# 14,000

Sometime on Thursday, the 6th of March, it appears I registered my 14,000th visitor to the math blog here. WordPress believes it to be someone from either the United States, France, Germany, Canada, or Australia, which at least covers a respectable number of possible time zones. The number’s a nice, big, round one, which I admit is about all I can think of that’s particularly interesting about it; even Wikipedia figures the most likely things you’re looking for if you look for 14,000 anything is either the ISO specification or an asteroid discovered in March of 1993 and apparently not even named yet. (It’s designated 1993 FZ55.) Well, at least asteroid 15,000 has a name.

Stare too hard at any one statistic, though, and you’ll start to wonder how reliable it is; I know for example that multiple of those 14,000 page views were me, testing neurotically to see whether the WordPress statistics counter was actually registering page views (particularly in the earliest days, when I was less self-confident and was using tags worse). Surely my just loading a page to see if it registers shouldn’t count as an actual page view, but, how can WordPress tell the difference?

Taking the WordPress statistics as if they meant what they purport to mean, though, indicates that apparently the most interesting thing I ever did was forget how to show why the area of a trapezoid was the trapezoid formula. My most-read article of all time is How Many Trapezoids I Can Draw, which is still standing at six by the way; some of the other articles which went into that (like Setting Out to Trap A Zoid and How Do You Make A Trapezoid Right) are also in the top five. All that’s based around working out how to work out a formula for the area of a trapezoid and convince yourself it’s right. For some reason the Reading the Comics post from September 11, 2012 also made the cut (I suppose the date is boosting it there). The early post How Many Numbers Have We Named is also fairly reliably popular.

Some of the most popular from the past year have included Something Neat About Triangles — since that post is only two months old I figure it’s got a good future ahead of it — and Solving The Price Is Right’s “Any Number” Game, which maybe solves the game less than explains why it’s usually a pretty good one to watch. Also popular is Counting From 52 to 11,108, and Inder J Taneja’s fascinating project in producing numbers using the digits one through nine in ascending or descending order.

# The Liquefaction of Gases – Part II

The CarnotCycle blog has a continuation of last month’s The Liquefaction of Gases, as you might expect, named The Liquefaction of Gases, Part II, and it’s another intriguing piece. The story here is about how the theory of cooling, and of phase changes — under what conditions gases will turn into liquids — was developed. There’s a fair bit of mathematics involved, although most of the important work is in in polynomials. If you remember in algebra (or in pre-algebra) drawing curves for functions that had x3 in them, and in finding how they sometimes had one and sometimes had three real roots, then you’re well on your way to understanding the work which earned Johannes van der Waals the 1910 Nobel Prize in Physics.

Originally posted on carnotcycle:

Future Nobel Prize winners both. Kamerlingh Onnes and Johannes van der Waals in 1908.

On Friday 10 July 1908, at Leiden in the Netherlands, Kamerlingh Onnes succeeded in liquefying the one remaining gas previously thought to be non-condensable – helium – using a sequential Joule-Thomson cooling technique to drive the temperature down to just 4 degrees above absolute zero. The event brought to a conclusion the race to liquefy the so-called permanent gases, following the revelation that all gases have a critical temperature below which they must be cooled before liquefaction is possible.

This crucial fact was established by Dr. Thomas Andrews, professor of chemistry at Queen’s College Belfast, in his groundbreaking study of the liquefaction of carbon dioxide, “On the Continuity of the Gaseous and Liquid States of Matter”, published in the Philosophical Transactions of the Royal Society of London in 1869.

As described in Part I of…

View original 1,997 more words

# A Wonder of Rationality

I’d like to talk about a neat little property of the rational numbers, which does involve there being infinitely many of them, and which isn’t about how there are just as many rational numbers as there are integers but there are more real numbers than there are rational numbers. (It’s true, but the point has already been well-covered by every mathematics blog ever.) Anyway, I’m laying the groundwork for something else.

Now, it’s common in mathematics to talk about the set of rational numbers, the numbers you get as one integer divided by another, as Q. The notation seems to trace back to the 1930s and the Bourbaki group which did so much to put mathematics on a basis of set theory, and the Q was chosen as it’s the start of “quotient”, which rational numbers after all are. (“R” was already called on to stand for the set of Real numbers.) I’m interested in two subsets of the rational numbers, the first of them, all the positive integers. For that I’ll write Q+. The other is just the rational numbers between zero and one. For that I’ll write Q(0, 1).

I can match every rational number between 0 and 1 to some rational number greater than zero. Here’s one way (there are many ways) to do it. Start out with some number, let me call it q, that’s in Q(0, 1). That’s a rational number between zero and one. Well, let me take its reciprocal: the result of one divided by q, which is going to be some rational number greater than 1. That’s a nice matching of the rational numbers between zero and one to the rational numbers greater than one, but I claimed I’d do this matching for rational numbers greater than zero. No matter; I can get there easily. Take that reciprocal and subtract one from it. This new number — let me call it p — is a rational number greater than zero, something in Q+. That is, each q (a rational between 0 and 1) can be matched with p (a positive rational), among other ways, by letting p equal (1/q) minus 1.

For example, let’s say, let q be 3/4. Then the reciprocal of that is 4/3, and subtracting one from that gets us a p of 1/3, which is certainly a positive number.

Or let’s say that q is 2/9. Then the reciprocal of q is 9/2, and subtracting one from that gets us a p of 7/2. (Some math teachers would want to change that 9/2 into 4 ½, and that 7/2 into 3 ½, but I don’t really know why they bother. I suppose the teachers are having fun and it’s quite easy to test, so, let them.)

If we start with a q of 3/32, then we go to its reciprocal, 32/3, and subtract one from that for a p of 29/3.

And I can run it the other way, too. Pick some rational number p, anything that’s positive. Add one to it, which will make it a rational number greater than 1. Take the reciprocal of this, and you have a rational number between 0 and 1. That is, p (a positive rational) can be matched with q (a rational between 0 and 1) by (again, among other ways) letting q equal 1/(p + 1).

For example, let’s let p be 3/5. Add one to that and we have 8/5, and the reciprocal of that is our q, 5/8, which is a rational number between zero and one.

Or let p be 14. Add one to that and we have 15, and the reciprocal of that is our q, 1/15, which is again between zero and one.

Or say that p is 39/7. Add one to that and we have 46/7, and the reciprocal of that is q, 7/46.

There are many ways to do this sort of matching. For example, you can match the rationals between 0 and 1 to the rationals between -1 and 1, or for that matter to all the rationals, positive and negative. It doesn’t have to be with a single rule, either; you’re allowed to set up a rule like “if q is less than one-half, find p by this rule; if q is greater than one-half, find p by that rule; if q is exactly one-half, do this other thing instead”. You can have a good bit of mental exercise by picking sets and trying to work out rules that match the numbers in one to the numbers in the other, and if I were smart I might try making a weekly puzzle section for that.

A reasonable person may point out that it’s absurd that you can match Q(0, 1) exactly to Q+. The rules I worked out give you one and only one p for each q, and vice-versa; but, the rationals between zero and one are all also positive rational numbers. That you can match the positive rational numbers to a subset of the positive rational numbers is counter-intuitive, at least when you first encounter it. It’s also the simplest definition for being “infinitely large” that I know of, though; if you can set up a one-to-one match of a set with a proper subset of itself, the set is considered to have an infinitely large cardinality, which is one of the ways mathematicians describe the sizes of things.

# Reading the Comics, March 1, 2014: Isn’t It One-Half X Squared Plus C? Edition

So the subject line references here a mathematics joke that I never have heard anybody ever tell, and only encounter in lists of mathematics jokes. It goes like this: a couple professors are arguing at lunch about whether normal people actually learn anything about calculus. One of them says he’s so sure normal people learn calculus that even their waiter would be able to answer a basic calc question, and they make a bet on that. He goes back and finds their waiter and says, when she comes with the check he’s going to ask her if she knows what the integral of x is, and she should just say, “why, it’s one-half x squared, of course”. She agrees. He goes back and asks her what the integral of x is, and she says of course it’s one-half x squared, and he wins the bet. As he’s paid off, she says, “But excuse me, professor, isn’t it one-half x squared plus C?”

Let me explain why this is an accurately structured joke construct and must therefore be classified as funny. “The integral of x”, as the question puts it, has not just one correct answer but rather a whole collection of correct answers, which are different from one another only by the addition of a constant whole number, by convention denoted C, and the inclusion of that “plus C” denotes that whole collection. The professor was being sloppy in referring to just a single example from that collection instead of the whole set, as the waiter knew to do. You’ll see why this is relevant to today’s collection of mathematics-themed comics.

Jef Mallet’s Frazz (February 22) points out one of the grand things about mathematics, that if you follow the proper steps in a mathematical problem you get to be right, and to be extraordinarily confident in that rightness. And that’s true, although, at least to me a good part of what’s fun in mathematics is working out what the proper steps are: figuring out what the important parts of something you want to study should be, and what follows from your representation of them, and — particularly if you’re trying to represent a complicated real-world phenomenon with a model — whether you’re representing the things you find interesting in the real-world phenomenon well. So, while following the proper steps gets you an answer that is correct within the limits of whatever it is you’re doing, you still get to work out whether you’re working on the right problem, which is the real fun.

Mark Pett’s Lucky Cow (February 23, rerun) uses that ambiguous place between mathematics and physics to represent extreme smartness. The equation the physicist brings to Neil is the (time-dependent) Schrödinger Equation, describing how probability evolves in time, and the answer is correct. If Neil’s coworkers at Lucky Cow were smarter they’d realize the scam, though: while the equation is impressively scary-looking to people not in the know, a particle physicist would have about as much chance of forgetting this as of forgetting the end of “E equals m c … ”.

Hilary Price’s Rhymes With Orange (February 24) builds on the familiar infinite-monkeys metaphor, but misses an important point. Price is right that yes, an infinite number of monkeys already did create the works of Shakespeare, as a result of evolving into a species that could have a Shakespeare. But the infinite monkeys problem is about selecting letters at random, uniformly: the letter following “th” is as likely to be “q” as it is to be “e”. An evolutionary system, however, encourages the more successful combinations in each generation, and discourages the less successful: after writing “th” Shakespeare would be far more likely to put “e” and never “q”, which makes calculating the probability rather less obvious. And Shakespeare was writing with awareness that the words mean things and they must be strings of words which make reasonable sense in context, which the monkeys on typewriters would not. Shakespeare could have followed the line “to be or not to be” with many things, but one of the possibilities would never be “carport licking hammer worbnoggle mrxl 2038 donkey donkey donkey donkey donkey donkey donkey”. The typewriter monkeys are not so selective.

Dan Thompson’s Brevity (February 26) is a cute joke about a number’s fashion sense.

Mark Pett’s Lucky Cow turns up again (February 28, rerun) for the Rubik’s Cube. The tolerably fun puzzle and astoundingly bad Saturday morning cartoon of the 80s can be used to introduce abstract algebra. When you rotate the nine little cubes on the edge of a Rubik’s cube, you’re doing something which is kind of like addition. Think of what you can do with the top row of cubes: you can leave it alone, unchanged; you can rotate it one quarter-turn clockwise; you can rotate it one quarter-turn counterclockwise; you can rotate it two quarter-turns clockwise; you can rotate it two quarter-turns counterclockwise (which will result in something suspiciously similar to the two quarter-turns clockwise); you can rotate it three quarter-turns clockwise; you can rotate it three quarter-turns counterclockwise.

If you rotate the top row one quarter-turn clockwise, and then another one quarter-turn clockwise, you’ve done something equivalent to two quarter-turns clockwise. If you rotate the top row two quarter-turns clockwise, and then one quarter-turn counterclockwise, you’ve done the same as if you’d just turned it one quarter-turn clockwise and walked away. You’re doing something that looks a lot like addition, without being exactly like it. Something odd happens when you get to four quarter-turns either clockwise or counterclockwise, particularly, but it all follows clear rules that become pretty familiar when you notice how much it’s like saying four hours after 10:00 will be 2:00.

Abstract algebra marks one of the things you have to learn as a mathematics major that really changes the way you start looking at mathematics, as it really stops being about trying to solve equations of any kind. You instead start looking at how structures are put together — rotations are seen a lot, probably because they’re familiar enough you still have some physical intuition, while still having significant new aspects — and, following this trail can get for example to the parts of particle physics where you predict some exotic new subatomic particle has to exist because there’s this structure that makes sense if it does.

Jenny Campbell’s Flo and Friends (March 1) is set off with the sort of abstract question that comes to mind when you aren’t thinking about mathematics: how many five-card combinations are there in a deck of (52) cards? Ruthie offers an answer, although — as the commenters get to disputing — whether she’s right depends on what exactly you mean by a “five-card combination”. Would you say that a hand of “2 of hearts, 3 of hearts, 4 of clubs, Jack of diamonds, Queen of diamonds” is a different one to “3 of hearts, Jack of diamonds, 4 of clubs, Queen of diamonds, 2 of hearts”? If you’re playing a game in which the order of the deal doesn’t matter, you probably wouldn’t; but, what if the order does matter? (I admit I don’t offhand know a card game where you’d get five cards and the order would be important, but I don’t know many card games.)

For that matter, if you accept those two hands as the same, would you accept “2 of clubs, 3 of clubs, 4 of diamonds, Jack of spades, Queen of spades” as a different hand? The suits are different, yes, but they’re not differently structured: you’re still three cards away from a flush, and two away from a straight. Granted there are some games in which one suit is worth more than another, in which case it matters whether you had two diamonds or two spades; but if you got the two-of-clubs hand just after getting the two-of-hearts hand you’d probably be struck by how weird it was you got the same hand twice in a row. You can’t give a correct answer to the question until you’ve thought about exactly what you mean when you say two hands of cards are different.

# February 2014′s Mathematics Blog Statistics

And so to the monthly data-tracking report. I’m sad to say that the total number of viewers dropped compared to January, although I have to admit given the way the month went — with only eight posts, one of them a statistics one — I can’t blame folks for not coming around. The number of individual viewers dropped from 498 to 423, and the number of unique visitors collapsed from 283 to 209. But as ever there’s a silver lining: the pages per viewer rose from 1.76 to 2.02, so, I like to think people are finding this more choice.

As usual the country sending me the most readers was the United States (235), with Canada in second (31) and Denmark, surprising to me, in third place (30). I suppose that’s a bit unreasonable on my part, since why shouldn’t Danes be interested in mathematics-themed comic strips, but, I’m used to the United Kingdom being there. Fourth place went to Austria (17) and I was again surprised by fifth place, Singapore (14), but happy to see someone from there reading, as I used to work there and miss the place, especially in the pits of winter. Sending me just a single reader each were: Albania, Argentina, Ecuador, Estonia, Ethiopia, Greece, Hungary, New Zealand, Peru, Saudia Arabia, South Korea, Thailand, United Arab Emirates, Uruguay, and Venezuela. Greece and South Korea are the only repeats from January 2013.

The most popular articles the past thirty days were:

1. Reading The Comics, February 1, 2014, my bread-and-butter subject for the blog.
2. How Many Trapezoids I Can Draw, which will be my immortal legacy.
3. Reading The Comics, February 11, 2014: Running Out Pi Edition, see above, although now I’m trying out something in putting particular titles on things.
4. The Liquefaction of Gases, Part I, referring to a real statistical mechanics post by CarnotCycle.
5. I Know Nothing Of John Venn’s Diagram Work, my confession of ignorance, or at least of casualness in thought, in the use of a valuable tool.

The most interesting search terms bringing people to me the past month were “comics strip about classical and modern physics”, “1,898,600,000,000,000,000,000,000,000 in words”, and “how much could a contestant win on the \$64.00 question”, which you’d superficially think would be a question you didn’t have to look up. (Of course, in the movie Take It Or Leave It, based on the radio quiz program, the amount of the gran jackpot is raised to a thousand dollars, for dramatic value. This is presumably not what the questioner was looking for.)

# Peer Gibberish

Well, this is an embarrassing thing to see: according to Nature, the Springer publishing and the Institute of Electrical and Electronic Engineers (IEEE) have had to withdraw at least 120 papers from their subscription services, because the papers were gibberish produced by a program, SCIgen, that strings together words and phrases into computer science-ish texts. SCIgen and this sort of thing are meant for fun (Nature also linked to arXiv vs snarXiv, which lets you try to figure out whether titles are actual preprints on the arXiv server or gibberish), but such nonsense papers have been accepted for conferences or published in, typically, poorly-reviewed forums, to general amusement and embarrassment when it’s noticed.

I’m also reminded of a bit of folklore from my grad school days, in a class on dynamical systems. That’s the study of physics-type problems, with the attention being not so much on actually saying what something will do from this starting point — for example, if you push this swing this hard, how long will it take to stop swinging — and more on what the different kinds of behavior are — can you make the swing just rock around a little bit, or loop around once and then rock to a stop, or loop around twice, or loop around four hundred times, or so on — and what it takes to change that behavior mode. The instructor referred us to a paper that was an important result but warned us to not bother trying to read it because nobody had ever understood it from the paper. Instead, it was understood — going back to the paper’s introduction — by people having the salient points explained by other people who’d had it taught to them in conversations, all the way back to the first understanders, who got it from the original authors, possibly in talking mathematics over while at the bar. I’m embarrassed to say I don’t remember which paper it was (it was a while ago and there are a lot of key results in the field), so I haven’t even been able to figure how to search for the paper or the lore around it.

# Reading the Comics, February 21, 2014: Circumferences and Monkeys Edition

And now to finish off the bundle of mathematic comics that I had run out of time for last time around. Once again the infinite monkeys situation comes into play; there’s also more talk about circumferences than average.

Brian and Ron Boychuk’s The Chuckle Brothers (February 13) does a little wordplay on how “circumference” sounds like it could kind of be a knightly name, which I remember seeing in a minor Bugs Bunny cartoon back in the day. “Circumference” the word derives from the Latin, “circum” meaning around and “fero” meaning “to carry”; and to my mind, the really interesting question is why do we have the words “perimeter” and “circumference” when it seems like either one would do? “Circumference” does have the connotation of referring to just the boundary of a circular or roughly circular form, but why should the perimeter of circular things be so exceptional as to usefully have its own distinct term? But English is just like that, I suppose.

Paul Trapp’s Thatababy (February 13) brings back the infinite-monkey metaphor. The infinite monkeys also appear in John Deering’s Strange Brew (February 20), which is probably just a coincidence based on how successfully tossing in lots of monkeys can produce giggling. Or maybe the last time Comic Strip Master Command issued its orders it sent out a directive, “more infinite monkey comics!”

Ruben Bolling’s Tom The Dancing Bug (February 14) delivers some satirical jabs about Biblical textual inerrancy by pointing out where the Bible makes mathematical errors. I tend to think nitpicking the Bible mostly a waste of good time on everyone’s part, although the handful of arithmetic errors are a fair wedge against the idea that the text can’t have any errors and requires no interpretation or even forgiveness, with the Ezra case the stronger one. The 1 Kings one is about the circumference and the diameter for a vessel being given, and those being incompatible, but it isn’t hard to come up with a rationalization that brings them plausibly in line (you have to suppose that the diameter goes from outer wall to outer wall, while the circumference is that of an inner wall, which may be a bit odd but isn’t actually ruled out by the text), which is why I think it’s the weaker.

Bill Whitehead’s Free Range (February 16) uses a blackboard full of mathematics as a generic “this is something really complicated” signifier. The symbols as written don’t make a lot of sense, although I admit it’s common enough while working out a difficult problem to work out weird bundles of partly-written expressions or abuses of notation (like on the middle left of the board, where a bracket around several equations is shown as being less than a bracket around fewer equations), just because ideas are exploding faster than they can be written out sensibly. Hopefully once the point is proven you’re able to go back and rebuild it all in a form which makes sense, either by going into standard notation or by discovering that you have soem new kind of notation that has to be used. It’s very exciting to come up with some new bit of notation, even if it’s only you and a couple people you work with who ever use it. Developing a good way of writing a concept might be the biggest thrill in mathematics, even better than proving something obscure or surprising.

Jonathan Lemon’s Rabbits Against Magic (February 18) uses a blackboard full of mathematics symbols again to give the impression of someone working on something really hard. The first two lines of equations on 8-Ball’s board are the time-dependent Schrödinger Equations, describing how the probability distribution for something evolves in time. The last line is Euler’s formula, the curious and fascinating relationship between pi, the base of the natural logarithm e, imaginary numbers, one, and zero.

Todd Clark’s Lola (February 20) uses the person-on-an-airplane setup for a word problem, in this case, about armrest squabbling. Interesting to me about this is that the commenters get into a squabble about how airplane speeds aren’t measured in miles per hour but rather in nautical miles, although nobody not involved in air traffic control really sees that. What amuses me about this is that what units you use to measure the speed of the plane don’t matter; the kind of work you’d do for a plane-travelling-at-speed problem is exactly the same whatever the units are. For that matter, none of the unique properties of the airplane, such as that it’s travelling through the air rather than on a highway or a train track, matter at all to the problem. The plane could be swapped out and replaced with any other method of travel without affecting the work — except that airplanes are more likely than trains (let’s say) to have an armrest shortage and so the mock question about armrest fights is one in which it matters that it’s on an airplane.

Bill Watterson’s Calvin and Hobbes (February 21) is one of the all-time classics, with Calvin wondering about just how fast his sledding is going, and being interested right up to the point that Hobbes identifies mathematics as the way to know. There’s a lot of mathematics to be seen in finding how fast they’re going downhill. Measuring the size of the hill and how long it takes to go downhill provides the average speed, certainly. Working out how far one drops, as opposed to how far one travels, is a trigonometry problem. Trying the run multiple times, and seeing how the speed varies, introduces statistics. Trying to answer questions like when are they travelling fastest — at a single instant, rather than over the whole run — introduce differential calculus. Integral calculus could be found from trying to tell what the exact distance travelled is. Working out what the shortest or the fastest possible trips introduce the calculus of variations, which leads in remarkably quick steps to optics, statistical mechanics, and even quantum mechanics. It’s pretty heady stuff, but I admit, yeah, it’s math.

# I Know Nothing Of John Venn’s Diagram Work

My Dearly Beloved, the professional philosopher, mentioned after reading the last comics review that one thing to protest in the Too Much Coffee Man strip — showing Venn diagram cartoons and Things That Are Funny as disjoint sets — was that the Venn diagram was drawn wrong. In philosophy, you see, they’re taught to draw a Venn diagram for two sets as two slightly overlapping circles, and then to black out any parts of the diagram which haven’t got any elements. If there are three sets, you draw three overlapping circles of equal size and again black out the parts that are empty.

I granted that this certainly better form, and indispensable if you don’t know anything about what sets, intersections, and unions have any elements in them, but that it was pretty much the default in mathematics to draw the loops that represent sets as not touching if you know the intersection of the sets is empty. That did get me to wondering what the proper way of doing things was, though, and I looked it up. And, indeed, according to MathWorld, I have been doing it wrong for a very long time. Per MathWorld (which is as good a general reference for this sort of thing as I can figure), to draw a Venn diagram reflecting data for N sets, the rules are:

1. Draw N simple, closed curves on the plane, so that the curves partition the plane into 2N connected regions.
2. Have each subset of the N different sets correspond to one and only one region formed by the intersection of the curves.

Partitioning the plane is pretty much exactly what you might imagine from the ordinary English meaning of the world: you divide the plane into parts that are in this group or that group or some other group, with every point in the plane in exactly one of these partitions (or on the border between them). And drawing circles which never touch mean that I (and Shannon Wheeler, and many people who draw Venn diagram cartoons) are not doing that first thing right: two circles that have no overlap the way the cartoon shows partition the plane into three pieces, not four.

I can make excuses for my sloppiness. For one, I learned about Venn diagrams in the far distant past and never went back to check I was using them right. For another, the thing I most often do with Venn diagrams is work out probability problems. One approach for figuring out the probability of something happen is to identify the set of all possible outcomes of an experiment — for a much-used example, all the possible numbers that can come up if you throw three fair dice simultaneously — and identify how many of those outcomes are in the set of whatever you’re interested in — say, rolling a nine total, or rolling a prime number, or for something complicated, “rolling a prime number or a nine”. When you’ve done this, if every possible outcome is equally likely, the probability of the outcome you’re interested in is the number of outcomes that satisfy what you’re looking for divided by the number of outcomes possible.

If you get to working that way, then, you might end up writing a list of all the possible outcomes and drawing a big bubble around the outcomes that give you nine, and around the outcomes that give you a prime number, and those aren’t going to touch for the reasons you’d expect. I’m not sure that this approach is properly considered a Venn diagram anymore, though, although I’d introduced it in statistics classes as such and seen it called that in the textbook. There might not be a better name for it, but it is doing violence to the Venn diagram concept and I’ll try to be more careful in future.

The Mathworld page, by the way, provides a couple examples of Venn diagrams for more than three propositions, down towards the bottom of the page. The last one that I can imagine being of any actual use is the starfish shape used to work out five propositions at once. That shows off 32 possible combinations of sets and I can barely imagine finding that useful as a way to visualize the relations between things. There are also representations based on seven sets, which have 128 different combinations, and for 11 propositions, a mind-boggling 2,048 possible combinations. By that point the diagram is no use for visualizing relationships of sets and is simply mathematics as artwork.

Something else I had no idea bout is that if you draw the three-circle Venn diagram, and set it so that the intersection of any two circles is at the center of the third, then the innermost intersection is a Reuleaux triangle, one of those oddball shapes that rolls as smoothly as a circle without actually being a circle. (MathWorld has an animated gif showing it rolling so.) This figure, it turns out, is also the base for something called the Henry Watt square drill bit. It can be used as a spinning drill bit to produce a (nearly) square hole, which is again pretty amazing as I make these things out, and which my father will be delighted to know I finally understand or have heard of.

In any case, the philosophy department did better teaching Venn diagrams properly than whatever math teacher I picked them up from did, or at least, my spouse retained the knowledge better than I did.

# Reading the Comics, February 11, 2014: Running Out Pi Edition

I’d figured I had enough mathematics comic strips for another of these entries, and discovered during the writing that I had much more to say about one than I had anticipated. So, although it’s no longer quite the 11th, or close to it, I’m going to exile the comics from after that date to the next of these entries.

Melissa DeJesus and Ed Power’s My Cage (February 6, rerun) makes another reference to the infinite-monkeys-with-typewriters scenario, which, since it takes place in a furry universe allows access to the punchline you might expect. I’ve written about that before, as the infinite monkeys problem sits at a wonderful intersection of important mathematics and captivating metaphors.

Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde (starting February 10) (and when am I going to make a macro for that credit and title?) has Cynthia given a slightly baffling homework lesson: to calculate the first ten digits of pi. The story continues through the 11th, the 12th, the 13th, finally resolving on the the 14th, in the way such stories must. I admit I’m not sure why exactly calculating the digits of π would be a suitable homework assignment; I can see working out division problems until the numbers start repeating, or doing a square root or something by hand until you’ve found enough digits.

π, though … well, there’s the question of why it’d be an assignment to start with, but also, what formula for generating π could be plausibly appropriate for an elementary school class. The one that seems obvious to me — π is equal to four times (1/1 minus 1/3 plus 1/5 minus 1/7 plus 1/9 minus 1/11 and so on and so on) — also takes way too long to work. If a little bit of coding is right, it takes something like 160 terms to get just the first two digits of π correct and that isn’t even stable. (The first 160 terms add to 3.135; the first 161 terms to 3.147.) Getting it to ten digits would take —

Well, I thought it might be as few was 10,000 terms, because it turns out the sum of the first ten thousand terms in that series is 3.1414926536, which looks dead-on until you notice that π is 3.1415926536. That’s a neat coincidence, though.

Anyway, obviously, that formula wouldn’t do, and we see on the strip of the 14th that Lucretia isn’t using that. There are a great many formulas that generate the value of π, any of which might be used for a project like this; some of them get the digits right quite rapidly, usually at a cost of being very complicated. The formula shown in the strip of the 14th, though, doesn’t seem to be right. Lucretia’s work uses the formula $\pi = \sqrt{12} \cdot \sum_{k = 0}^{\infty} \frac{(-3)^{-k}}{2k + 1}$, which takes only about 21 terms to get to the demanded ten digits of accuracy. I don’t want to guess how many pages of work it would take to get to 13,908 places.

If I don’t miss my guess the formula used here is one by Abraham Sharp, an astronomer and mathematician who worked for the Royal Observatory at Greenwich and set a record by calculating π to 72 decimal digits. He was also an instrument-maker, of rather some skill, and I found a page purporting to show his notes of how to cut some complicated polyhedrons out of a block of wood, so, if my father wants to carve a 120-sided figure, here’s his chance. Sharp seems to have started with Leibniz’s formula (yes, that Leibniz) — that the arctangent of a number x is equal to x minus one-third x cubed plus one-fifth x to the fifth power minus one-seventh x to the seventh power, et cetera — with the knowledge that the arctangent of the square root of one-third is equal to one-sixth π and produced this series that looks a lot like the one we started with, but which gets digits correct so very much more quickly.

Darrin Bell’s Candorville (February 13) is primarily a bit of guys insulting friends, but what do you know and π makes a cameo appearance here.

Shannon Wheeler’s Too Much Coffee Man (February 10) is a Venn Diagram cartoon in the service of arguing that Venn Diagram cartoons aren’t funny. Putting aside the smoke and sparks popping out of the Nomad space probe which Kirk and Spock are rushing to the transporter room, I don’t think it’s quite fair: the ease the Venn diagram gives to grouping together concepts and showing how they relate helps organize one’s understanding of concepts and can be a really efficient way to set up a joke. Granting that, perhaps Wheeler’s seen too many Venn Diagram cartoons that fail, a complaint I’m sympathetic to.

Bill Amend’s FoxTrot (February 11, rerun) was one of those strips trying to be taped to the math teacher’s door, with the pun-based programming for the Math Channel.

# The Liquefaction of Gases – Part I

I know, or at least I’m fairly confident, there’s a couple readers here who like deeper mathematical subjects. It’s fine to come up with simulated Price is Right games or figure out what grades one needs to pass the course, but those aren’t particularly challenging subjects.

But those are hard to write, so, while I stall, let me point you to CarnotCycle, which has a nice historical article about the problem of liquefaction of gases, a problem that’s not just steeped in thermodynamics but in engineering. If you’re a little familiar with thermodynamics you likely won’t be surprised to see names like William Thomson, James Joule, or Willard Gibbs turn up. I was surprised to see in the additional reading T O’Conor Sloane show up; science fiction fans might vaguely remember that name, as he was the editor of Amazing Stories for most of the 1930s, in between Hugo Gernsback and Raymond Palmer. It’s often a surprising world.

Originally posted on carnotcycle:

Photo credit: Scientific American

On Monday 3 December 1877, the French Academy of Sciences received a letter from Louis Cailletet, a 45 year-old physicist from Châtillon-sur-Seine. The letter stated that Cailletet had succeeded in liquefying both carbon monoxide and oxygen.

Liquefaction as such was nothing new to 19th century science, it should be said. The real news value of Cailletet’s announcement was that he had liquefied two gases previously considered ‘non condensable’.

While a number of gases such as chlorine, carbon dioxide, sulfur dioxide, hydrogen sulfide, ethylene and ammonia had been liquefied by the simultaneous application of pressure and cooling, the principal gases comprising air – nitrogen and oxygen – together with carbon monoxide, nitric oxide, hydrogen and helium, had stubbornly refused to liquefy, despite the use of pressures up to 3000 atmospheres. By the mid-1800s, the general opinion was that these gases could not be converted into liquids under…

View original 1,430 more words