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  • Joseph Nebus 3:00 pm on Monday, 30 November, 2015 Permalink | Reply
    Tags: , , university   

    A Timeline Of Mathematics Education 

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

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

  • Joseph Nebus 4:00 pm on Saturday, 28 November, 2015 Permalink | Reply
    Tags: artwork, AV Club, brute force, Fermi Problems, , , , rectangles, secrets   

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

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

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

    'The painting (of interlocking rectangles) is really a secret message left by an informant. It reveals the address of a house where stolen artwork is being stashed. The title, Riverside, is the street name, and the total amount of rectangles is the house number. Where will Slylock Fox find the stolen artwork?

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

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

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

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

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

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

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

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

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

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

    • BunKaryudo 12:20 pm on Sunday, 29 November, 2015 Permalink | Reply

      I’m feeling pretty pleased with myself after reading this post since I’d actually heard of the Fermi Paradox before. I know it basically just boils down to, “Many scientists estimate that intelligent life should be common in the universe, so where is everyone?” Nevertheless, I’m puffing my chest out and strutting around like a mathematical genius this week.


    • Garfield Hug 1:12 pm on Monday, 30 November, 2015 Permalink | Reply

      Congrats on achieving a new statistic on your blog! Happy 30,000 views! 👍👏


  • Joseph Nebus 3:00 pm on Wednesday, 25 November, 2015 Permalink | Reply
    Tags: balls, boundaries, , , , ,   

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

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

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


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

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

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

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

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

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

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

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

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

    • ivasallay 4:38 pm on Wednesday, 25 November, 2015 Permalink | Reply

      I didn’t know any of this before, but it was well written and easy enough to understand. Thanks.

      Liked by 1 person

      • Joseph Nebus 6:23 am on Saturday, 28 November, 2015 Permalink | Reply

        Thank you. I’m most glad to hear it. I’m surprised how many of this sequence I keep finding I should write.

        Liked by 1 person

  • Joseph Nebus 6:00 pm on Sunday, 22 November, 2015 Permalink | Reply
    Tags: holograms, , , ,   

    Reading the Comics, November 21, 2015: Communication Edition 

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

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

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

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

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

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

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

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

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

  • Joseph Nebus 5:00 pm on Thursday, 19 November, 2015 Permalink | Reply
    Tags: , , , , , wrestling   

    Reading the Comics, November 18, 2015: All Caught Up Edition 

    Yes, I feel a bit bad that I didn’t have anything posted yesterday. I’d had a nice every-other-day streak going for a couple weeks there. But I had honestly expected more mathematically themed comic strips, and there just weren’t enough in my box by the end of the 17th. So I didn’t have anything to schedule for a post the 18th. The 18th came through, though, and now I’ve got enough to talk about. And that before I get to reading today’s comics. So, please, enjoy.

    Scott Adams’s Dilbert Classics for the 16th of November (originally published the 21st of September, 1992) features Dilbert discovering Bell’s Theorem. Bell’s Theorem is an important piece of our understanding of quantum mechanics. It’s a theorem that excites people who first hear about it. It implies quantum mechanics can’t explain reality unless it can allow information to be transmitted between interacting particles faster than light. And quantum mechanics does explain reality. The thing is, and the thing that casual readers don’t understand, is that there’s no way to use this to send a signal. Imagine that I took two cards, one an ace and one an eight, seal them in envelopes, and gave them to astronauts. The astronauts each travel to ten light-years away from me in opposite directions. (They took extreme offense at something I said and didn’t like one another anyway.) Then one of them opens her envelope, finding that she’s got the eight. Then instantly, even though they’re twenty light-years apart, she knows the other astronaut has an ace in her envelope. But there is no way the astronauts can use this to send information to one another, which is what people want Bell’s Theorem to tell us. (My example is not legitimate quantum mechanics and do not try to use it to pass your thesis defense. It just shows why Bell’s Theorem does not give us a way to send information we care about faster than light.) The next day Dilbert’s Garbageman, the Smartest Man in the World, mentions Dilbert’s added something to Bell’s Theorem. It’s the same thing everybody figuring they can use quantum entanglement to communicate adds to the idea.

    Tom Thaves’ Frank and Ernest for the 16th of November riffs on the idea of a lottery as a “tax on people who are bad at math”. Longtime readers here know that I have mixed feelings about that, and not just because I’m wary of cliché. If the jackpot is high enough, you can reach the point where the expectation value of the prize is positive. That is, you would expect to make money if you played the game under the same conditions often enough. But that chance is still vanishingly small. Even playing a million times would not make it likely you would more earn money than you spent. I’m not dogmatic enough to say what your decision should be, at least if the prize is big enough. (And that’s not considering the value placed on the fun of playing. One may complain that it shouldn’t be any fun to buy a soon-to-be-worthless ticket. But many people do enjoy it and I can’t bring myself to say they’re all wrong about feeling enjoyment.)

    And it happens that on the 18th Brant Parker and Johnny Hart’s Wizard of Id Classics (originally run the 20th of November, 1965) did a lottery joke. That one is about a lottery one shouldn’t play, except that the King keeps track of who refuses to buy a ticket. I know when we’re in a genre.

    Peter Mann’s The Quixote Syndrome for the 16th of November explores something I had never known but that at least the web seems to think is true. Apparently in 1958 Samuel Beckett knew the 12-year-old André Roussimoff. People of my age cohort have any idea who that is when they hear Roussimoff became pro wrestling star André the Giant. And Beckett drove the kid to school. Mann — taking, I think, a break from his usual adaptations of classic literature — speculates on what they might have talked about. His guess: Beckett attempting to ease one of his fears through careful study and mathematical treatment. The problem is goofily funny. But the treatment is the sort of mathematics everyone understands needing and understands using.

    John Deering’s Strange Brew for the 17th of November tells a rounding up joke. Scott Hilburn’s The Argyle Sweater told it back in August. I suspect the joke is just in the air. Most jokes were formed between 1922 and 1978 anyway, and we’re just shuffling around the remains of that fruitful era.

    Tony Cochrane’s Agnes for the 18th of November tells a resisting-the-word-problem joke. I admit expecting better from Cochrane. But casting arithmetic problems into word problems is fraught with peril. It isn’t enough to avoid obsolete references. (If we accept trains as obsolete. I’m from the United States Northeast, where subways and even commuter trains are viable things.) The problem also has to ask something the problem-solver can imagine wanting to know. It may not matter whether the question asks how far apart two trains, two cars, or two airplanes are, if the student can’t see their distance as anything but trivia. We may need better practice in writing stories if we’re to write story problems.

    • ivasallay 7:09 pm on Thursday, 19 November, 2015 Permalink | Reply

      “One may complain that it shouldn’t be any fun to buy a soon-to-be-worthless ticket.” You may not want to tell people that, but I think it’s a very good point. My favorite, believe it or not, was the rounding up comic.


      • Joseph Nebus 4:18 am on Friday, 20 November, 2015 Permalink | Reply

        I certainly believe you about the rounding up comic. It’s one of those kinds of jokes that puts the punch line so close to the setup that you have to go back and notice where thing happened, and that’s reliably disorienting and fun.

        I understand the reasoning that a lottery ticket should be a completely irrational purchase and that one shouldn’t get pleasure from buying one. But I’m not sure I can draw a distinction between buying a ticket and spending one or two dollars on any other short-lived consumable item. We don’t regard it as inherently stupid that someone might, say, buy a pack of toy gun blasting caps and throw them on the ground to make a couple bangs. Making it a purchase of a chance of money somehow offends people who don’t share the thrill.

        Liked by 1 person

  • Joseph Nebus 3:00 pm on Monday, 16 November, 2015 Permalink | Reply
    Tags: , , , ,   

    The Set Tour, Part 8: Balls, Only Made Harder 

    I haven’t forgotten or given up on the Set Tour, don’t worry or celebrate. I just expected there to be more mathematically-themed comic strips the last couple days. Really, three days in a row without anything at ComicsKingdom or GoComics to talk about? That’s unsettling stuff. Ah well.


    We are also starting to get into often-used domains that are a bit stranger. We are going to start seeing domains that strain the imagination more. But this isn’t strange quite yet. We’re looking at the surface of a sphere.

    The surface of a sphere we call S2. The “S” suggests a sphere. The “2” means that we have a two-dimensional surface, which matches what we see with the surface of the Earth, or a beach ball, or a soap bubble. All these are sphere enough for our needs. If we want to say where we are on the surface of the Earth, it’s most convenient to do this with two numbers. These are a latitude and a longitude. The latitude is the angle made between the point we’re interested in and the equator. The longitude is the angle made between the point we’re interested in and a reference prime longitude.

    There are some variations. We can replace the latitude, for example, with the colatitude. That’s the angle between our point and the north pole. Or we might replace the latitude with the cosine of the colatitude. That has some nice analytic properties that you have to be well into grad school to care about. It doesn’t matter. The details may vary but it’s all the same. We put in a number for the east-west distance and another for the north-south distance.

    It may seem pompous to use the same system to say where a point is on the surface of a beach ball. But can you think of a better one? Pointing to the ball and saying “there”, I suppose. But that requires we go around with the beach ball pointing out spots. Giving two numbers saves us having to go around pointing.

    (Some weenie may wish to point out that if we were clever we could describe a point exactly using only a single number. This is true. Nobody does that unless they’re weenies trying to make a point. This essay is long enough without describing what mathematicians really mean by “dimension”. “How many numbers normal people use to identify a point in it” is good enough.)

    S2 is a common domain. If we talk about something that varies with your position on the surface of the earth, we’re probably using S2 as the domain. If we talk about the temperature as it varies with position, or the height above sea level, or the population density, we have functions with a domain of S2 and a range in R. If we talk about the wind speed and direction we have a function with domain of S2 and a range in R3, because the wind might be moving in any direction.

    Of course, I wrote down Sn rather than just S2. As with Rn and with Rm x n, there is really a family of similar domains. They are common enough to share a basic symbol, and the superscript is enough to differentiate them.

    What we mean by Sn is “the collection of points in Rn+1 that are all the same distance from the origin”. Let me unpack that a little. The “origin” is some point in space that we pick to measure stuff from. On the number line we just call that “zero”. On your normal two-dimensional plot that’s where the x- and y-axes intersect. On your normal three-dimensional plot that’s where the x- and y- and z-axes intersect.

    And by “the same distance” we mean some set, fixed distance. Usually we call that the radius. If we don’t specify some distance then we mean “1”. In fact, this is so regularly the radius I’m not sure how we would specify a different one. Maybe we would write Snr for a radius of “r”. Anyway, Sn, the surface of the sphere with radius 1, is commonly called the “unit sphere”. “Unit” gets used a fair bit for shapes. You’ll see references to a “unit cube” or “unit disc” or so on. A unit cube has sides length 1. A unit disc has radius 1. If you see “unit” in a mathematical setting it usually means “this thing measures out at 1”. (The other thing it may mean is “a unit of measure, but we’re not saying which one”. For example, “a unit of distance” doesn’t commit us to saying whether the distance is one inch, one meter, one million light-years, or one angstrom. We use that when we don’t care how big the unit is, and only wonder how many of them we have.)

    S1 is an exotic name for a familiar thing. It’s all the points in two-dimensional space that are a distance 1 from the origin. Real people call this a “circle”. So do mathematicians unless they’re comparing it to other spheres or hyperspheres.

    This is a one-dimensional figure. We can identify a single point on it easily with just one number, the angle made with respect to some reference direction. The reference direction is almost always that of the positive x-axis. That’s the line that starts at the center of the circle and points off to the right.

    S3 is the first hypersphere we encounter. It’s a surface that’s three-dimensional, and it takes a four-dimensional space to see it. You might be able to picture this in your head. When I try I imagine something that looks like the regular old surface of the sphere, only it has fancier shading and maybe some extra lines to suggest depth. That’s all right. We can describe the thing even if we can’t imagine it perfectly. S4, well, that’s something taking five dimensions of space to fit in. I don’t blame you if you don’t bother trying to imagine what that looks like exactly.

    The need for S4 itself tends to be rare. If we want to prove something about a function on a hypersphere we usually make do with Sn. This doesn’t tell us how many dimensions we’re working with. But we can imagine that as a regular old sphere only with a most fancy job of drawing lines on it.

    If we want to talk about Sn aloud, or if we just want some variation in our prose, we might call it an n-sphere instead. So the 2-sphere is the surface of the regular old sphere that’s good enough for everybody but mathematicians. The 1-sphere is the circle. The 3-sphere and so on are harder to imagine. Wikipedia asserts that 3-spheres and higher-dimension hyperspheres are sometimes called “glomes”. I have not heard this word before, and I would expect it to start a fight if I tried to play it in Scrabble. However, I do not do mathematics that often requires discussion of hyperspheres. I leave this space open to people who do and who can say whether “glome” is a thing.

    Something that all these Sn sets have in common are that they are the surfaces of spheres. They are just the boundary, and omit the interior. If we want a function that’s defined on the interior of the Earth we need to find a different domain.

    • BunKaryudo 4:39 pm on Monday, 16 November, 2015 Permalink | Reply

      Don’t let those weenies try explaining how to identify spots on the bubble with just one number. I was proud that my non-mathematical brain managed to more or less follow how to do it with two numbers. I don’t want a bunch of weenies spoiling it all and throwing me into confusion again.


      • Joseph Nebus 4:16 am on Tuesday, 17 November, 2015 Permalink | Reply

        Two numbers is easy. If you’ve done latitude and longitude you’ve gotten the idea. Everything else is an implementation detail.

        If you feel like a bit of a puzzle, you can work out how to go from the latitude and longitude to a single number that does represent a point on the surface of the sphere. Or vice-versa, from one big number to a latitude and longitude. (There are a lot of ways to do this. But there’s at least one really easy way.)

        Liked by 1 person

        • BunKaryudo 4:42 am on Tuesday, 17 November, 2015 Permalink | Reply

          Yes, I like the two numbers way. I can understand that one. The one number version sounds harder. If there’s a really easy way to do it, that’s the one I’d use. I’m not confident I could follow anything more complicated.


          • Joseph Nebus 4:05 am on Friday, 20 November, 2015 Permalink | Reply

            It does seem like two numbers is the natural way to represent points. Of course, that natural-ness reflects a cultural heritage. We’ve gotten very comfortable representing stuff with pairs of numbers, thanks to things like latitude-and-longitude, or cities with rectangular-grid layouts such as midtown Manhattan.

            Liked by 1 person

            • BunKaryudo 11:47 am on Friday, 20 November, 2015 Permalink | Reply

              That’s interesting. I never thought of it being part of our cultural heritage before. Of course, the thing about ideas that come from our common cultural heritage is that when you’re actually part of the culture they can be rather difficult to spot. They just seem like common sense.


              • Joseph Nebus 6:57 am on Saturday, 21 November, 2015 Permalink | Reply

                They do, yes. They seem like common sense, or even more insidiously they don’t even seem to be ideas at all. I feel that different most starkly when I look at things like those South American nations that would use webs of tied strings to represent numbers. Even seeing how they’re supposed to be read, I feel wholly lost. And that’s just numerals, almost the first thing you can do with mathematics.


  • Joseph Nebus 3:00 pm on Saturday, 14 November, 2015 Permalink | Reply
    Tags: antifreeze,   

    How Antifreeze Works 

    I hate to report this but Peter Mander’s CarnotCycle blog has reached its last post for the calendar year. It’s a nice, practical one, though, explaining how antifreeze works. What’s important about antifreeze to us is that we can add something to water so that its freezing point is at some more convenient temperature. The logic of why it works is there in statistical mechanics, and the mathematics of it can be pretty simple. One of the things which awed me in high school chemistry class was learning how to use the formulas to describe how much different solutions would shift the freezing temperature around. It seemed all so very simple, and so practical too.

  • Joseph Nebus 3:00 pm on Thursday, 12 November, 2015 Permalink | Reply
    Tags: emojis, hauntings, , ,   

    Reading the Comics, November 10, 2015: Symbols And Meanings Edition 

    Eric the Circle for the 5th of November, by “andei”, is a mathematics-vocabulary pun. Ellipses are measured with a property called eccentricity. It measures, in a sense, how far any conic section is from being a circle. A circle has an eccentricity of zero. An ellipse, other than a circle, has an eccentricity between 0 and 1. The smaller the eccentricity the harder it is to tell the ellipse from a circle. The larger the eccentricity the longer one direction of the ellipse is compared to the other. For example, the Earth’s orbit around the sun, a very circular thing, has an eccentricity of about 0.0167 these days. Halley’s Comet, which gets closer to the Sun than Venus does, and farther from the sun than Neptune does, has an eccentricity of about 0.967. An eccentricity of exactly 1 means the shape is a parabola. An eccentricity of greater than 1 means the shape is a hyperbola.

    Mark Pett’s Mr Lowe for the 5th of November (originally the 2nd of November, 2000) gives a lousy reason to learn long division. I admit I’m not sure I can give a good reason anyone needs to know long division now that calculators are a well-proven technology. Perhaps the best reason is that long division works like much of computational mathematics does. You make a best guess for an answer, and test it, and improve it as necessary. Needing to improve an answer does not mean one started out wrong. It just means that we can approximate and modify solutions.

    Russell Myers’s Broom Hilda for the 6th of November is almost this entry’s anthropomorphic numerals joke. I’m not sure just how to categorize it. Perhaps “literal” is the best to be done.

    Mark Anderson’s Andertoons for the 8th of November is a joke about turning a wrong answer into a “teach the controversy!” special plea. There are mathematical controversies. But I think the only ones thriving are in fields too abstract for the average person to know or care about. But we can look to controversies of the past. An example an elementary school kid might understand is “should 1 be considered a prime number?” It’s generally not regarded as a prime number. If it were, it would add special cases or extra words to many theorems about prime numbers. That would add boring parts to a lot of work. If we move the number 1 off to its own category (a “unit”), then we can talk about prime numbers and composite numbers more easily. Is that good enough reason? If it isn’t, then what would be a good enough reason?

    Bill Amend’s FoxTrot for the 8th of November (a new strip, not a rerun) is a subverted word problem joke. It does contain a mention of curves (of happiness) going to infinity, and how they might do that. There’s some interesting linguistics at work here. A plot of a function — call it f(x), for convenience — is a graph that shows sets of values where the equation y = f(x) is true. We talk about functions “going to infinity”, although properly speaking they don’t “go” anywhere at all, any more than a photograph in a paper book moves.

    But it’s hard to resist the image we get from imagining drawing the curve. The eye follows the pen that sweeps, usually left to right, fluttering up and down. And near some points the pen goes soaring off the top (or bottom) of the page. If we imagine zooming out, again and again, the pen still soars off the edge of the page. So we call that “going to infinity”. What we mean is there are some values in the domain which the function matches to numbers in the range that are greater than any finite number. (Or less than any finite but negative number, if we’re going off to negative infinity.)

    We can even talk about how cuves “go to” infinity. If the function y = f(x) becomes infinitely large at some point, what does the function f(x)/x do? If that function stays finite we can say f(x) grows to infinity in the same way than x does. If f(x)/x grows infinitely large we can say that f(x) grows to infinity faster than x does. If f(x)/ex stays finite, we can say that f(x) grows to infinity in the same way that the exponential function ex does.

    Rates of growth may seem like a dull thing to worry about. They become more obviously relevant if we’re interested in functions that measure, for example, how much of a resource is required to do something. Suppose we have different ways to find the best choice out of a set of things. How long finding that takes depends on how many things there are to look through. If we are looking at scalability — how well we’ll be able to find the best choice out of a much larger set of things — then the rate of growth of these functions can be quite important. If doubling the set of things to look through means searching takes ten thousand times longer, we know we’re probably searching wrong, and should find a better way to do it. If doubling the set of things to look through means we have to take one-and-a-half times as long to find what we want, we’re probably using a good approach.

    Greg Evans and Karen Evans’s Luann for the 8th of November builds its joke on the idea that mathematical symbols are funny-looking things you have to interpret, just the same way emojis are. Gunther gives his best shot at explaining the various symbols. The grouping of them makes me wonder exactly what mathematics class he’s taking, though. I can’t think offhand of one that would have all of these in the same textbook.

    There’s also an actual mistake right up front. He identifies “(f, g)” as the inner product. The “inner product” is a name we give to a collection of functions, all with different domains but all with the range of real numbers. It allows us to describe a “norm”, or size, of whatever kind of thing we have. It also allows us to describe something that works like an angle between two things, and from it, orthogonality. If we’re looking at vectors, then this inner product is also known as the dot product. The mistake, though, is that the inner product is normally written with angled braces, as <f, g> instead. Normal parentheses usually mean we are giving a set of coordinates or an n-tuple. They can also mean that we are taking a Cartesian product, which looks a lot like giving a set of coordinates or an n-tuple. Probably the writer or artist made an understandable mistake while transcribing notes.

    The talk of an inner product suggests more than anything else that the subject is linear algebra. The reference to “Dim(U)” is consistent with this. If U is a matrix, we can talk about its dimension. This is a measure of how many of the rows of the matrix U cannot be made as the sum of scalar products of other rows. That’s useful because it tells us how many of the rows are “linearly independent”, or in a way, tell us something that we can’t get from other rows. So this is linear algebra work.

    φ is indeed the Golden Ratio, the number approximately 1.618. It’s a famous number but it’s really got no mathematical significance. Its reciprocal, 1/φ, is about 0.618, and that’s pretty, but that’s all. Many have tried to imbue the Golden Ratio with biological or aesthetic significance, and have failed, because it has none. In mathematics, the Golden Ratio is one of those celebrities who’s famous for no discernable reason or accomplishment.

    Δ is the Delta symbol, yes. It’s often used as a shorthand for “change in”. So “Δ x” means “the change in x”. We usually take this to mean a small but noticeable change. If we mean a much smaller change, or a perturbation from what we originally wanted, we might switch to a lowercase “δ x”. If we mean an incredibly tiny change we go to “dx”. This is important in calculus and analysis, as well as in many numerical methods classes.

    ∝ does mean proportional to. We use it to say one quantity varies as the other one does. For example, that the distance you go in an hour is proportional to how fast you go. Go twice as fast, you go twice as far. This turns up in analysis some, and in applied mathematics that tries to model real-world phenomena. We may be unsure of the precise relationship between two things, but we can say how we expect one thing to affect the other. ∝ is a symbol that lets us talk about qualitative relationships among things.

    The equals sign with a triangle above it baffled me, and I had to search about for it. It seems to baffle a modest number of people. Apparently it’s used as a way of saying “is defined as”. That is, the term on the left side of this symbol is by definition equal to whatever appears on the right side. I don’t remember seeing it before, and I don’t get what role it serves that the three-line equals sign ≡ doesn’t already do. I’m not saying the Evanses are wrong to use it, just that it’s not one I’m familiar with.

    But you see why I can’t figure what course Gunther is taking. Two of the symbols make sense for linear algebra. One fits in almost anywhere in calculus or applied mathematics. One is mostly an applied mathematics term. One is useless. The last is obscure, anyway. What do they have in common? And what could Tiffany’s message showing a heart-eyed smiley face, pizza, and two check marks mean? “I love to watch pizza voting”?

    Dave Kellett’s science fiction/humor comic Drive for the 9th of November reveals the probability of a catastrophe has been mis-reported. The choice of numbers is amusing. It’s hard to have an instinctive feel for the difference between a chance of 1-in-600 and a chance of 1-in-400. The difference makes itself known after a few hundred attempts, at least.

    Chris Giarrusso’s absurdist G-Man Webcomics for the 9th of November takes literally the problem of haunting, mysterious shapes.

    Gary Wise and Lance Aldrich’s Real Life Adventures asks how to find the area of a trapezoid. I couldn’t dare say.

    • ivasallay 7:55 pm on Thursday, 12 November, 2015 Permalink | Reply

      I didn’t know anything about eccentricity before I read this post. Thank you.


      • Joseph Nebus 12:21 am on Friday, 13 November, 2015 Permalink | Reply

        That surprises me. I thought eccentricity was one of the standard things taught about representing ellipses. It’s certainly one of the most easily testable things about them.

        Of course I don’t remember exactly when I learned about eccentricity, and my learning was surely contaminated by its use in orbital dynamics. It’s the same meaning of the word — orbits are basically ellipses — but the eccentricity is a useful and easy-to-understand quantity when you’re talking about planets and satellites and spaceships.


  • Joseph Nebus 3:00 pm on Tuesday, 10 November, 2015 Permalink | Reply
    Tags: , Dear Abby, Johnny Carson, Ramsey Theory   

    Why Was Someone Upset With Ramsey Theory In 1979? 

    I mentioned a couple weeks back reading John Stillwell’s Roads To Infinity: The Mathematics of Truth and Proof and how it stirred my desire to do mathematical logic. Besides that it reminded me of a baffling thing I’d read sometime around 1980. My memory is too vague to pin down the year nearer than that, but it was surely sometime between 1979 and 1983.

    It draws on some newspaper column, I think a letter to Dear Abby or Dear Ann Landers. The letter-writer was complaining about ivory-tower academicians such as (to paraphrase) “mathematicians who work on how many people can be at a dinner party without three knowing each other instead of on solving world hunger”. The complaint struck me as unfair as a kid. The skills that make good mathematicians don’t have to have anything to do with feeding people. And it struck me even back then there was probably enough food produced. It was just not getting to hungry people for reasons that were likely, at heart, evil. (Today, I hold basically the same view.) Still the letter struck me as weird because … … Well, even granting the argument that mathematicians could be working on world hunger instead, what is Dear Abby supposed to do about it? (That I don’t remember Dear Abby’s response suggests maybe it was some other feature, or perhaps the letters to the editor. Or that she had no good answer.)

    Roads To Infinity brings this old complaint back to mind because among its pages it discusses Ramsey Theory. This is a section of mathematics interested in combinatorics and graph theory. Its questions are like: how many ways can you arrange things that connect to one another with certain restrictions? And the dinner-party thing is the one piece of Ramsey Theory that any normal, non-mathematician might have heard of. This is because it’s a theorem that can be put into an immediately accessible, immediately understandable form. Even a seven-year-old can understand the question. The seven-year-old could even follow a demonstration of why the proof is true. The seven-year-old might even follow the proof, because it’s easier than you might guess.

    The problem alluded to by the Dear Abby(?) complainer, and discussed in Roads To Infinity, is: what is the smallest number of people you must invite to a party to be sure that either at least three of them all know one another, or that at least three of them do not know one another? This is the simplest interesting example of the “party problem”. It asks how many things you need to gather so as to be sure that either some number m share a property, or some number n of them do not. I’ll not spoil the fun for people who want to work out this particular case.

    What’s interesting about the result is that it suggests you can’t avoid structure. Get together enough things that can either have or not-have a relationship between pairs. Furthermore you will get relationships among bigger groups. We could interpret this as a reason there must be coincidences; logic compels them. The field speaks to us about how things must relate to one another.

    But here’s what has me baffled: why was the Dear Abby(?) letter-writer aware of Ramsey Theory? What was going on in United States pop culture of the late-70s or early-80s that this might have been on the complainer’s mind? Why not something at least as abstract and more accessible, like the Goldbach conjecture? (That’s the notion any even number greater than two can be written as the sum of two primes?) Did something tell people this dinner-party problem was something mathematicians had worked on? Did Johnny Carson make a monologue joke about it?

    The original problem, as best I can figure, was solved in 1930. Perhaps there was a surprising improvement in the proof that made it newsworthy at the time. I don’t know the history of mathematics in the 1970s in the right detail for that. Was it a recreational-mathematics challenge going around, the way a couple months ago everybody was worked up about that Singapore Birthday problem? Was there a good bio-pic of Frank Plumpton Ramsey that came out around that time?

    I don’t know what motivated the letter-writer to start. Nor do I know why the memory of that letter should have lasted in my mind. I am curious if someone can suggest why the subject ever entered the realm of people complaining in newspapers, though.

    • ivasallay 4:44 pm on Tuesday, 10 November, 2015 Permalink | Reply

      Maybe the letter writer and a mathematician were at a dinner party together. Maybe they knew each other or they didn’t know each other and the mathematician commented on it. Maybe the dinner party was ruined because of it.


      • Joseph Nebus 6:04 pm on Wednesday, 11 November, 2015 Permalink | Reply

        All quite possible. I wonder if they knew each other before the dinner party.


    • John Friedrich 1:01 am on Wednesday, 11 November, 2015 Permalink | Reply

      n = 4 is not it. If you have four people where only A knows B, B knows C, and C knows D, the requirements are met.

      I suspect n = 5 is the solution because I can’t come up with a counterexample, but I lack a proof.


    • John Friedrich 1:06 am on Wednesday, 11 November, 2015 Permalink | Reply

      Bah, I have failed to recognize a trivial case, n = 5 is not correct either.


      • Joseph Nebus 6:04 pm on Wednesday, 11 November, 2015 Permalink | Reply

        Yeah, 5 is just short of doing it. But you’re getting there. If you get to n = 260 you’ve gone too far.


    • Jason Dyer 2:17 pm on Wednesday, 11 November, 2015 Permalink | Reply

      If I recall the story correctly, there was a newspaper article about the Ramsey theory discovery. They were simply responding to that.

      The weird thing is, if any kind of mathematics is going to help solve world hunger, it is graph theory, which is enormously helpful in logistics.


      • Joseph Nebus 6:07 pm on Wednesday, 11 November, 2015 Permalink | Reply

        Ah, now, this is interesting and maybe I can find some leads in newspaper archives. Thank you.

        And yeah, it is a weird complaint. For all the beautiful abstractness of graph theory, it also has this immediate application to anything that networks. The dinner-party example is a good case of that. There’s whole fields of mathematics that can’t be made as immediately understandable as that.


    • Tony 3:14 pm on Wednesday, 25 November, 2015 Permalink | Reply

      Hey Joseph – I’ve got some (possibly?) interesting info about this for you, but it’s rather long, and the email address I’ve successfully used for you before is giving me server-lever “does not exist” errors. If/when you see this (and if you are interested!) shoot me an email from an address you prefer and I’ll send the info along. Happy Thanksgiving – Tony


      • Joseph Nebus 6:24 am on Saturday, 28 November, 2015 Permalink | Reply

        Oh, my, thank you. Shall send an e-mail shortly. Thanks and I hope yours was a happy Thanksgiving.


  • Joseph Nebus 4:00 pm on Sunday, 8 November, 2015 Permalink | Reply
    Tags: , , , horse racing,   

    Reading the Comics, November 4, 2015: Gambling Edition 

    I don’t presume to guess why. But Comic Strip Master Command sent out orders one lead-time ago to have everybody do jokes that relate to gambling. We see the consequences here.

    John Rose’s Barney Google and Snuffy Smith for the 2nd of November builds its joke on the idea that the mathematics of gambling is all anyone really needs. It’s a better-than-average crack about the usefulness of mathematics. It’s also truer than average. Much of how we make decisions is built on the expectation value, a core concept of probability. If we do this, what can we expect to gain or lose? If we do that instead, what would we expect? If we can place a value — even a loose, approximate value — on our time, our money, our experiences, we gain a new tool for making decisions.

    After showing that Jughaid can count to 21 (in cards) and the common currency demoninations, Snuffy concludes that Jughaid does know all the mathematics he could possibly *need*.

    John Rose’s Barney Google and Snuffy Smith for the 2nd of November, 2015. I am actually more comfortable with Snuffy’s sentiments than you might think. However, I think he might want Jughaid to also get comfortable with card-counting techniques, and the conditional probabilities that they imply. (Given that a 10, an 8, and a 9 are already showing, what is the probability that the dealer will go over?)
    Unanswered question: where did Snuffy Smith get a twenty-dollar bill from?

    Probability runs through the history of mathematics. That’s euphemistic. Gambling runs through the history of mathematics. Quite a bit of what we call probability derives from people who wanted to better understand games of chance, and to get an edge in the bets they might place. A question like “how many ways can three dice come up?” is a good homework problem today. It was once a subject of serious study and argument. We realize it’s still a good question when we wonder if the first die coming up 6, the second 3, and and third 1 is a different outcome from the first die coming up 3, the second 1, and the third 6.

    Fully understanding the mathematics of gambling requires not just counting and not just fractions. It will bring us to algebra, to calculus, and to all the tools that let us understand thermodynamics and quantum mechanics. If that isn’t everything, that is a good rough approximation.

    Scott Adams’s Dilbert Classic for the 2nd of November originally ran the 8th of September, 1992. It’s about a sadly common kind of nerd behavior, the desire to one-up one’s stories of programming hardship. In this one the generic guy — a different figure from Adams’s current model of generic guy — asserts he goes back to before binary numbers, even. I admit skepticism. Certainly you could list different numbers by making the same symbol often enough. We do that when we resort to tally marks. But we need some second symbol to note the end of a number. With tally marks we can do that with physical space. A computer’s memory, though? That needs something else.

    Kevin Fagan’s Drabble began a story about the logic of buying a lottery ticket this week. (The story goes on several days past this.) This is another probability, that is gambling, problem. Large jackpots present a pretty good philosophical challenge. It’s possible the jackpot will be so large that the expected value of buying a ticket is positive. This would seem to imply you should buy a ticket. But your chance of winning will be, as ever, vanishingly small. One chance in 200 million or more. You will not win. This would seem to imply you should not buy a ticket. Both are hard arguments to refute. I admit that when the jackpot gets sufficiently large, I’ll buy one or two tickets. I don’t expect to win the $200 million jackpot or anything like that, though. I’ll be content if I can secure a cozy little $25,000 minor prize. But I might just get a long john doughnut instead.

    Larry Wright’s Motley for the 2nd of November originally ran that day in 1987. It name-drops E = mc2 as shorthand for genius, the equation’s general role.

    Doug Bratton’s Pop Culture Shock Therapy for the 3rd of November doesn’t mention E = mc2, but it is an Albert Einstein joke. It doesn’t build on the comforting but dubious legend of Einstein being a poor student. That’s an unusual direction.

    Eric the Circle for the 3rd of November is by “Shane”. It’s a cute joke: if Eric were in a horserace, how would his lead be measured? Obviously, by comparison to his diameter. I doubt the race caller would need so many digits past the decimal, though. If cartoons and old-time radio sitcoms about horseracing haven’t led me wrong, distances are measured in a couple common fractions of a horse length — a half, a quarter, three-quarters and so on. So surely Eric would be called “about seven radii” or “three and a half diameters” ahead. It would make sense if his lead were measured by circumferences, if he’s rolling along. But it can be surprisingly hard to estimate by eye what the circumference of a circle is. Diameters are easier.

    Jonathan Lemon’s Rabbits Against Magic for the 4th of November has a M&oum;bius strip joke. Obviously, though, what’s taking so long is that Eightball’s spare tire isn’t even on the rim. This is bad.

    John Zakour and Scott Roberts’s Working Daze for the 4th of November is a variation on the joke about mathematicians being lousy at arithmetic. Here it’s an accountant who’s bad. I am reminded of the science fiction great Arthur C Clarke mentioning his time as an accounts auditor. He supposed that as long as figures added up approximately, to something like one percent, then there probably wasn’t anything requiring further scrutiny going on. He was able to finish his day’s work quickly, and went on to other jobs in time. Bob Newhart also claimed to not demand too much precision in the accounts he was overseeing. He then went on to sell comedy records to radio stations for a fair bit less than they cost to produce, so perhaps he was better off not working on the money side of things.

    • ivasallay 6:19 pm on Monday, 9 November, 2015 Permalink | Reply

      The Möbius strip tire was especially good!


      • Joseph Nebus 6:03 pm on Wednesday, 11 November, 2015 Permalink | Reply

        I was tickled by it too.

        I have heard of attempts to apply Möbius strips in actual production — making belts and such with a twist in them, so that any wear on the surface is spread out over double the area — but don’t know if that’s really done or is just something mathematicians tell each other is a useful application of the things.


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