moon and the Sun based on observing, with the naked eye, the half-moon; and his work on problems like finding the value of pi get surprisingly close to integral calculus and would probably be a better introduction to the subject than pre-calculus courses are. It’s quite easy in considering how big (and how old) the universe is to get to numbers that are really difficult to envision, so, trying to reduce that by imagining stars as grains of salt might help, if you can imagine a ball of salt eight miles across.

Reading the Comics, December 12, 2013

It’s a bit of a shame there weren’t quite enough comics to run my little roundup on the 11th of December, for that nice 11/12/13 sequence, but I’m not in charge of arranging these things. For this week’s gathering of mathematically themed comic strips there’s not any deeper theme than they mention mathematic points, but at least the first couple of them have some real meat to the subject matter. (It feels to me like if one of the gathered comics inspires an essay, it’s usually one of the first couple in a collection. That might indicate that I get tired while writing these out, or it might reflect a biased recollection of when I do break out an essay.)

John Allen’s Nest Heads (December 5) is built around a kid not understanding a probability distribution: how many days in a row does it take to get the chance of snow to be 100 percent? The big flaw here is the supposition that the chance of snow is a (uhm) cumulative thing, so that if the snow didn’t happen yesterday or the day before it’s the more likely to happen today or tomorrow. As we actually use weather forecasts, though, they’re … well, I’m not sure I’d say they’re independent, that yesterday’s 30 percent chance of snow has nothing to do with today’s 25 percent chance, since it seems to me plausible that whether it snowed yesterday affects whether it snows today. But they don’t just add up until we get a 100 percent chance of snow when things start to drop.

Meteors and Money Management

I probably heard of Wethersfield, Connecticut, although I forgot about it until teaching a statistics course last academic year. The town vanished from my memory shortly thereafter, because as far as I know I’ve never been there or known anyone who had. The rather exciting meteor strike in Russia last week brought it back to mind, though, because the town worked its way into a probability book I was using for reference.

Here’s the setup: the town is about 14 square miles in area, out of something like 200,000,000 square miles of land and water on the surface of the Earth. Something like three meteors of appreciable size strike the surface of the Earth, somewhere, three times a day. Suppose that every spot on the planet is equally likely to get a meteor strike. So, what’s the probability that Wethersfield should get struck in any one year?