## Reading the Comics, March 9, 2016: Mathematics Recreation Edition

I haven’t been skipping the comics, even with the effort of keeping up on the Leap Day 2016 A To Z Glossary. I just try to keep to the pace which Comic Strip Master Command sets.

The kids-information feature Short Cuts, by Jeff Harris, got ahead of “Pi Day” last Sunday. I imagine the feature gets run mid-week in some features, so that it’s better to run a full week before March 14th. But here’s a bundle of trivia, some jokes, some activities, that sort of thing. I am curious about one of Harris’s trivias, that Pi “plays an important role in some of the equations used in Einstein’s famous general theory of relativity”. That’s true, but it’s not as if general relativity is a rare appearance for pi in physics. Maybe Harris chose it on aesthetic grounds. General relativity has a familiar name and exotic concepts. And it allowed him to put in an equation that’s mysterious yet attractive-looking.

Samson’s Dark Side Of The Horse for the 7th of March made me wonder how many sudoku puzzles there are. The answer is — well, you have to start thinking carefully about what you mean by “how many”. For example: start with one puzzle. Swap out every appearance of a 1 with a 2, and a 2 with a 1. Is this new one actually a different puzzle? You can make a case for yes or for no. And that’s before we get into the question of how many clues to give to solve the puzzle. If I’m not misreading Wikipedia’s “Mathematics of Sudoku” page, the number of different nine-by-nine combinations of digits that can be legitimate sudoku puzzle solutions is 6,670,903,752,021,072,936,960. This was worked out in 2005 by Bertram Felgenhauer and Frazer Jarvis. They worked it out partly by logic, partly by brute force. Brute force is trying all the possibilities to see what works. It’s a method that rewards endurance. We like that we can turn it over to computers now. Or cartoon horses, whichever. They’re good at endurance.

Jef Mallett’s Frazz started a sequence about problem-writing on the 7th of March. Caulfield’s setup, complaining about trains and apple bushels, suggests he was annoyed by mathematics problems. I understand. Much of real mathematics starts with curiosity about something (how many sudoku puzzles are there?). Then it’s working out what computation might answer that question. Then it’s doing that calculation. And then it’s verifying that the calculation is right. Mathematics educators have to teach ways to do a calculation, and test that. And to teach how to know what calculation to do, and test that. That’s challenging enough. Add to that working out something to be curious about and you understand the appeal of stock setups. Maybe mathematics should include some courses in creative writing and short-short fiction. (Verification is, in my experience, the part nobody cares about. This is a shame. The hardest part of doing numerical mathematics is making sure your computation makes any sense.)

Richard Thompson’s Richard’s Poor Almanac rerun the 7th of March features the Non-Euclidean Creeper. It’s a plant perhaps related to the Cubist Fir Christmas tree and to the Otterloops’ troublesome non-Euclidean tree. Non-Euclidean geometry will probably always sound more intimidating and exotic. Euclidean geometry describes the way objects on the human scale behave. Shapes that fit on the table, or in your garden, follow Euclidean rules. But non-Euclidean isn’t magic; it’s the way that shapes on the surface of a globe work, for example. And the idea of drawing a thing like a square on the surface of the Earth isn’t so bizarre.

Paul Trap’s Thatababy for the 7th makes sport of geometry.

Jim Toomey’s Sherman’s Lagoon for the 9th of March, 2016. This by the way followed a storyline about the resident turtle catching bioluminescence, the way that turtle species noticed last year did. Certain comic strips can be sources of surprisingly reliable science news. Note: the mathematical kind of ‘ring’ is not meant here.

My love and I were talking the other day about Jim Toomey’s Sherman’s Lagoon. It’s a bit odd as comic strips go. It’s been around forever, for one, but nobody talks about it. It’s stayed reliably funny. Comic strips that’ve been around forever tend to … you know … not be. The strip’s done as a work-and-home strip except the cast is all sea life. And the thing is, Toomey keeps paying attention to new discoveries in sea life, and other animal research. And this is a fantastic era for discoveries in sea life, aside from how humans have now eaten all of it and we don’t have any left. I am not joking when I say the comic strip is an effortless way to keep up with new discoveries about the oceans.

I missed it when in December the discovery was announced to the world. But the setup, about the common name being given by a group of kids, is apparently quite correct. So we should expect from Toomey. (The scientific name is Etmopterus benchleyi. The last name refers to Peter Benchley, repentant Jaws novelist.) LiveScience.com’s article says lead author Dr Vicky Vásquez had to “scale them back” from their starting point, the “super ninja”. This differs from Hawthorne’s claim that the kids started from the “math stinks” shark, but it’s still a delight anyway.

## From ElKement: May The Force Field Be With You

I’m derelict in mentioning this but ElKement’s blog, Theory And Practice Of Trying To Combine Just Anything, has published the second part of a non-equation-based description of quantum field theory. This one, titled “May The Force Field Be With You: Primer on Quantum Mechanics and Why We Need Quantum Field Theory”, is about introducing the idea of a field, and a bit of how they can be understood in quantum mechanics terms.

A field, in this context, means some quantity that’s got a defined value for every point in space and time that you’re studying. As ElKement notes, the temperature is probably the most familiar to people. I’d imagine that’s partly because it’s relatively easy to feel the temperature change as one goes about one’s business — after all, gravity is also a field, but almost none of us feel it appreciably change — and because weather maps make the changes of that in space and in time available in attractive pictures.

The thing the field contains can be just about anything. The temperature would be just a plain old number, or as mathematicians would have it a “scalar”. But you can also have fields that describe stuff like the pull of gravity, which is a certain amount of pull and pointing, for us, toward the center of the earth. You can also have fields that describe, for example, how quickly and in what direction the water within a river is flowing. These strengths-and-directions are called “vectors” [1], and a field of vectors offers a lot of interesting mathematics and useful physics. You can also plunge into more exotic mathematical constructs, but you don’t have to. And you don’t need to understand any of this to read ElKement’s more robust introduction to all this.

[1] The independent student newspaper for the New Jersey Institute of Technology is named The Vector, and has as motto “With Magnitude and Direction Since 1924”. I don’t know if other tech schools have newspapers which use a similar joke.

• #### elkement 6:22 am on Thursday, 3 October, 2013 Permalink | Reply

Thanks again for your kind pingback and publicity :-)
I need to get to vectors and tensors in the next post(s) but I am still trying to figure out how to do this without mentioning those terms. Fluid dynamics is often a good starting point, e.g. to introduce, ‘derive’ or better motivate Schrödinger’s equation. On the other hand Feynman used to plunge directly into path integrals – presenting them as a rule along the lines of “This is the way nature works – live with it” – and deriving Schrödinger’s equation later.

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• #### Joseph Nebus 3:20 am on Saturday, 5 October, 2013 Permalink | Reply

I’m not quite sure how I’d do either. I think I could probably explain vectors without having to use mathematical symbolism, since the idea of stuff moving at particular speeds in directions can call on physical intuition. Tensors I don’t know how I’d try to explain in popular terms, partly because I’m not really as proficient in them as I should be. I probably need to think seriously about my own understanding of them.

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• #### elkement 6:26 pm on Monday, 7 October, 2013 Permalink | Reply

I have also always considered easier to imagine the different aspects of a vector – the abstract object and the ‘arrow’ as it lives in a specific base. But how do you really imagine the ‘abstract tensor object’ – in contrast to a ‘matrix’ (with more than 3 dimensions probably…)

I have started to read about general relativity (… will finish after I have finally understood the Higgs…) and it took me quite a while to comprehend that you are not allowed to shift a vector in curved space as you shift the ‘arrow’. Actually it made me think about vectors in a new way…

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• #### Joseph Nebus 2:46 am on Friday, 18 October, 2013 Permalink | Reply

(I’m embarrassed that I lost this comment somehow.)

I can sort of reconstruct the process when I think I started to get vectors as a concept, particularly in thinking of them as not tied to some particular point, or even containing information about a point, but somehow floating freely off that. If I get around to trying to explain vectors I might even be able to make all that explicit again.

Tensors I keep feeling like I’m on the verge of having that intuitive leap to where I have some mental model for how they work but I keep finding I don’t quite do enough work with them that it gets past following the rules and into really understanding the rules.

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## From ElKement: Space Balls, Baywatch, and the Geekiness of Classical Mechanics

Over on Elkement’s blog, Theory and Practice of Trying To Combine Just Anything, is the start of a new series about quantum field theory. Elke Stangl is trying a pretty impressive trick here in trying to describe a pretty advanced field without resorting to the piles of equations that maybe are needed to be precise, but, which also fill the page with piles of equations.

The first entry is about classical mechanics, and contrasting the familiar way that it gets introduced to people —- the whole forceequalsmasstimesacceleration bit — and an alternate description, based on what’s called the Principle of Least Action. This alternate description is as good as the familiar old Newton’s Laws in describing what’s going on, but it also makes a host of powerful new mathematical tools available. So when you get into serious physics work you tend to shift over to that model; and, if you want to start talking Modern Physics, stuff like quantum mechanics, you pretty nearly have to start with that if you want to do anything.

So, since it introduces in clear language a fascinating and important part of physics and mathematics, I’d recommend folks try reading the essay. It’s building up to an explanation of fields, as the modern physicist understands them, too, which is similarly an important topic worth being informed about.

• #### elkement 11:03 am on Thursday, 19 September, 2013 Permalink | Reply

Thanks a lot, Joseph – I am really honored :-) I hope I will be able to meet the expectations raised by your post :-D

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• #### Joseph Nebus 2:45 am on Friday, 20 September, 2013 Permalink | Reply

Well, thank you, and I’m confident in you.

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• #### elkement 11:06 am on Thursday, 19 September, 2013 Permalink | Reply

Reblogged this on Theory and Practice of Trying to Combine Just Anything and commented:
This is self-serving, but I can’t resist reblogging Joseph Nebus’ endorsement of my posts on Quantum Field Theory. Joseph is running a great blog on mathematics, and he manages to explain math in an accessible and entertaining way. I hope I will be able to do the same to theoretical physics!

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## Feynman Online Physics

Likely everybody in the world has already spotted this before, but what the heck: CalTech and the Feynman Lectures Website have put online an edition of volume one of The Feynman Lectures on Physics. This is an HTML 5 edition, so older web browsers might not be able to read it sensibly.

The Feynman Lectures are generally regarded as one of the best expositions of basic physics; they started as part of an introduction to physics class that spiralled out of control and that got nearly all the freshmen who were trying to take it lost. I know the sense of being lost; when I was taking introductory physics I turned to them on the theory they might help me understand what the instructor was going on about. It didn’t help me.

This isn’t because Feynman wasn’t explaining well what was going on. It’s just that he approached things with a much deeper, much broader perspective than were really needed for me to figure out my problems in — oh, I’m not sure, probably something like how long a block needs to slide down a track or something like that. Here’s a fine example, excerpted from Chapter 5-2, “Time”:

## Gibbs’ Elementary Principles in Statistical Mechanics

I had another discovery from the collection of books at archive.org, now that I thought to look for it: Josiah Willard Gibbs’s Elementary Principles in Statistical Mechanics, originally published in 1902 and reprinted 1960 by Dover, which gives you a taste of Gibbs’s writings by its extended title, Developed With Especial Reference To The Rational Foundation of Thermodynamics. Gibbs was an astounding figure even in a field that seems to draw out astounding figures, and he’s a good candidate for the title of “greatest scientist to come from the United States”.

He lived in walking distance of Yale (where his father and then he taught) nearly his whole life, working nearly isolated but with an astounding talent for organizing the many complex and confused ideas in the study of thermodynamics into a neat, logical science. Some great scientists have the knack for finding important work to do; some great scientists have the knack for finding ways to express work so the masses can understand it. Gibbs … well, perhaps it’s a bit much to say the masses understand it, but the language of modern thermodynamics and of quantum mechanics is very much the language he spoke a century-plus ago.

My understanding is he published almost all his work in the journal Transactions of the Connecticut Philosophical Society, in a show of hometown pride which probably left the editors baffled but, I suppose, happy to print something this fellow was very sure about.

To give some idea why they might have found him baffling, though, consider the first paragraph of Chapter 1, which is accurate and certainly economical:

We shall use Hamilton’s form of the equations of motion for a system of n degrees of freedom, writing $q_1, \cdots q_n$ for the (generalized) coördinates, $\dot{q}_1, \cdots \dot{q}_n$ for the (generalized) velocities, and

$F_1 q_1 + F_2 q_2 + \cdots + F_n q_n$ [1]

for the moment of the forces. We shall call the quantities $F_1, \cdots F_n$ the (generalized) forces, and the quantities $p_1 \cdots p_n$, defined by the equations

$p_1 = \frac{d\epsilon_p}{d\dot{q}_1}, p_2 = \frac{d\epsilon_p}{d\dot{q}_2}, etc.,$ [2]

where $\epsilon_p$ denotes the kinetic energy of the system, the (generalized) momenta. The kinetic energy is here regarded as a function of the velocities and coördinates. We shall usually regard it as a function of the momenta and coördinates, and on this account we denote it by $\epsilon_p$. This will not prevent us from occasionally using formulas like [2], where it is sufficiently evident the kinetic energy is regarded as function of the $\dot{q}$‘s and $q$‘s. But in expressions like $d\epsilon_p/dq_1$, where the denominator does not determine the question, the kinetic energy is always to be treated in the differentiation as function of the p’s and q’s.

(There’s also a footnote I skipped because I don’t know an elegant way to include it in WordPress.) Your friend the physics major did not understand that on first read any more than you did, although she probably got it after going back and reading it a touch more slowly. And his writing is just like that: 240 pages and I’m not sure I could say any of them could be appreciably tightened.

Also, I note I finally reached 9,000 page views! Thank you; I couldn’t have done it without at least twenty of you, since I’m pretty sure I’ve obsessively clicked on my own pages at minimum 8,979 times.

• #### Peter Mander 8:05 pm on Thursday, 21 March, 2013 Permalink | Reply

Fully agree with your assessment of Gibbs’ greatness. The US should be immensely proud of him.

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## 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?

• #### Geoffrey Brent (@GeoffreyBrent) 3:46 am on Wednesday, 20 February, 2013 Permalink | Reply

Working the numbers, rounding a lot, and skipping some second-order issues: Wethersfield is about 1/14,000,000 of the Earth’s surface, and 3 meteors/day = 12,000 over an 11-year period. For any given Wethersfield-sized area, the chance of getting hit by just one of those meteors is 12000/14000000, i.e. about 1/1000.

Of those 12,000 meteors there are about (12,000^2)/2 pairs, and for each pair there’s a (1/14,000,000^2) chance that BOTH will hit Wethersfield. So the chance Wethersfield will get hit by two meteors in a given 11-year period is about (12,000/14,000,000)^2 which works out at a bit under one in a million.

BUT if you divided the surface of the Earth up into Wethersfield-sized areas, you’d have fourteen million of them. Or about four million if you exclude the oceans. That means that you can expect about four such areas to experience a double-hit in that particular 11-year period.

A complication here is that we’re defining the intervals of interest (spacial and temporal) AFTER observing where the meteors hit, which means we may have cherry-picked those choices in a way that’s more likely to produce “coincidences”.

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• #### Joseph Nebus 4:44 am on Wednesday, 20 February, 2013 Permalink | Reply

I agree with you down the line, there, depending on just how you want to work out “just a bit under” one in a million.

The book I got it from — and I am trying to figure out where in my notes I wrote the source down, although I plundered the problem for homework — placed this in binomial distributions, where the young students returning to college after not thinking about algebra for years can try to imagine how you would even calculate “$0.99993^{11,999}$”, and then gives it a cameo appearance for Poisson distributions, which can be at least as terrifying to the calculator.

And, certainly, naming Wethersfield after the meteor hit makes it look like a longer shot than it should be. Saying that there should be about four Wethersfield-sized areas struck by two meteors every decade … well, that doesn’t quite make the idea any more accessible, since I still haven’t seen the town, and only occasionally appreciate just how big the Earth really is, but it feels more evocative.

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## Counting To 52

fluffy brought to my attention a cute, amusing little bit from the Annals of Improbable Research, itself passing on some work by one Inder J Taneja. Taneja worked out a paper, available from arxiv.org, which lists results to the sort of mathematical puzzle that’s open to anyone with some paper and a pencil and some desire to do some recreational stuff.

• #### fluffy 9:57 pm on Sunday, 17 February, 2013 Permalink | Reply

Presumably, the largest constructable descending number would be 987654321 (which is somewhat larger than the largest constructable ascending number, 123456789

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• #### fluffy 9:58 pm on Sunday, 17 February, 2013 Permalink | Reply

Argh, WordPress ate my superscripts.

1^(2^(3^(4^(5^(6^(7^(8^9))))))) etc.

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• #### fluffy 10:01 pm on Sunday, 17 February, 2013 Permalink | Reply

er, and obviously that should be (1+2)^3^4^5^6^7^8^9. 1^2^3… etc. isn’t very interesting.

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• #### Chiaroscuro 2:27 am on Monday, 18 February, 2013 Permalink | Reply

Fluffy: Larger would be
12^(3^(4^(5^(6^(7^(8^9))))))))

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• #### fluffy 6:05 am on Monday, 18 February, 2013 Permalink | Reply

Oh, right. Also 9^8^7^6^5^4^321 beats the other descending one.

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• #### Joseph Nebus 4:47 am on Wednesday, 20 February, 2013 Permalink | Reply

Yeah, I think for getting at extremes you have to look at where sequences are bigger than exponentials, and that’s mostly going to turn out to be $4^{321}$ and $12^{3}$.

I haven’t figured how to turn on a preview mode yet, but you should be able to do all sorts of nice LaTeX symbols by entering $and the word latex, then the symbols, and then close with a final$. I also haven’t figured how to add a footnote saying this is possible.

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• #### fluffy 5:15 am on Wednesday, 20 February, 2013 Permalink | Reply

So like $12^3^4^5^6^7^8^9$ or whatnot?

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• #### fluffy 5:17 am on Wednesday, 20 February, 2013 Permalink | Reply

Hooray for comment previews.

$12^[3^[4^[5^[6^[7^[8^9]]]]]]$

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## Arthur Christmas and the End of Time

In working out my little Arthur Christmas-inspired problem, I argued that if the reindeer take some nice rational number of hours to complete one orbit of the Earth, eventually they’ll meet back up with Arthur and Grand-Santa stranded on the ground. And if the reindeer take an irrational number of hours to make one orbit, they’ll never meet again, although if they wait long enough, they’ll get pretty close together, eventually.

So far this doesn’t sound like a really thrilling result: the two parties, moving on their own paths, either meet again, or they don’t. Doesn’t sound quite like I earned the four-figure income I got from mathematics work last year. But here’s where I get to be worth it: if the reindeer and Arthur don’t meet up again, but I can accept their being very near one another, then they will get as close as I like. I only figured how long it would take for the two to get about 23 centimeters apart, but if I wanted, I could wait for them to be two centimeters apart, or two millimeters, or two angstroms if I wanted. I’d pay for this nearer miss with a longer wait. And this gives me my opening to a really stunning bit of mathematics.

## Fun With General Physics

I’m sure to let my interest in the Internet Archive version of Landau, Akhiezer, and Lifshiz General Physics wane soon enough. But for now I’m still digging around and finding stuff that delights me. For example, here, from the end of section 58 (Solids and Liquids):

As the temperature decreases, the specific heat of a solid also decreases and tends to zero at absolute zero. This is a consequence of a remarkable general theorem (called Nernst’s theorem), according to which, at sufficiently low temperatures, any quantity representing a property of a solid or liquid becomes independent of temperature. In particular, as absolute zero is approached, the energy and enthalpy of a body no longer depend on the temperature; the specific heats cp and cV, which are the derivatives of these quantities with respect to temperature, therefore tend to zero.

It also follows from Nernst’s theorem that, as $T \rightarrow 0$, the coefficient of thermal expansion tends to zero, since the volume of the body ceases to depend on the temperature.

## General Physics from the Internet Archive

Mir Books is this company that puts out downloadable, translated copies of mostly Soviet mathematics and physics books. As often happens I started reading them kind of on a whim and kept following in the faith that someday I’d see a math text I just absolutely had to have. It hasn’t quite reached that, although a post from today identified one I do like which, naturally enough, they aren’t publishing. It’s from the Internet Archive instead.

The book is General Physics, by L D Landau, A I Akhiezer, and E M Lifshiz. The title is just right; it gets you from mechanics to fields to crystals to thermodynamics to chemistry to fluid dynamics in about 370 pages. The scope and size probably tell you this isn’t something for the mass audience; the book’s appropriate for an upper-level undergraduate or a grad student, or someone who needs a reference for a lot of physics.

So I can’t recommend this for normal readers, but if you’re the sort of person who sees beauty in a quote like:

Putting r here equal to the Earth’s radius R, we find a relation between the densities of the atmosphere at the Earth’s surface (nE) and at infinity (n):

$n_{\infty} = n_E e^{-\frac{GMm}{kTR}}$

then by all means read on.

• #### elkement 11:56 am on Sunday, 3 February, 2013 Permalink | Reply

Thanks a lot for this pointer – downloaded immediately! I do indulge in recapturing physics basics incl. all fields I am not concerned with on a daily basis. This book seems to be concise and really comprising many different sub-fields in physics. I am also a Landau-Lifshitz fan in general and theoretically I “ought to” own their land mark books on theoretical physics – but buying these would be quite an investment.

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• #### Joseph Nebus 2:06 am on Monday, 4 February, 2013 Permalink | Reply

I’ve certainly downloaded my own copy. I remember calling on this book a couple of times in my thesis work, although I’m sorry to say I didn’t understand it as well as I really should have. I like to think I know it better now, though.

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## Six Minutes Off

Let me return, reindeer-like, to my problem, pretty well divorced from the movie at this point, of the stranded Arthur Christmas and Grand-Santa, stuck to wherever they happen to be on the surface of the Earth, going around the Earth’s axis of rotation every 86,164 seconds, while their reindeer and sleigh carry on orbiting the planet’s center once every $\sqrt{2}$ hours. That’s just a touch more than every 5,091 seconds. This means, sadly, that the reindeer will never be right above Arthur again, or else the whole system of rational and irrational numbers is a shambles. Still, they might come close.

After all, one day after being stranded, Arthur and Grand-Santa will be right back to the position where they started, and the reindeer will be just finishing up their seventeenth loop around the Earth. To be more nearly exact, after 86,164 seconds the reindeer will have finished just about 16.924 laps around the planet. If Arthur and Grand-Santa just hold out for another six and a half minutes (very nearly), the reindeer will be back to their line of latitude, and they’ll just be … well, how far away from that spot depends on just where they are. Since this is my problem, I’m going to drop them just a touch north of 30 degrees north latitude, because that means they’ll be travelling a neat 400 meters per second due to the Earth’s rotation and I certainly need some nice numbers here. Any nice number. I’m putting up with a day of 86,164 seconds, for crying out loud.

## How Fast Is The Earth Spinning?

To get to my next point about Arthur Christmas I needed to know how fast an arbitrary point on the Earth is moving, as the Earth rotates. This required me getting out a sheet of paper and doing some sketches, so, I figured it’s worth a side article to explain what I was doing.

The first thing was that I simplified stuff. In particular, I decided the Earth is near enough a sphere that I’m not bothering with the fact that it isn’t. The difference between an actual sphere and the geoid is not worth bothering with unless you’re timing the retrofire for a ballistically-reentering space capsule. That’s … actually fairly close to the problem I want, about how long it might take the reindeer and sleigh to get back to Arthur Christmas and Grand-Santa, but that’s also too much work for the improvement in the answer I’d get.

## Returning to Arthur Christmas

As promised, since I’ve got the chance, I want to return to the question of the reindeer behavior as shown in the Aardman movie Arthur Christmas, and what would ultimately happen to them if the reindeer carry on as Grand-Santa claims they will. (Again, this does require spoiling a plot point of the film and so I tuck the rest behind a cut.)

## What Is The Most Common Jeopardy! Response?

Happy New Year!

I want to bring a pretty remarkable project to people’s attention. Dan Slimmon here has taken the archive of Jeopardy! responses (you know, the answers, only the ones given in the form of a question) from the whole Jeopardy! fan archive, http://www.j-archive.com, and analyzed them. He was interested not just in the most common response — which turns out to be “What is Australia?” — but in the expectation value of the responses.

Expectation value I’ve talked about before, and for that matter, everyone mentioning probability or statistics has. Slimmon works out approximately what the expectation value would be for each clue. That is, imagine this: if you ignored the answer on the board entirely and just guessed to every answer either responded absolutely nothing or else responded “What is Australia?”, some of the time you’d be right, and you’d get whatever that clue was worth. How much would you expect to get if you just guessed that answer? Responses that turn up often, such as “Australia”, or that turn up more often in higher-value squares, are worth more. Responses that turn up rarely, or only in low-value squares, have a lower expectation value.

Simmons goes on to list, based on his data, what the 1000 most frequent Jeopardy! responses are, and what the 1000 responses with the highest expectation value are. I’m so delighted to discover this work that I want to bring folks’ attention to it. (I do have a reservation about his calculations, but I need some time to convince myself that I understand exactly his calculation, and my reservation, before I bother anyone with it.)

The comments at his page include a discussion of a technical point about the expectation value calculation which has an interesting point about the approximations often useful, or inevitable, in this kind of work, but that’ll take a separate essay to quite explain that I haven’t the time for just today.

[ Edit: I initially misunderstood Slimmon’s method and have amended the article to reflect the calculation’s details. Specifically I misunderstood him at first to have calculated the expectation value of giving a particular response, and either having it be right or wrong. Slimmon assumed that one would either give the response or not at all; getting the answer wrong costs the contestant money and so has a negative value, while not answering has no value. ]

## Could “Arthur Christmas” Happen In Real Life?

If you haven’t seen the Aardman Animation movie Arthur Christmas, first, shame on you as it’s quite fun. But also you may wish to think carefully before reading this entry, and a few I project to follow, as it takes one plot point from the film which I think has some interesting mathematical implications, reaching ultimately to the fate of the universe, if I can get a good running start. But I can’t address the question without spoiling a suspense hook, so please do consider that. And watch the film; it’s a grand one about the Santa family.

The premise — without spoiling more than the commercials did — starts with Arthur, son of the current Santa, and Grand-Santa, father of the current fellow, and a linguistic construct which perfectly fills a niche I hadn’t realized was previously vacant, going off on their own to deliver a gift accidentally not delivered to one kid. To do this they take the old sleigh, as pulled by the reindeer, and they’re off over the waters when something happens and there I cut for spoilers.

• #### fluffy 11:36 pm on Monday, 24 December, 2012 Permalink | Reply

For example they could try maintaining a precise angle to the Northwest and end up circling around the North pole in ever-tighter circles until they eventually converge at the North pole and simply spin in place (or explode due to division by zero).

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• #### Joseph Nebus 5:07 am on Thursday, 27 December, 2012 Permalink | Reply

Yes, that’s the loxodrome shape. Unless they’re heading exactly in one of the cardinal directions, they would end up in an infinitely long spiral, or at least until they get near enough a pole that some other navigational scheme breaks things up.

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• #### Rocket the Pony (@Blue_Pony) 3:39 pm on Tuesday, 25 December, 2012 Permalink | Reply

I suppose ultimately it depends on how reindeer navigate. If they use uncorrected magnetic navigation, that might happen. If they use inertial navigation, they’re more likely to behave like an orbiting satellite, albeit probably with less westward deviation as seen from the ground, since the air is going to help carry them along with the motion of the earth to a greater or lesser degree. If that’s the case, then reindeer should pass withim sight, if not directly overhead, eventually.

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• #### Joseph Nebus 5:08 am on Thursday, 27 December, 2012 Permalink | Reply

Yeah, that’s one of the points to be refined here: depending on different interpretations of what it means to keep going in the same direction, different outcomes are possible. I mean to get to the “orbiting satellite” alternative next.

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## Keep The Change

I’m sorry to have fallen quiet for so long; the week has been a busy one and I haven’t been able to write as much as I want. I did want to point everyone to Geoffrey Brent’s elegant solution of my puzzle about loose change, and whether one could have different types of coin without changing the total number of value of those coins. It’s a wonderful proof and one I can’t see a way to improve on, including an argument for the smallest number of coins that allow this ambiguity. I want to give it some attention.

The proof that there is some ambiguous change amount is a neat sort known as an existence proof, which you likely made it through mathematics class without seeing. In an existence proof one doesn’t particularly care whether one finds a solution to the problem, but instead bothers trying to show whether a solution exists. In mathematics classes for people who aren’t becoming majors, the existence of a solution is nearly guaranteed, except when a problem is poorly proofread (I recall accidentally forcing an introduction-to-multivariable-calculus class to step into elliptic integrals, one of the most viciously difficult fields you can step into without requiring grad school backgrounds), or when the instructor wants to see whether people are just plugging numbers into formulas without understanding them. (I mean the formulas, although the numbers can be a bit iffy too.) (Spoiler alert: they have no idea what the formulas are for, but using them seems to make the instructor happy.)

• #### Geoffrey Brent (@GeoffreyBrent) 11:07 pm on Saturday, 22 December, 2012 Permalink | Reply

Well, I managed to improve on it by fixing another error of mine ;-) Fortunately not a fatal one!

I’m glad you liked it. It’s been so long since I’ve done an existence proof (back in uni, one of my calculus lecturers gave me a funny look when I went the other way and gave a constructive proof to an existence question).

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• #### Chiaroscuro 6:36 am on Sunday, 23 December, 2012 Permalink | Reply

I also quite appreciated it; I figured you’d take my setup and to good things with it. (As a horribly lapsed math-fan, I’m resigned to such occasional nudges.) :)

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## Reading the Comics, September 11, 2012

Since the last installment of these mathematics-themed comic strips there’s been a steady drizzle of syndicated comics touching on something mathematical. This probably reflects the back-to-school interests that are naturally going to interest the people drawing either Precocious Children strips or Three Generations And A Dog strips.

• #### Tim Erickson 6:18 am on Saturday, 15 September, 2012 Permalink | Reply

Nice review! Alas, too many mainstream comics usually mention math as, “its’ horrible, it’s hard, kids hate it, parents couldn’t do it.”

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• #### Joseph Nebus 4:58 am on Sunday, 16 September, 2012 Permalink | Reply

Thank you. Unfortunately, yeah, “we can’t understand it” is the obvious and easiest joke to make about mathematics, and not enough people try to dig for a deeper gag.

I am always genuinely delighted to find a comic strip making a joke about the New Math, possibly because I grew up learning math more or less the New way and felt no terror at it, possibly because I’m always delighted by jokes that start out 30 years past their expiration date.

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## Reading the Comics, August 27, 2012

I’m also surprised to find it’s been about a month since my last roundup of mathematics-themed comic strips, but that’s about how it worked out. There was a long stretch of not many syndicated comics touching on any subjects at all and then a rush as cartoonists noticed that summer vacation is on the verge of ending. (I understand in some United States school districts it already has ended, but I grew up in a state where school simply never started before Labor Day, so the idea of school in August feels fundamentally implausible.)

## Reading the Comics, July 14, 2012

I hope everyone’s been well. I was on honeymoon the last several weeks and I’ve finally got back to my home continent and new home so I’ll try to catch up on the mathematics-themed comics first and then plunge into new mathematics content. I’m splitting that up into at least two pieces since the comics assembled into a pretty big pile while I was out. And first, I want to offer the link to the July 2 Willy and Ethel, by Joe Martin, since even though I offered it last time I didn’t have a reasonably permanent URL for it.

• #### Chiaroscuro 1:15 am on Sunday, 22 July, 2012 Permalink | Reply

So what’s going to happen to those two boxes of leftover cobras?

..on less mongoosey take, even Martin Gardner wouldn’t give a life-or-death logic puzzle to a monkey. That’s kind of messed up.

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• #### Joseph Nebus 5:04 am on Sunday, 22 July, 2012 Permalink | Reply

Yeah, the life-or-death thing is a little weird, but it’s only a little bit out of bounds considering how the guy in Magic In A Minute treats his monkey pal. (Also, logic puzzles are not, properly speaking, magic tricks, at least not without much more setup.)

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• #### Donna 3:03 am on Sunday, 22 July, 2012 Permalink | Reply

Awwww! Congratulations!! Happy Happy Joy Joy!!! I love happy weddings, happy couples, happy lives. Enjoy!

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• #### Joseph Nebus 5:01 am on Sunday, 22 July, 2012 Permalink | Reply

Thank you! The wedding came through in quite good order, and the honeymoon was grand, despite a lot of raining.

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## Reading the Comics, July 1, 2012

This will be a hastily-written installment since I married just this weekend and have other things occupying me. But there’s still comics mentioning math subjects so let me summarize them for you. The first since my last collection of these, on the 13th of June, came on the 15th, with Dave Whamond’s Reality Check, which goes into one of the minor linguistic quirks that bothers me: the claim that one can’t give “110 percent,” since 100 percent is all there is. I don’t object to phrases like “110 percent”, though, since it seems to me the baseline, the 100 percent, must be to some standard reference performance. For example, the Space Shuttle Main Engines routinely operated at around 104 percent, not because they were exceeding their theoretical limits, but because the original design thrust was found to be not quite enough, and the engines were redesigned to deliver more thrust, and it would have been far too confusing to rewrite all the documentation so that the new design thrust was the new 100 percent. Instead 100 percent was the design capacity of an engine which never flew but which existed in paper form. So I’m forgiving of “110 percent” constructions, is the important thing to me.

• #### bug 3:41 am on Tuesday, 3 July, 2012 Permalink | Reply

Oh man, I should read this more !

While it would be simple enough to justify negative numbers through nuclear physics (i.e. every particle having an antiparticle), it’s also not that hard to consider them as deficits (“Tim lacks 3 apples”) rather than “anti-assets”. That way, they don’t actually represent anything physical, but instead a difference (ha) from one’s expectation of a physical state. This also makes a lot more sense considering their use in accounting.

Also, I’ve never heard that engineers dislike complex numbers. They’re practically essential…

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• #### Joseph Nebus 10:09 pm on Thursday, 5 July, 2012 Permalink | Reply

Treating negative numbers as positive numbers in the other direction was historically the intermediate step between just working with negative numbers. Accountants seem to have been there first, with geometers following close behind. Descartes’ original construction of the coordinate system divided the plane into the four quadrants we still have, with positive numbers in each of them, representing “right and up” in the first quadrant, “left and up” in the second, “left and down” in the third, and “right and down” in the fourth. But this ends up being a nuisance and making do with a negative sign rather than a separate tally gets to be easier fast.

I can’t speak about the truth of electrical engineers disliking complex numbers, but it is certainly a part of mathematics folklore that if any students are going to have trouble with complex numbers, or reject them altogether, it’s more likely to be the electrical engineers. I note also the lore of the Salem Hypothesis, about the apparent predilection of engineers, particularly electrical engineers, to nutty viewpoints. (Petr Beckmann is probably the poster child for this, as he spent considerable time telling everyone Relativity was a Fraud, and he was indeed an electrical engineer.)

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