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  • Joseph Nebus 6:00 pm on Friday, 3 February, 2017 Permalink | Reply
    Tags: , January, , ,   

    How January 2017 Treated My Mathematics Blog 


    My mathematics blog finally broke the psychologically important barrier of 1,000 page views in January! It’s an important barrier to me. WordPress’s statistics say I drew 1,031 page views in January, the largest number since July. In December 2016 I’d puttered around 956, and November 923. This came from only 586 distinct readers, about the same as in December (589) and November (575), but that just implies more archive-binging going on.

    What’s baffling and a tiny bit disheartening about this is it came on one of my least-written months. I posted only 11 articles in January, compared to 21 in December and November. This was my laziest month since September. I have no idea what’s the most economical balance between time spent writing and instances of being read. But given two of the top-five articles this past month I suspect I got identified as an authority on a couple of questions.

    Part of why I suspect that’s so: there were only 97 pages liked around here in January. That’s the lowest count I’ve seen in the past twelve months, and it’s down a fair bit from December’s 136 and November’s 157. Maybe I need a couple more posts per month to encourage reader engagement. Or maybe not. There were 33 comments in January, not that different to December’s 29 and November’s 35. I think that a lot of January’s comments were examinations of December’s readership review. That counts, of course, although it suggests people have more fun talking about blogging than they do about mathematics. I can’t fault them; there’s a natural limit to how much there is to say about a comic strip filling a blackboard full of mathematics symbols.

    According to Insights my most popular day for page views here was Thursday, which throws me. It’s usually Sundays, when there’s always a Reading the Comics post. But for January it was Thursdays, with 16 percent of page views. That’s not very much above 1/7th of the days, though, so I suspect there’s not much link between what day it is and whether anybody reads me. The most popular hour, with 10 percent of page views, was yet again 6:00 pm, which I’m assuming is 6 pm Universal Time. I set most posts to appear at 6:00 pm Universal Time.

    So here’s what was popular around here in January:

    Here’s always-liked list of countries and number of page views from each. And for another month running India’s a top-five country; I don’t know why that should satisfy me so. Singapore comes in pretty high too, but I can explain why that satisfies me. I used to work there.

    Country Views
    United States 598
    United Kingdom 94
    Hong Kong SAR China 33
    India 33
    Philippines 30
    Germany 24
    Singapore 22
    Canada 19
    Austria 16
    Slovenia 13
    Spain 12
    France 11
    Taiwan 10
    Australia 7
    Puerto Rico 7
    Japan 6
    Israel 5
    Russia 5
    Pakistan 4
    South Africa 4
    Bosnia & Herzegovina 3
    Egypt 3
    Greece 3
    New Zealand 3
    Norway 3
    Poland 3
    Portugal 3
    Sweden 3
    Ukraine 3
    Brazil 2
    Croatia 2
    Denmark 2
    Indonesia 2
    Ireland 2
    Nepal 2
    Nigeria 2
    Northern Mariana Islands 2
    Saudi Arabia 2
    Switzerland 2
    Thailand 2
    Bangladesh 1 (*)
    Belgium 1 (*)
    Estonia 1
    Finland 1
    Italy 1
    Kuwait 1 (*)
    Lithuania 1
    Malaysia 1
    Mexico 1
    Netherlands 1
    Paraguay 1
    South Korea 1
    Trinidad and Tobago 1

    Bangladesh, Belgium, and Kuwait were single-reader countries last month. No country’s on a three-month single-reader streak. There were 53 countries altogether sending me readers, up from December’s 50 and November’s 46. I make that out as 13 single-view countries, technically down from December and November’s 15. The mysterious “European Union” reader is gone again.

    Search terms were the usual meager set of things, including:

    • comics strip of production function
    • little iodine comics
    • 5 color map theory (way easier than the four-color map theorem, plus it let me rag on New England so I’m glad someone was looking)
    • how to do pinball league and how does pinball league work (get some players and some pinball machines, and have them play each other. It’s easy and fun! Try to get it in a bar somewhere, as that’s good for giving the place a pleasant casual air; but there’s interesting probability stuff going on in the topic)
    • origin is the gateway to your entire gaming universe (with and without a period on the end)
    • what engineering taught in school dy/dx what society expect him to do mason image (um … I don’t know?)
    • urban legend venn diagram (I know of no urban legends about Venn diagrams and would be delighted if someone shared one. I also don’t know any Venn diagrams showing elements in common among various urban legends, but that would probably be a neat way of organizing at least some tales and I’d be glad at least for seeing those).

    February starts with my blog having 45,135 page views from 19,475 admitted distinct viewers. (My first year or so WordPress didn’t record unique visitors in any way they’ve told us about.) I seem to start February with 646 WordPress.com followers and I don’t know how that compares to the start of January. I failed to keep track of that. I do wonder how many of those are active yet.

    If you’d like to follow my blog here please click the buttons on the upper-right corner of the page. You can have new posts e-mailed to you, or you can follow in the WordPress reader, which gives me the chance to fix my stupid typoes. And I’m on Twitter as @Nebus, with usually just a couple posts a day. I don’t understand those folks who have 86 things to tweet about every hour day and night either. Thank you, won’t you please?

     
  • Joseph Nebus 6:00 pm on Wednesday, 1 February, 2017 Permalink | Reply
    Tags: cables, , , podcasts   

    Mathematics Stuff To Read Or Listen To 


    I concede January was a month around here that could be characterized as “lazy”. Not that I particularly skimped on the Reading the Comics posts. But they’re relatively easy to do: the comics tell me what to write about, and I could do a couple paragraphs on most anything, apparently.

    While I get a couple things planned out for the coming month, though, here’s some reading for other people.

    The above links to a paper in the Proceedings of the National Academy of Sciences. It’s about something I’ve mentioned when talking about knot before. And it’s about something everyone with computer cables or, like the tweet suggests, holiday lights finds. The things coil up. Spontaneous knotting of an agitated string by Dorian M Raymer and Douglas E Smith examines when these knots are likely to form, and how likely they are. It’s not a paper for the lay audience, but there are a bunch of fine pictures. The paper doesn’t talk about Christmas lights, no matter what the tweet does, but the mathematics carries over to this.

    MathsByAGirl, meanwhile, had a post midmonth listing a couple of mathematics podcasts. I’m familiar with one of them, BBC Radio 4’s A Brief History of Mathematics, which was a set of ten-to-twenty-minute sketches of historically important mathematics figures. I’ll trust MathsByAGirl’s taste on other podcasts. I’d spent most of this month finishing off a couple of audio books (David Hackett Fischer’s Washington’s Crossing which I started listening to while I was in Trenton for a week, because that’s the sort of thing I think is funny, and Robert Louis Stevenson’s Doctor Jekyll and Mister Hyde And Other Stories) and so fell behind on podcasts. But now there’s some more stuff to listen forward to.

    And then I’ll wrap up with this from KeplerLounge. It looks to be the start of some essays about something outside the scope of my Why Stuff Can Orbit series. (Which I figure to resume soon.) We start off talking about orbits as if planets were “point masses”. Which is what the name suggests: a mass that fills up a single point, with no volume, no shape, no features. This makes the mathematics easier. The mathematics is just as easy if the planets are perfect spheres, whether hollow or solid. But real planets are not perfect spheres. They’re a tiny bit blobby. And they’re a little lumpy as well. We can ignore that if we’re doing rough estimates of how orbits work. But if we want to get them right we can’t ignore that anymore. And this essay describes some of how we go about dealing with that.

     
  • Joseph Nebus 6:00 pm on Sunday, 29 January, 2017 Permalink | Reply
    Tags: , , , , , , , Randy Glasbergen, ,   

    Reading the Comics, January 28, 2017: Chuckle Brothers Edition 


    The week started out quite busy and I was expecting I’d have to split my essay again. It didn’t turn out that way; Comic Strip Master Command called a big break on mathematically-themed comics from Tuesday on. And then nobody from Comics Kingdom or from Creators.com needed inclusion either. I just have a bunch of GoComics links and a heap of text here. I bet that changes by next week. Still no new Jumble strips.

    Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 22nd was their first anthropomorphic numerals joke of the week.

    Kevin Fagan’s Drabble for the 22nd uses arithmetic as the sort of problem it’s easy to get clearly right or clearly wrong. It’s a more economical use of space than (say) knowing how many moons Saturn’s known to have. (More than we thought there were as long ago as Thursday.) I do like that there’s a decent moral to this on the way to the punch line.

    Bill Amend’s FoxTrot for the 22nd has Jason stand up for “torus” as a better name for doughnuts. You know how nerdy people will like putting a complicated word onto an ordinary thing. But there are always complications. A torus ordinarily describes the shape made by rotating a circle around an axis that’s in the plane of the circle. The result is a surface, though, the shell of a doughnut and none of the interior. If we’re being fussy. I don’t know of a particular name for the torus with its interior and suspect that, if pressed, a mathematician would just say “torus” or maybe “doughnut”.

    We can talk about toruses in two dimensions; those look just like circles. The doughnut-shell shape is a torus in three dimensions. There’s torus shapes made by rotating spheres, or hyperspheres, in four or more dimensions. I’m not going to draw them. And we can also talk about toruses by the number of holes that go through them. If a normal torus is the shape of a ring-shaped pool toy, a double torus is the shape of a two-seater pool toy, a triple torus something I don’t imagine exists in the real world. A quadruple torus could look, I imagine, like some pool toys Roller Coaster Tycoon allows in its water parks. I’m saying nothing about whether they’re edible.

    Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 23rd was their second anthropomorphic numerals joke of the week. I suppose sometimes you just get an idea going.

    Mikael Wulff and Anders Morgenthaler’s TruthFacts for the 23rd jokes about mathematics skills versus life. The growth is fine enough; after all, most of us are at, or get to, our best at something while we’re training in it or making regular use of it. So the joke peters out into the usual “I never use mathematics in real life” crack, which, eh. I agree it’s what I feel like my mathematics skills have done ever since I got my degree, at any rate.

    Teresa Burritt’s Frog Applause for the 24th describes an extreme condition which hasn’t been a problem for me. I’m not an overindulgey type.

    Randy Glasbergen’s Glasbergen Cartoons rerun for the 26th is the pie chart joke for this week.

    Michael Fry’s Committed rerun for the 28th just riffs on the escalation of hyperbole, and what sure looks like an exponential growth of hyperbolic numbers. There’s a bit of scientific notation in the last panel. The “1 x” part isn’t necessary. It doesn’t change the value of the expression “1 x 1026”. But it might be convenient to use the “1 x” anyway. Scientific notation is about separating the size of the number from the interesting digits that the number has. Often when you compare numbers you’re interested in the size or else you’re interested in the important digits. Get into that habit and it’s not worth making an exception just because the interesting digits turn out to be boring in this case.

     
  • Joseph Nebus 6:00 pm on Thursday, 26 January, 2017 Permalink | Reply
    Tags: , Clear Blue Water, Hi and Lois, , , One Big Family, ,   

    Reading the Comics, January 21, 2017: Homework Edition 


    Now to close out what Comic Strip Master Command sent my way through last Saturday. And I’m glad I’ve shifted to a regular schedule for these. They ordered a mass of comics with mathematical themes for Sunday and Monday this current week.

    Karen Montague-Reyes’s Clear Blue Water rerun for the 17th describes trick-or-treating as “logarithmic”. The intention is to say that the difficulty in wrangling kids from house to house grows incredibly fast as the number of kids increases. Fair enough, but should it be “logarithmic” or “exponential”? Because the logarithm grows slowly as the number you take the logarithm of grows. It grows all the slower the bigger the number gets. The exponential of a number, though, that grows faster and faster still as the number underlying it grows. So is this mistaken?

    I say no. It depends what the logarithm is, and is of. If the number of kids is the logarithm of the difficulty of hauling them around, then the intent and the mathematics are in perfect alignment. Five kids are (let’s say) ten times harder to deal with than four kids. Sensible and, from what I can tell of packs of kids, correct.

    'Anne has six nickels. Sue has 41 pennies. Who has more money?' 'That's not going to be easy to figure out. It all depends on how they're dressed!'

    Rick Detorie’s One Big Happy for the 17th of January, 2017. The section was about how the appearance and trappings of wealth matter for more than the actual substance of wealth so everyone’s really up to speed in the course.

    Rick Detorie’s One Big Happy for the 17th is a resisting-the-word-problem joke. There’s probably some warning that could be drawn about this in how to write story problems. It’s hard to foresee all the reasonable confounding factors that might get a student to the wrong answer, or to see a problem that isn’t meant to be there.

    Bill Holbrook’s On The Fastrack for the 19th continues Fi’s story of considering leaving Fastrack Inc, and finding a non-competition clause that’s of appropriate comical absurdity. As an auditor there’s not even a chance Fi could do without numbers. Were she a pure mathematician … yeah, no. There’s fields of mathematics in which numbers aren’t all that important. But we never do without them entirely. Even if we exclude cases where a number is just used as an index, for which Roman numerals would be almost as good as regular numerals. If nothing else numbers would keep sneaking in by way of polynomials.

    'Uh, Fi? Have you looked at the non-compete clause in your contract?' 'I wouldn't go to one of Fastrack's competitors.' 'No, but, um ... you'd better read this.' 'I COULDN'T USE NUMBERS FOR TWO YEARS???' 'Roman numerals would be okay.'

    Bill Holbrook’s On The Fastrack for the 19th of January, 2017. I feel like someone could write a convoluted story that lets someone do mathematics while avoiding any actual use of any numbers, and that it would probably be Greg Egan who did it.

    Dave Whamond’s Reality Check for the 19th breaks our long dry spell without pie chart jokes.

    Mort Walker and Dik Browne’s Vintage Hi and Lois for the 27th of July, 1959 uses calculus as stand-in for what college is all about. Lois’s particular example is about a second derivative. Suppose we have a function named ‘y’ and that depends on a variable named ‘x’. Probably it’s a function with domain and range both real numbers. If complex numbers were involved then the variable would more likely be called ‘z’. The first derivative of a function is about how fast its values change with small changes in the variable. The second derivative is about how fast the values of the first derivative change with small changes in the variable.

    'I hope our kids are smart enough to win scholarships for college.' 'We can't count on that. We'll just have to save the money!' 'Do you know it costs about $10,000 to send one child through college?!' 'That's $40,000 we'd have to save!' Lois reads to the kids: (d^2/dx^2)y = 6x - 2.

    Mort Walker and Dik Browne’s Vintage Hi and Lois for the 27th of July, 1959. Fortunately Lois discovered the other way to avoid college costs: simply freeze the ages of your children where they are now, so they never face student loans. It’s an appealing plan until you imagine being Trixie.

    The ‘d’ in this equation is more of an instruction than it is a number, which is why it’s a mistake to just divide those out. Instead of writing it as \frac{d^2 y}{dx^2} it’s permitted, and common, to write it as \frac{d^2}{dx^2} y . This means the same thing. I like that because, to me at least, it more clearly suggests “do this thing (take the second derivative) to the function we call ‘y’.” That’s a matter of style and what the author thinks needs emphasis.

    There are infinitely many possible functions y that would make the equation \frac{d^2 y}{dx^2} = 6x - 2 true. They all belong to one family, though. They all look like y(x) = \frac{1}{6} 6 x^3 - \frac{1}{2} 2 x^2 + C x + D , where ‘C’ and ‘D’ are some fixed numbers. There’s no way to know, from what Lois has given, what those numbers should be. It might be that the context of the problem gives information to use to say what those numbers should be. It might be that the problem doesn’t care what those numbers should be. Impossible to say without the context.

     
    • Joshua K. 6:26 am on Monday, 30 January, 2017 Permalink | Reply

      Why is the function in the Hi & Lois discussion stated as y(x) = (1/6)6x^3 – (1/2)2x^2 + Cx +D? Why not just y(x) = x^3 – x^2 + Cx + D?

      Like

      • Joseph Nebus 5:43 pm on Friday, 3 February, 2017 Permalink | Reply

        Good question! I actually put a fair bit of thought into this. If I were doing the problem myself I’d have cut right to x^3 – x^2 + Cx + D. But I thought there’s a number of people reading this for whom calculus is a perfect mystery and I thought that if I put an intermediate step it might help spot the pattern at work, that the coefficients in front of the x^3 and x^2 terms don’t vanish without cause.

        That said, I probably screwed up by writing them as 1/6 and 1/2. That looks too much like I’m just dividing by what the coefficients are. If I had taken more time to think out the post I should have written 1/(23) and 1/(12). This might’ve given a slightly better chance at connecting the powers of x and the fractions in the denominator. I’m not sure how much help that would give, since I didn’t describe how to take antiderivatives here. But I think it’d be a better presentation and I should remember that in future situations like that.

        Like

  • Joseph Nebus 6:00 pm on Sunday, 22 January, 2017 Permalink | Reply
    Tags: , BC, dummy variables, Edge City, , , , ,   

    Reading the Comics, January 16, 2017: Numerals Edition 


    Comic Strip Master Command decreed that last week should be busy again. So I’m splitting its strips into two essays. It’s a week that feels like it had more anthropomorphic numerals jokes than usual, but see if I actually count these things.

    2 asks 4: 'Six, six, six, can't you think of anything but six?'

    Mike Peters’s Mother Goose and Grimm for the 15th of January, 2017. I understand that sometimes you just have to use the idea you have instead of waiting for something that can best use the space available, but really, a whole Sunday strip for a single panel? And a panel that’s almost a barren stage?

    Mike Peters’s Mother Goose and Grimm for the 15th I figured would be the anthropomorphic numerals joke for the week. Shows what I know. It is an easy joke, but I do appreciate the touch of craft involved in picking the numerals. The joke is just faintly dirty if the numbers don’t add to six. If they were a pair of 3’s, there’d be the unwanted connotations of a pair of twins talking about all this. A 6 and a 0 would make at least one character weirdly obsessed. So it has to be a 4 and a 2, or a 5 and a 1. I imagine Peters knew this instinctively, at this point in his career. It’s one of the things you learn in becoming an expert.

    Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 15th is mostly physical comedy, with a touch of — I’m not sure what to call this kind of joke. The one where a little arithmetic error results in bodily harm. In this sort of joke it’s almost always something not being carried that’s the error. I suppose that’s a matter of word economy. “Forgot to carry the (number)” is short, and everybody’s done it. And even if they don’t remember making this error, the phrasing clarifies to people that it’s a little arithmetic mistake. I think in practice mistaking a plus for a minus (or vice-versa) is the more common arithmetic error. But it’s harder to describe that clearly and concisely.

    Jef Mallett’s Frazz for the 15th puzzled me. I hadn’t heard this thing the kid says about how if you can “spew ten random lines from a classic movie” to convince people you’ve seen it. (I don’t know the kid’s name; it happens.) I suppose that it would be convincing, though. I certainly know a couple lines from movies I haven’t seen, what with living in pop culture and all that. But ten would be taxing for all but the most over-saturated movies, like any of the Indiana Jones films. (There I’m helped by having played the 90s pinball machine a lot.) Anyway, knowing ten random mathematics things isn’t convincing, especially since you can generate new mathematical things at will just by changing a number. But I would probably be convinced that someone who could describe what’s interesting about ten fields of mathematics had a decent understanding of the subject. That requires remembering more stuff, but then, mathematics is a bigger subject than even a long movie is.

    In Bill Holbrook’s On The Fastrack for the 16th Fi speaks of tallying the pluses and minuses of her life. Trying to make life into something that can be counted is an old decision-making technique. I think Benjamin Franklin explained how he found it so useful. It’s not a bad approach if a choice is hard. The challenging part is how to weight each consideration. Getting into fractions seems rather fussy to me, but some things are just like that. There is the connotation here that a fraction is a positive number smaller than 1. But the mathematically-trained (such as Fi) would be comfortable with fractions larger than 1. Or also smaller than zero. “Fraction” is no more bounded than “real number”. So, there’s the room for more sweetness here than might appear to the casual reader.

    'In a couple of weeks I'm getting married, so I'm taking stock of my life, adding up the pluses and minuses that factor into my goals.' 'Am I a positive or a negative integer?' 'You're a fraction.' 'How presumptuous of me.'

    Bill Holbrook’s On The Fastrack for the 16th of January, 2017. Were I in Dethany’s position I would have asked about being a positive or negative number, but then that would leave Holbrook without a third panel. Dethany knows what her author needs most.

    Scott Hilburn’s The Argyle Sweater for the 16th is the next anthropomorphic numerals joke for this week. I’m glad Hilburn want to be in my pages more. 5’s concern about figuring out x might be misplaced. We use variables for several purposes. One of them is as a name to give a number whose value we don’t know but wish to work out, and that’s how we first see them in high school algebra. But a variable might also be a number whose value we don’t particularly care about and will never try to work out. This could be because the variable is a parameter, with a value that’s fixed for a problem but not what we’re interested in. We don’t typically use ‘x’ for that, though; usually parameter are something earlier in the alphabet. That’s merely convention, but it is convention that dates back to René Descartes. Alternatively, we might use ‘x’ as a dummy variable. A dummy variable serves the same role that falsework on a building or a reference for an artistic sketch does. We use dummy variables to organize and carry out work, but we don’t care what its values are and we don’t even see the dummy variable in the final result. A dummy variable can be any name, but ‘x’ and ‘t’ are popular choices.

    Terry LaBan and Patty LaBan’s Edge City rerun for the 16th plays on the idea that mathematics people talk in algebra. Funny enough, although, “the opposing defense is a variable of 6”? That’s an idiosyncratic use of “variable”. I’m going to suppose that Charles is just messing with Len’s head because, really, it’s fun doing a bit of that.

     
  • Joseph Nebus 6:00 pm on Thursday, 19 January, 2017 Permalink | Reply
    Tags: , , , , , , , The Daily Drawing, The Elderberries, Ziggy   

    Reading the Comics, January 14, 2017: Maybe The Last Jumble? Edition 


    So now let me get to the other half of last week’s comics. Also, not to spoil things, but this coming week is looking pretty busy so I may have anothe split-week Reading the Comics coming up. The shocking thing this time is that the Houston Chronicle has announced it’s discontinuing its comics page. I don’t know why; I suppose because they’re fed up with people coming loyally to a daily feature. I will try finding alternate sources for the things I had still been reading there, but don’t know if I’ll make it.

    I’m saddened by this. Back in the 90s comics were just coming onto the Internet. The Houston Chronicle was one of a couple newspapers that knew what to do with them. It, and the Philadelphia Inquirer and the San Jose Mercury-News, had exactly what we wanted in comics: you could make a page up of all the strips you wanted to read, and read them on a single page. You could even go backwards day by day in case you missed some. The Philadelphia Inquirer was the only page that let you put the comics in the order you wanted, as opposed to alphabetical order by title. But if you were unafraid of opening up URLs you could reorder the Houston Chronicle page you built too.

    And those have all faded away. In the interests of whatever interest is served by web site redesigns all these papers did away with their user-buildable comics pages. The Chronicle was the last holdout, but even they abolished their pages a few years ago, with a promise for a while that they’d have a replacement comics-page scheme up soon. It never came and now, I suppose, never will.

    Most of the newspapers’ sites had become redundant anyway. Comics Kingdom and GoComics.com offer user-customizable comics pages, with a subscription model that makes it clear that money ought to be going to the cartoonists. I still had the Chronicle for a few holdouts, like Joe Martin’s strips or the Jumble feature. And from that inertia that attaches to long-running Internet associations.

    So among the other things January 2017 takes away from us, it is taking the last, faded echo of the days in the 1990s when newspapers saw comics as awesome things that could be made part of their sites.

    Lorie Ransom’s The Daily Drawing for the 11th is almost but not quite the anthropomorphized-numerals joke for this installment. It’s certainly the most numerical duck content I’ve got on record.

    Tom II Wilson’s Ziggy for the 11th is an Early Pi Day joke. Cosmically there isn’t any reason we couldn’t use π in take-a-number dispensers, after all. Their purpose is to give us some certain order in which to do things. We could use any set of numbers which can be put in order. So the counting numbers work. So do the integers. And the real numbers. But practicality comes into it. Most people have probably heard that π is a bit bigger than 3 and a fair bit smaller than 4. But pity the two people who drew e^{\pi} and \pi^{e} figuring out who gets to go first. Still, I won’t be surprised if some mathematics-oriented place uses a gimmick like this, albeit with numbers that couldn’t be confused. At least not confused by people who go to mathematics-oriented places. That would be for fun rather than cake.

    CTEFH -OOO-; ITODI OOO--; RAWDON O--O-O; FITNAN OO--O-. He wanted to expand his collection and the Mesopotamian abacus would make a OOOO OOOOOOOO.

    the Jumble for the 11th of January, 2017. This link’s all but sure to die the 1st of February, so, sorry about that. Mesopotamia did have the abacus, although I don’t know that the depiction is anything close to what the actual ones looked like. I’d imagine they do, at least within the limits of what will be an understandable drawing.

    I can’t promise that the Jumble for the 11th is the last one I’ll ever feature here. I might find where David L Hoyt and Jeff Knurek keep a linkable reference to their strips and point to them. But just in case of the worst here’s an abacus gag for you to work on.

    Corey Pandolph, Phil Frank, and Joe Troise’s The Elderberries for the 12th is, I have to point out, a rerun. So if you’re trying to do the puzzle the reference to “the number of the last president” isn’t what you’re thinking of. It is an example of the conflation of intelligence with skill at arithmetic. It’s also an example the conflation of intelligence with a mastery of trivia. But I think it leans on arithmetic more. I am not sure when this strip first appeared. “The last president” might have been Bill Clinton (42) or George W Bush (43). But this means we’re taking the square root of either 33 or 34. And there’s no doing that in your head. The square root of a whole number is either a whole number — the way the square root of 36 is — or else it’s an irrational number. You can work out the square root of a non-perfect-square by hand. But it’s boring and it’s worse than just writing “\sqrt{33} ” or “\sqrt{34} ”. Except in figuring out if that number is larger than or smaller than five or six. It’s good for that.

    Dave Blazek’s Loose Parts for the 13th is the actuary joke for this installment. Actuarial studies are built on one of the great wonders of statistics: that it is possible to predict how often things will happen. They can happen to a population, as in forecasts of how many people will be in traffic accidents or fires or will lose their jobs or will move to a new city. We may have no idea to whom any of these will happen, and they may have no way of guessing, but the enormous number of people and great number of things that can combine to make a predictable state of affairs. I suppose it’s imaginable that a group could study its dynamics well enough to identify who screws up the most and most seriously. So they might be able to say what the odds are it is his fault. But I imagine in practice it’s too difficult to define screw-ups or to assign fault consistently enough to get the data needed.

    Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th is another multiverse strip, echoing the Dinosaur Comics I featured here Sunday. I’ll echo my comments then. If there is a multiverse — again, there is not evidence for this — then there may be infinitely many versions of every book of the Bible. This suggests, but it does not mandate, that there should be every possible incarnation of the Bible. And a multiverse might be a spendthrift option anyway. Just allow for enough editions, and the chance that any of them will have a misprint at any word or phrase, and we can eventually get infinitely many versions of every book of the Bible. If we wait long enough.

     
  • Joseph Nebus 6:00 pm on Tuesday, 17 January, 2017 Permalink | Reply
    Tags: capitals, , , ,   

    48 Altered States 


    I saw this intriguing map produced by Brian Brettschneider.

    He made it on and for Twitter, as best I can determine. I found it from a stray post in Usenet newsgroup soc.history.what-if, dedicated to ways history could have gone otherwise. It also covers ways that it could not possibly have gone otherwise but would be interesting to see happen. Very different United States state boundaries are part of the latter set of things.

    The location of these boundaries is described in English and so comes out a little confusing. It’s hard to make concise. Every point in, say, this alternate Missouri is closer to Missouri’s capital of … uhm … Missouri City than it is to any other state’s capital. And the same for all the other states. All you kind readers who made it through my recent A To Z know a technical term for this. This is a Voronoi Diagram. It uses as its basis points the capitals of the (contiguous) United States.

    It’s an amusing map. I mean amusing to people who can attach concepts like amusement to maps. It’d probably be a good one to use if someone needed to make a Risk-style grand strategy game map and didn’t want to be to beholden to the actual map.

    No state comes out unchanged, although a few don’t come out too bad. Maine is nearly unchanged. Michigan isn’t changed beyond recognition. Florida gets a little weirder but if you showed someone this alternate shape they’d recognize the original. No such luck with alternate Tennessee or alternate Wyoming.

    The connectivity between states changes a little. California and Arizona lose their border. Washington and Montana gain one; similarly, Vermont and Maine suddenly become neighbors. The “Four Corners” spot where Utah, Colorado, New Mexico, and Arizona converge is gone. Two new ones look like they appear, between New Hampshire, Massachusetts, Rhode Island, and Connecticut; and between Pennsylvania, Maryland, Virginia, and West Virginia. I would be stunned if that weren’t just because we can’t zoom far enough in on the map to see they’re actually a pair of nearby three-way junctions.

    I’m impressed by the number of borders that are nearly intact, like those of Missouri or Washington. After all, many actual state boundaries are geographic features like rivers that a Voronoi Diagram doesn’t notice. How could Ohio come out looking anything like Ohio?

    The reason comes to historical subtleties. At least once you get past the original 13 states, basically the east coast of the United States. The boundaries of those states were set by colonial charters, with boundaries set based on little or ambiguous information about what the local terrain was actually like, and drawn to reward or punish court factions and favorites. Never mind the original thirteen (plus Maine and Vermont, which we might as well consider part of the original thirteen).

    After that, though, the United States started drawing state boundaries and had some method to it all. Generally a chunk of territory would be split into territories and later states that would be roughly rectangular, so far as practical, and roughly similar in size to the other states carved of the same area. So for example Missouri and Alabama are roughly similar to Georgia in size and even shape. Louisiana, Arkansas, and Missouri are about equal in north-south span and loosely similar east-to-west. Kansas, Nebraska, South Dakota, and North Dakota aren’t too different in their north-to-south or east-to-west spans.

    There’s exceptions, for reasons tied to the complexities of history. California and Texas get peculiar shapes because they could. Michigan has an upper peninsula for quirky reasons that some friend of mine on Twitter discovers every three weeks or so. But the rough guide is that states look a lot more similar to one another than you’d think from a quick look. Mark Stein’s How The States Got Their Shapes is an endlessly fascinating text explaining this all.

    If there is a loose logic to state boundaries, though, what about state capitals? Those are more quirky. One starts to see the patterns when considering questions like “why put California’s capital in Sacramento instead of, like, San Francisco?” or “Why Saint Joseph instead Saint Louis or Kansas City?” There is no universal guide, but there are some trends. Generally states end up putting their capitals in a city that’s relatively central, at least to the major population centers around the time of statehood. And, generally, not in one of the state’s big commercial or industrial centers. The desire to be geographically central is easy to understand. No fair making citizens trudge that far if they have business in the capital. Avoiding the (pardon) first tier of cities has subtler politics to it; it’s an attempt to get the government somewhere at least a little inconvenient to the money powers.

    There’s exceptions, of course. Boston is the obviously important city in Massachusetts, Salt Lake City the place of interest for Utah, Denver the equivalent for Colorado. Capitals relocated; Atlanta is Georgia’s eighth(?) I think since statehood. Sometimes they were weirder. Until 1854 Rhode Island rotated between five cities, to the surprise of people trying to name a third city in Rhode Island. New Jersey settled on Trenton as compromise between the East and West Jersey capitals of Perth Amboy and Burlington. But if you look for a city that’s fairly central but not the biggest in the state you get to the capital pretty often.

    So these are historical and cultural factors which combine to make a Voronoi Diagram map of the United States strange, but not impossibly strange, compared to what has really happened. Things are rarely so arbitrary as they seem at first.

     
    • Matthew Wright 6:49 pm on Tuesday, 17 January, 2017 Permalink | Reply

      New Zealand’s provincial borders were devised at much the same time as the midwestern and western US and in much the same way. Some guy with a map that only vaguely showed rivers, and a ruler. Well, when I say ‘some guy’ I mean George Grey, Edward Eyre and their factotum, Alfred Domett among only a handful of others. Early colonial New Zealand was like that. The civil service consisted of about three people (all of them Domett) and because the franchise system meant some voting districts might have as few as 25 electors, anybody had at least a 50/50 chance of becoming Prime Minister.

      Like

      • Joseph Nebus 3:45 pm on Saturday, 21 January, 2017 Permalink | Reply

        I am intrigued and delighted to learn this! For all that I do love maps and seeing how borders evolve over time I’m stronger on United States and Canadian province borders; they’re just what was easily available when I grew up. (Well, and European boundaries, but I don’t think there’s a single one of them that’s based on anything more than “this is where the armies stood on V-E Day”.)

        Would you have a recommendation on a pop history of New Zealand for someone who knows only, mostly, that I guess confederation with Australia was mooted in 1900 but refused since the islands are actually closer to the Scilly Isles than they are Canberra for crying out loud?

        Liked by 1 person

    • Matthew Wright 8:43 pm on Saturday, 21 January, 2017 Permalink | Reply

      Europe has had so many boundary changes since Roman times that I wouldn’t be surprised if there’s a tradition for governments to issue people with an eraser and pot of paint to update their maps – and, no question, their history IS the history of those boundary changes. Certainly it explains their wars…

      On matters NZ, I wrote just such a book – it was first published in 2004 and has been through a couple of editions (I updated it in 2012). My publishers, Bateman, put it up on Kindle:

      It’s ‘publisher priced’ but I’d thoroughly recommend it! :-) The parallels between NZ’s settler period and the US ‘midwestern’ expansion through to California at the same time are direct.

      The reasons why NZ never joined Australia in 1900 have been endlessly debated and never answered but probably had something to do with the way NZ was socially re-identifying itself with Britain at the time. The British ignored the whole thing for defence/strategic purposes, deploying just one RN squadron to Sydney as the ‘mid point’ of Australasia. Sydney-siders liked it, but everybody from Perth to Wellington was annoyed. I wrote my thesis on the political outcome, way back when.

      Like

      • Joseph Nebus 6:19 am on Saturday, 28 January, 2017 Permalink | Reply

        Aw, thank you kindly! I’d thought you might have something suitable.

        The organizing of territory that white folks told themselves was unsettled is a process I find interesting, I suppose because I’ve always wondered about how one goes about establishing systems. I think it’s similar to my interest in how nations devastated by wars get stuff like trash collection and fire departments and regional power systems running again. The legal system for at least how the United States organized territory is made clear enough in public schools (at least to students who pay attention, like me), but it isn’t easy to find the parallel processes in other countries. Now and then I try reading about Canada and how two of every seven sections of land in (now) Quebec and Ontario was reserved to the church and then I pass out and by the time I wake up again they’re making infrastructure promises to Prince Edward Island.

        I’m not surprised that from the British side of things the organization of New Zealand and Australia amounted to a bit of afterthought and trusting things would work out all right. I have read a fair bit (for an American) about the British Empire and it does feel like all that was ever thought about was India and the route to India and an ever-widening corridor of imagined weak spots on the route to India. The rest of the world was, pick some spot they had already, declare it “the Gibraltar of [ Geographic Region ]” and suppose there’d be a ship they could send there if they really had to.

        Like

  • Joseph Nebus 6:00 pm on Sunday, 15 January, 2017 Permalink | Reply
    Tags: , Barkeater Lake, combinatorics, , Dinette Set, , Nancy, , Reply All, , Slylock Fox,   

    Reading the Comics, January 14, 2017: Redeye and Reruns Edition 


    So for all I worried about the Gocomics.com redesign it’s not bad. The biggest change is it’s removed a side panel and given the space over to the comics. And while it does show comics you haven’t been reading, it only shows one per day. One week in it apparently sticks with the same comic unless you choose to dismiss that. So I’ve had it showing me The Comic Strip That Has A Finale Every Day as a strip I’m not “reading”. I’m delighted how thisbreaks the logic about what it means to “not read” an “ongoing comic strip”. (That strip was a Super-Fun-Pak Comix offering, as part of Ruben Bolling’s Tom the Dancing Bug. It was turned into a regular Gocomics.com feature by someone who got the joke.)

    Comic Strip Master Command responded to the change by sending out a lot of comic strips. I’m going to have to divide this week’s entry into two pieces. There’s not deep things to say about most of these comics, but I’ll make do, surely.

    Julie Larson’s Dinette Set rerun for the 8th is about one of the great uses of combinatorics. That use is working out how the number of possible things compares to the number of things there are. What’s always staggering is that the number of possible things grows so very very fast. Here one of Larson’s characters claims a science-type show made an assertion about the number of possible ideas a brain could hold. I don’t know if that’s inspired by some actual bit of pop science. I can imagine someone trying to estimate the number of possible states a brain might have.

    And that has to be larger than the number of atoms in the universe. Consider: there’s something less than a googol of atoms in the universe. But a person can certainly have the idea of the number 1, or the idea of the number 2, or the idea of the number 3, or so on. I admit a certain sameness seems to exist between the ideas of the numbers 2,038,412,562,593,604 and 2,038,412,582,593,604. But there is a difference. We can out-number the atoms in the universe even before we consider ideas like rabbits or liberal democracy or jellybeans or board games. The universe never had a chance.

    Or did it? Is it possible for a number to be too big for the human brain to ponder? If there are more digits in the number than there are atoms in the universe we can’t form any discrete representation of it, after all. … Except that we kind of can. For example, “the largest prime number less than one googolplex” is perfectly understandable. We can’t write it out in digits, I think. But you now have thought of that number, and while you may not know what its millionth decimal digit is, you also have no reason to care what that digit is. This is stepping into the troubled waters of algorithmic complexity.

    Shady Shrew is selling fancy bubble-making wands. Shady says the crazy-shaped wands cost more than the ordinary ones because of the crazy-shaped bubbles they create. Even though Slylock Fox has enough money to buy an expensive wand, he bought the cheaper one for Max Mouse. Why?

    Bob Weber Jr’s Slylock Fox and Comics for Kids for the 9th of January, 2017. Not sure why Shady Shrew is selling the circular wands at 50 cents. Sure, I understand wanting a triangle or star or other wand selling at a premium. But then why have the circular wands at such a cheap price? Wouldn’t it be better to put them at like six dollars, so that eight dollars for a fancy wand doesn’t seem that great an extravagance? You have to consider setting an appropriate anchor point for your customer base. But, then, Shady Shrew isn’t supposed to be that smart.

    Bob Weber Jr’s Slylock Fox and Comics for Kids for the 9th is built on soap bubbles. The link between the wand and the soap bubble vanishes quickly once the bubble breaks loose of the wand. But soap films that keep adhered to the wand or mesh can be quite strangely shaped. Soap films are a practical example of a kind of partial differential equations problem. Partial differential equations often appear when we want to talk about shapes and surfaces and materials that tug or deform the material near them. The shape of a soap bubble will be the one that minimizes the torsion stresses of the bubble’s surface. It’s a challenge to solve analytically. It’s still a good challenge to solve numerically. But you can do that most wonderful of things and solve a differential equation experimentally, if you must. It’s old-fashioned. The computer tools to do this have gotten so common it’s hard to justify going to the engineering lab and getting soapy water all over a mathematician’s fingers. But the option is there.

    Gordon Bess’s Redeye rerun from the 28th of August, 1970, is one of a string of confused-student jokes. (The strip had a Generic Comedic Western Indian setting, putting it in the vein of Hagar the Horrible and other comic-anachronism comics.) But I wonder if there are kids baffled by numbers getting made several different ways. Experience with recipes and assembly instructions and the like might train someone to thinking there’s one correct way to make something. That could build a bad intuition about what additions can work.

    'I'm never going to learn anything with Redeye as my teacher! Yesterday he told me that four and one make five! Today he said, *two* and *three* make five!'

    Gordon Bess’s Redeye rerun from the 28th of August, 1970. Reprinted the 9th of January, 2017. What makes the strip work is how it’s tied to the personalities of these kids and couldn’t be transplanted into every other comic strip with two kids in it.

    Corey Pandolph’s Barkeater Lake rerun for the 9th just name-drops algebra. And that as a word that starts with the “alj” sound. So far as I’m aware there’s not a clear etymological link between Algeria and algebra, despite both being modified Arabic words. Algebra comes from “al-jabr”, about reuniting broken things. Algeria comes from Algiers, which Wikipedia says derives from `al-jaza’ir”, “the Islands [of the Mazghanna tribe]”.

    Guy Gilchrist’s Nancy for the 9th is another mathematics-cameo strip. But it was also the first strip I ran across this week that mentioned mathematics and wasn’t a rerun. I’ll take it.

    Donna A Lewis’s Reply All for the 9th has Lizzie accuse her boyfriend of cheating by using mathematics in Scrabble. He seems to just be counting tiles, though. I think Lizzie suspects something like Blackjack card-counting is going on. Since there are only so many of each letter available knowing just how many tiles remain could maybe offer some guidance how to play? But I don’t see how. In Blackjack a player gets to decide whether to take more cards or not. Counting cards can suggest whether it’s more likely or less likely that another card will make the player or dealer bust. Scrabble doesn’t offer that choice. One has to refill up to seven tiles until the tile bag hasn’t got enough left. Perhaps I’m overlooking something; I haven’t played much Scrabble since I was a kid.

    Perhaps we can take the strip as portraying the folk belief that mathematicians get to know secret, barely-explainable advantages on ordinary folks. That itself reflects a folk belief that experts of any kind are endowed with vaguely cheating knowledge. I’ll admit being able to go up to a blackboard and write with confidence a bunch of integrals feels a bit like magic. This doesn’t help with Scrabble.

    'Want me to teach you how to add and subtract, Pokey?' 'Sure!' 'Okay ... if you had four cookies and I asked you for two, how many would you have left?' 'I'd still have four!'

    Gordon Bess’s Redeye rerun from the 29th of August, 1970. Reprinted the 10th of January, 2017. To be less snarky, I do like the simply-expressed weariness on the girl’s face. It’s hard to communicate feelings with few pen strokes.

    Gordon Bess’s Redeye continued the confused-student thread on the 29th of August, 1970. This one’s a much older joke about resisting word problems.

    Ryan North’s Dinosaur Comics rerun for the 10th talks about multiverses. If we allow there to be infinitely many possible universes that would suggest infinitely many different Shakespeares writing enormously many variations of everything. It’s an interesting variant on the monkeys-at-typewriters problem. I noticed how T-Rex put Shakespeare at typewriters too. That’ll have many of the same practical problems as monkeys-at-typewriters do, though. There’ll be a lot of variations that are just a few words or a trivial scene different from what we have, for example. Or there’ll be variants that are completely uninteresting, or so different we can barely recognize them as relevant. And that’s if it’s actually possible for there to be an alternate universe with Shakespeare writing his plays differently. That seems like it should be possible, but we lack evidence that it is.

     
  • Joseph Nebus 6:00 pm on Thursday, 12 January, 2017 Permalink | Reply
    Tags: , December, , , ,   

    How December 2016 Treated My Mathematics Blog 


    I’m getting back to normal. And getting to suspect WordPress just isn’t sending out “Fireworks” reports on how the year for my blog went. Fine then; I’ll carry on. Going back to the Official WordPress statistics page and sharing it for whatever value that has we find that … apparently I just held November 2016 all over again. Gads what a prospect.

    As ever I exaggerate, and as ever, not by much. There were 956 page views from 589 distinct readers in December. In November there were 923 page views from 575 distinct readers. There were 21 posts in December, compared to 21 posts in November. Both are up from October, 907 page views from 536 visitors, although that was a nice and easy month with only 13 posts published. I’m a little disappointed to fall under a thousand page views for four months running, but, like, I tried posting stuff more often. What else can I do, besides answer comments the same year they’re posted and chat with people on their blogs? You know?

    There were 136 pages liked in December, down from November’s 157 and up from October’s 115. Comments were down to 29 from November’s 35, and while that’s up from October’s 24 I should point out some of January’s comments are really me answering December comments. I had a lot of things slurping up time and energy. That doesn’t mean I’m not going to count the comments I wrote in January as anything other than January’s comments, though.

    According to Insights, my most popular day for reading is Sunday, with 17 percent of page views coming then. I expected that; Sunday’s been the most popular day the last few months. It’s only slightly most popular, though. 17 percent (18 percent last month) is about what you’d expect for people reading here without any regard for the day of the week. 6 pm was the most popular hour, barely, with 9 percent of page views then. That’s the hour I’ve settled on for posting stuff. But that hour’s down from being 14 percent of page views in November. I don’t know what that signifies.

    My roster of countries and the page views from them looks like this. I’m curiously delighted that India’s becoming a regular top-five country.

    Country Views
    United States 587
    United Kingdom 61
    India 47
    Canada 44
    Germany 25
    Austria 22
    Slovenia 15
    Philippines 13
    Netherlands 10
    Spain 9
    Australia 9
    Italy 7
    Puerto Rico 7
    Finland 6
    Norway 6
    Singapore 6
    France 5
    Ireland 5
    Switzerland 5
    Indonesia 4
    Sweden 4
    Thailand 4
    Bahrain 3
    Barbados 3
    Estonia 3
    Israel 3
    Turkey 3
    Chile 2
    Greece 2
    New Zealand 2
    Nigeria 2
    Peru 2
    Poland 2
    Sri Lanka 2
    United Arab Emirates 2
    Bangladesh 1
    Belgium 1
    Denmark 1 (*)
    Egypt 1
    European Union 1
    Japan 1 (**)
    Kuwait 1
    Lebanon 1
    Luxembourg 1
    Nepal 1
    Pakistan 1
    Romania 1
    Saudi Arabia 1 (**)
    Slovakia 1
    South Africa 1 (**)

    There’s 50 countries altogether that sent me viewers, if we take “European Union” as a country. That’s up from November’s 46. There were 15 single-view countries, the same as in November. Denmark was a single-view country last month. Japan, Saudia Arabia, and South Africa are on three-month single-view streaks. “European Union” is back after a brief absence.

    For the second month in a row none of my most popular posts were Reading the Comics essays. They instead were split between the A To Z, some useful-mathematics stuff, and some idle trivia. The most popular stuff in December here was:

    There weren’t many specific search terms; most were just “unknown”. Of the search terms that could be known I got this bunch that started out normal enough and then got weird.

    • comics strip of production function
    • comics of production function theory
    • comics about compound event in math
    • comics trip math probability
    • example of probability comics trip
    • population of charlotte nc 1975
    • a to z image 2017
    • mathematics dark secrets

    I, um, maybe have an idea what that last one ought to find.

    January starts with my mathematics blog having gotten 44,104 page views total from 18,889 distinct known visitors. That’s still a little page view lead on my humor blog, but that’s going to be lost by the start of February. My humor blog’s been more popular consistently the several months, and the humor blog got some little wave of popularity the past couple days. Why should it have had that? My best guess: I’m able to use that platform to explain what’s going on in Judge Parker, which I can’t quite justify here. Maybe next month.

    If you’d like to follow my mathematics blog, please, click the buttons in the upper-right corner of the page to follow the blog on WordPress or by e-mail. You can also find me on Twitter as @nebusj where I try not to be one of those people who somehow has fifty tweets or retweets every hour of the day. But I haven’t done any livetweeting of a bad cartoon in ages. Might change.

     
    • mathtuition88 8:04 am on Friday, 13 January, 2017 Permalink | Reply

      Nice number of United States views!

      Like

      • Joseph Nebus 3:18 pm on Saturday, 21 January, 2017 Permalink | Reply

        Thank you. I’m always surprised by how the numbers concentrate in a particular region. I’d naively expect to be about equally read anywhere in the English-speaking world, although perhaps my heavy focus on United States-syndicated comic strips does something to attract more United States readers and shoo off non-US-audiences. It’s a curious effect, anyway.

        Liked by 1 person

        • mathtuition88 3:29 pm on Saturday, 21 January, 2017 Permalink | Reply

          I have a similar case, most of my viewership (90% in fact) is from Singapore even though 70%-80% of my content should be considered country-neutral.

          Like

          • Joseph Nebus 3:48 pm on Saturday, 21 January, 2017 Permalink | Reply

            Yeah, the country links are weird. I would understand time-zone-based links; something that appears at 3 am local time is not going to be read nearly as much as the same thing at 3 pm. So with most of my mathematics posts here appearing in late-morning/early-afternoon United States time, and early-evening European time, I would expect more chances for readers there. But that there seem to be correlations across national boundaries even for places that haven’t got time zone differences is weird: why not as many Hong Kong readers as Singaporean ones? Or shouldn’t India’s large English-reading audience balance out a couple hours’ difference in time zone? Something I don’t understand is going on here.

            Liked by 1 person

            • mathtuition88 3:54 pm on Saturday, 21 January, 2017 Permalink | Reply

              Have you tried Google Webmasters? There is a way to set your website’s target location (they call it geotargeting). It didn’t work much for me, but it is worth a try.

              Like

              • Joseph Nebus 5:21 am on Saturday, 28 January, 2017 Permalink | Reply

                I have not! I haven’t even thought about it, actually, but it’s worth at least investigating. I suppose that insofar as I have a location the United States is fair enough; my comic strip posts are irredeemably America-centric. But I’d like other people to feel welcome around here.

                Liked by 1 person

    • elkement (Elke Stangl) 10:37 am on Sunday, 15 January, 2017 Permalink | Reply

      It would be interesting to see statistics for a large number of WordPress.com blog and about how views have been changed over recent years. More and more blogs are started, but on the other hand the life time of most blogs seems to be alive only for about 1-2 years. Recently somebody ‘from the past’ commented on my blog: He came back to his abandoned blog after a few years and found that I was the only blogger ‘still alive’ from the crowd he once followed.

      I think views are not increased significantly if you blog more. E.g. in the last year I had about 1400 views per month – despite I blog only twice a month. Most of the views are generated by a small set of posts, some of them as old as 2012. In 2014 I blogged more than twice as much and had about 30% more views. But this included some pronounced spikes which I attributed to bot-like behavior as the clicks over time were so regular. WordPress support could not confirm this but could not refute it either.
      It somehow feels as if an ‘established’ blog is given a certain share of internet attention, and it will not change no matter what you do :-)

      Do you see some long-term trend in views per year? Does it correlate with posts per year?

      Liked by 1 person

      • elkement (Elke Stangl) 10:39 am on Sunday, 15 January, 2017 Permalink | Reply

        (… and I wished there would be an editor… ‘ the life time of most blogs seems to be alive’ … one time would have been enough… I guess you know that I mean ;-))

        Liked by 1 person

        • Joseph Nebus 3:29 pm on Saturday, 21 January, 2017 Permalink | Reply

          Oh, yes, understood easily. … It is a little surprising there’s not at least the chance to edit the first five minutes after posting something.

          Liked by 1 person

      • Joseph Nebus 3:27 pm on Saturday, 21 January, 2017 Permalink | Reply

        I admit a part of my posting these numbers is that I’m curious what other people’s readership patterns are like. I’m shameless and happy to admit my actual exact numbers as best as I can know them. But I’d like to know about overall trends. After all, it was only by comparing numbers that we worked out there seems to have been some strange drop like a year and a half ago that we think reflected mobile-device numbers no longer being counted.

        The expiration of older blogs is another of those strange phenomena. I mean some free weekend to go through my blog and cut out sites that haven’t updated in, like, two years. But why I should do that I don’t know; if they aren’t posting it isn’t as though they’re crowding out space. Just some sense that my readership list ought to be faintly in touch with what’s current.

        I’ve got a few perennial posts. The count of how many grooves are on a record’s side (or, really, how many times the groove intersects a radial line on a record). How to figure what you need on the final. The Arthur Christmas series. The latter two I try to promote at appropriate times, though. Past that it’s usually my comic strip posts that get readers, I suppose because people like to look up when curious mathematics stuff appears in Luann and they wonder if that makes any sense.

        Now, my long-term, year-long trends … I’m not sure. I have got five full years (wow) of numbers to work with so I can make something that looks faintly like a linear regression study. Might do that and see if there’s any correlation with post count.

        Liked by 2 people

      • mathtuition88 3:51 pm on Saturday, 21 January, 2017 Permalink | Reply

        That is true. For my blog 10% of the posts generate 90% of the views. And unfortunately those 10% are probably the least mathematical of the posts (for instance discussion/information of the Singapore education system). Pareto principle holds true.

        Liked by 1 person

        • Joseph Nebus 5:16 am on Saturday, 28 January, 2017 Permalink | Reply

          You know, discussions of what are the popular versus the most-worked-on versus the most common posts people have reminds me of something from Walt Kelly’s masterpiece comic strip Pogo. The irascible Porky Pine warned, I think, Pogo, “If the public decides it’s gonna honor you they ain’t gonna let your feelings get in the way.”

          Liked by 1 person

  • Joseph Nebus 6:00 pm on Sunday, 8 January, 2017 Permalink | Reply
    Tags: , , Birdbrains, , Elderberries, Grand Avenue, , Pot Shots, , Quincy   

    Reading the Comics, January 7, 2016: Just Before GoComics Breaks Everything Edition 


    Most of the comics I review here are printed on GoComics.com. Well, most of the comics I read online are from there. But even so I think they have more comic strips that mention mathematical themes. Anyway, they’re unleashing a complete web site redesign on Monday. I don’t know just what the final version will look like. I know that the beta versions included the incredibly useful, that is to say dumb, feature where if a particular comic you do read doesn’t have an update for the day — and many of them don’t, as they’re weekly or three-times-a-week or so — then it’ll show some other comic in its place. I mean, the idea of encouraging people to find new comics is a good one. To some extent that’s what I do here. But the beta made no distinction between “comic you don’t read because you never heard of Microcosm” and “comic you don’t read because glancing at it makes your eyes bleed”. And on an idiosyncratic note, I read a lot of comics. I don’t need to see Dude and Dude reruns in fourteen spots on my daily comics page, even if I didn’t mind it to start.

    Anyway. I am hoping, desperately hoping, that with the new site all my old links to comics are going to keep working. If they don’t then I suppose I’m just ruined. We’ll see. My suggestion is if you’re at all curious about the comics you read them today (Sunday) just to be safe.

    Ashleigh Brilliant’s Pot-Shots is a curious little strip I never knew of until GoComics picked it up a few years ago. Its format is compellingly simple: a little illustration alongside a wry, often despairing, caption. I love it, but I also understand why was the subject of endless queries to the Detroit Free Press (Or Whatever) about why was this thing taking up newspaper space. The strip rerun the 31st of December is a typical example of the strip and amuses me at least. And it uses arithmetic as the way to communicate reasoning, both good and bad. Brilliant’s joke does address something that logicians have to face, too. Whether an argument is logically valid depends entirely on its structure. If the form is correct the reasoning may be excellent. But to be sound an argument has to be correct and must also have its assumptions be true. We can separate whether an argument is right from whether it could ever possibly be right. If you don’t see the value in that, you have never participated in an online debate about where James T Kirk was born and whether Spock was the first Vulcan in Star Fleet.

    Thom Bluemel’s Birdbrains for the 2nd of January, 2017, is a loaded-dice joke. Is this truly mathematics? Statistics, at least? Close enough for the start of the year, I suppose. Working out whether a die is loaded is one of the things any gambler would like to know, and that mathematicians might be called upon to identify or exploit. (I had a grandmother unshakably convinced that I would have some natural ability to beat the Atlantic City casinos if she could only sneak the underaged me in. I doubt I could do anything of value there besides see the stage magic show.)

    Jack Pullan’s Boomerangs rerun for the 2nd is built on the one bit of statistical mechanics that everybody knows, that something or other about entropy always increasing. It’s not a quantum mechanics rule, but it’s a natural confusion. Quantum mechanics has the reputation as the source of all the most solid, irrefutable laws of the universe’s working. Statistical mechanics and thermodynamics have this musty odor of 19th-century steam engines, no matter how much there is to learn from there. Anyway, the collapse of systems into disorder is not an irrevocable thing. It takes only energy or luck to overcome disorderliness. And in many cases we can substitute time for luck.

    Scott Hilburn’s The Argyle Sweater for the 3rd is the anthropomorphic-geometry-figure joke that’s I’ve been waiting for. I had thought Hilburn did this all the time, although a quick review of Reading the Comics posts suggests he’s been more about anthropomorphic numerals the past year. This is why I log even the boring strips: you never know when I’ll need to check the last time Scott Hilburn used “acute” to mean “cute” in reference to triangles.

    Mike Thompson’s Grand Avenue uses some arithmetic as the visual cue for “any old kind of schoolwork, really”. Steve Breen’s name seems to have gone entirely from the comic strip. On Usenet group rec.arts.comics.strips Brian Henke found that Breen’s name hasn’t actually been on the comic strip since May, and D D Degg found a July 2014 interview indicating Thompson had mostly taken the strip over from originator Breen.

    Mark Anderson’s Andertoons for the 5th is another name-drop that doesn’t have any real mathematics content. But come on, we’re talking Andertoons here. If I skipped it the world might end or something untoward like that.

    'Now for my math homework. I've got a comfortable chair, a good light, plenty of paper, a sharp pencil, a new eraser, and a terrific urge to go out and play some ball.'

    Ted Shearer’s Quincy for the 14th of November, 1977, and reprinted the 7th of January, 2017. I kind of remember having a lamp like that. I don’t remember ever sitting down to do my mathematics homework with a paintbrush.

    Ted Shearer’s Quincy for the 14th of November, 1977, doesn’t have any mathematical content really. Just a mention. But I need some kind of visual appeal for this essay and Shearer is usually good for that.

    Corey Pandolph, Phil Frank, and Joe Troise’s The Elderberries rerun for the 7th is also a very marginal mention. But, what the heck, it’s got some of your standard wordplay about angles and it’ll get this week’s essay that much closer to 800 words.

     
  • Joseph Nebus 6:00 pm on Thursday, 5 January, 2017 Permalink | Reply
    Tags: , , , , mathematics history, recap,   

    What I Learned Doing The End 2016 Mathematics A To Z 


    The slightest thing I learned in the most recent set of essays is that I somehow slid from the descriptive “End Of 2016” title to the prescriptive “End 2016” identifier for the series. My unscientific survey suggests that most people would agree that we had too much 2016 and would have been better off doing without it altogether. So it goes.

    The most important thing I learned about this is I have to pace things better. The A To Z essays have been creeping up in length. I didn’t keep close track of their lengths but I don’t think any of them came in under a thousand words. 1500 words was more common. And that’s fine enough, but at three per week, plus the Reading the Comics posts, that’s 5500 or 6000 words of mathematics alone. And that before getting to my humor blog, which even on a brief week will be a couple thousand words. I understand in retrospect why November and December felt like I didn’t have any time outside the word mines.

    I’m not bothered by writing longer essays, mind. I can apparently go on at any length on any subject. And I like the words I’ve been using. My suspicion is between these A To Zs and the Theorem Thursdays over the summer I’ve found a mode for writing pop mathematics that works for me. It’s just a matter of how to balance workloads. The humor blog has gotten consistently better readership, for the obvious reasons (lately I’ve been trying to explain what the story comics are doing), but the mathematics more satisfying. If I should have to cut back on either it’d be the humor blog that gets the cut first.

    Another little discovery is that I can swap out equations and formulas and the like for historical discussion. That’s probably a useful tradeoff for most of my readers. And it plays to my natural tendencies. It is very easy to imagine me having gone into history than into mathematics or science. It makes me aware how mediocre my knowledge of mathematics history is, though. For example, several times in the End 2016 A To Z the Crisis of Foundations came up, directly or in passing. But I’ve never read a proper history, not even a basic essay, about the Crisis. I don’t even know of a good description of this important-to-the-field event. Most mathematics history focuses around biographies of a few figures, often cribbed from Eric Temple Bell’s great but unreliable book, or a couple of famous specific incidents. (Newton versus Leibniz, the bridges of Köningsburg, Cantor’s insanity, Gödel’s citizenship exam.) Plus Bourbaki.

    That’s not enough for someone taking the subject seriously, and I do mean to. So if someone has a suggestion for good histories of, for example, how Fourier series affected mathematicians’ understanding of what functions are, I’d love to know it. Maybe I should set that as a standing open request.

    In looking over the subjects I wrote about I find a pretty strong mix of group theory and real analysis. Maybe that shouldn’t surprise. Those are two of the maybe three legs that form a mathematics major’s education. So anyone wanting to understand mathematicians would see this stuff and have questions about it. (There are more things mathematics majors learn, but there are a handful of things almost any mathematics major is sure to spend a year being baffled by.)

    The third leg, I’d say, is differential equations. That’s a fantastic field, but it’s hard to describe without equations. Also pictures of what the equations imply. I’ve tended towards essays with few equations and pictures. That’s my laziness. Equations are best written in LaTeX, a typesetting tool that might as well be the standard for mathematicians writing papers and books. While WordPress supports a bit of LaTeX it isn’t quite effortless. That comes back around to balancing my workload. I do that a little better and I can explain solving first-order differential equations by integrating factors. (This is a prank. Nobody has ever needed to solve a first-order differential equation by integrating factors except for mathematics majors being taught the method.) But maybe I could make a go of that.

    I’m not setting any particular date for the next A-To-Z, or similar, project. I need some time to recuperate. And maybe some time to think of other running projects that would be fun or educational for me. There’ll be something, though.

     
  • Joseph Nebus 6:00 pm on Tuesday, 3 January, 2017 Permalink | Reply
    Tags: , , , ,   

    The End 2016 Mathematics A To Z Roundup 


    As is my tradition for the end of these roundups (see Summer 2015 and then Leap Day 2016) I want to just put up a page listing the whole set of articles. It’s a chance for people who missed a piece to easily see what they missed. And it lets me recover that little bit extra from the experience. Run over the past two months were:

     
  • Joseph Nebus 6:00 pm on Sunday, 1 January, 2017 Permalink | Reply
    Tags: , , Bad Machinery, Buni, Daily Drawing, , Madeline L'Engle, , , , Speechless, Wrong Hands   

    Reading the Comics, December 30, 2016: New Year’s Eve Week Edition 


    So last week, for schedule reasons, I skipped the Christmas Eve strips and promised to get to them this week. There weren’t any Christmas Eve mathematically-themed comic strips. Figures. This week, I need to skip New Year’s Eve comic strips for similar schedule reasons. If there are any, I’ll talk about them next week.

    Lorie Ransom’s The Daily Drawing for the 28th is a geometry wordplay joke for this installment. Two of them, when you read the caption.

    John Graziano’s Ripley’s Believe It or Not for the 28th presents the quite believable claim that Professor Dwight Barkley created a formula to estimate how long it takes a child to ask “are we there yet?” I am skeptical the equation given means all that much. But it’s normal mathematician-type behavior to try modelling stuff. That will usually start with thinking of what one wants to represent, and what things about it could be measured, and how one expects these things might affect one another. There’s usually several plausible-sounding models and one has to select the one or ones that seem likely to be interesting. They have to be simple enough to calculate, but still interesting. They need to have consequences that aren’t obvious. And then there’s the challenge of validating the model. Does its description match the thing we’re interested in well enough to be useful? Or at least instructive?

    Len Borozinski’s Speechless for the 28th name-drops Albert Einstein and the theory of relativity. Marginal mathematical content, but it’s a slow week.

    John Allison’s Bad Machinery for the 29th mentions higher dimensions. More dimensions. In particular it names ‘ana’ and ‘kata’ as “the weird extra dimensions”. Ana and kata are a pair of directions coined by the mathematician Charles Howard Hinton to give us a way of talking about directions in hyperspace. They echo the up/down, left/right, in/out pairs. I don’t know that any mathematicians besides Rudy Rucker actually use these words, though, and that in his science fiction. I may not read enough four-dimensional geometry to know the working lingo. Hinton also coined the “tesseract”, which has escaped from being a mathematician’s specialist term into something normal people might recognize. Mostly because of Madeline L’Engle, I suppose, but that counts.

    Samson’s Dark Side of the Horse for the 29th is Dark Side of the Horse‘s entry this essay. It’s a fun bit of play on counting, especially as a way to get to sleep.

    John Graziano’s Ripley’s Believe It or Not for the 29th mentions a little numbers and numerals project. Or at least representations of numbers. Finding other orders for numbers can be fun, and it’s a nice little pastime. I don’t know there’s an important point to this sort of project. But it can be fun to accomplish. Beautiful, even.

    Mark Anderson’s Andertoons for the 30th relieves us by having a Mark Anderson strip for this essay. And makes for a good Roman numerals gag.

    Ryan Pagelow’s Buni for the 30th can be counted as an anthropomorphic-numerals joke. I know it’s more of a “ugh 2016 was the worst year” joke, but it parses either way.

    John Atkinson’s Wrong Hands for the 30th is an Albert Einstein joke. It’s cute as it is, though.

     
  • Joseph Nebus 6:00 pm on Saturday, 31 December, 2016 Permalink | Reply
    Tags: 19th Century, , Axiom of Choice, continuum hypothesis, , , , , ZFC   

    The End 2016 Mathematics A To Z: Zermelo-Fraenkel Axioms 


    gaurish gave me a choice for the Z-term to finish off the End 2016 A To Z. I appreciate it. I’m picking the more abstract thing because I’m not sure that I can explain zero briefly. The foundations of mathematics are a lot easier.

    Zermelo-Fraenkel Axioms

    I remember the look on my father’s face when I asked if he’d tell me what he knew about sets. He misheard what I was asking about. When we had that straightened out my father admitted that he didn’t know anything particular. I thanked him and went off disappointed. In hindsight, I kind of understand why everyone treated me like that in middle school.

    My father’s always quick to dismiss how much mathematics he knows, or could understand. It’s a common habit. But in this case he was probably right. I knew a bit about set theory as a kid because I came to mathematics late in the “New Math” wave. Sets were seen as fundamental to why mathematics worked without being so exotic that kids couldn’t understand them. Perhaps so; both my love and I delighted in what we got of set theory as kids. But if you grew up before that stuff was popular you probably had a vague, intuitive, and imprecise idea of what sets were. Mathematicians had only a vague, intuitive, and imprecise idea of what sets were through to the late 19th century.

    And then came what mathematics majors hear of as the Crisis of Foundations. (Or a similar name, like Foundational Crisis. I suspect there are dialect differences here.) It reflected mathematics taking seriously one of its ideals: that everything in it could be deduced from clearly stated axioms and definitions using logically rigorous arguments. As often happens, taking one’s ideals seriously produces great turmoil and strife.

    Before about 1900 we could get away with saying that a set was a bunch of things which all satisfied some description. That’s how I would describe it to a new acquaintance if I didn’t want to be treated like I was in middle school. The definition is fine if we don’t look at it too hard. “The set of all roots of this polynomial”. “The set of all rectangles with area 2”. “The set of all animals with four-fingered front paws”. “The set of all houses in Central New Jersey that are yellow”. That’s all fine.

    And then if we try to be logically rigorous we get problems. We always did, though. They’re embodied by ancient jokes like the person from Crete who declared that all Cretans always lie; is the statement true? Or the slightly less ancient joke about the barber who shaves only the men who do not shave themselves; does he shave himself? If not jokes these should at least be puzzles faced in fairy-tale quests. Logicians dressed this up some. Bertrand Russell gave us the quite respectable “The set consisting of all sets which are not members of themselves”, and asked us to stare hard into that set. To this we have only one logical response, which is to shout, “Look at that big, distracting thing!” and run away. This satisfies the problem only for a while.

    The while ended in — well, that took a while too. But between 1908 and the early 1920s Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem paused from arguing whose name would also be the best indie rock band name long enough to put set theory right. Their structure is known as Zermelo-Fraenkel Set Theory, or ZF. It gives us a reliable base for set theory that avoids any contradictions or catastrophic pitfalls. Or does so far as we have found in a century of work.

    It’s built on a set of axioms, of course. Most of them are uncontroversial, things like declaring two sets are equivalent if they have the same elements. Declaring that the union of sets is itself a set. Obvious, sure, but it’s the obvious things that we have to make axioms. Maybe you could start an argument about whether we should just assume there exists some infinitely large set. But if we’re aware sets probably have something to teach us about numbers, and that numbers can get infinitely large, then it seems fair to suppose that there must be some infinitely large set. The axioms that aren’t simple obvious things like that are too useful to do without. They assume stuff like that no set is an element of itself. Or that every set has a “power set”, a new set comprised of all the subsets of the original set. Good stuff to know.

    There is one axiom that’s controversial. Not controversial the way Euclid’s Parallel Postulate was. That’s ugly one about lines crossing another line meeting on the same side they make angles smaller than something something or other. That axiom was controversial because it read so weird, so needlessly complicated. (It isn’t; it’s exactly as complicated as it must be. Or better, it’s as simple as it could possibly be and still be useful.) The controversial axiom of Zermelo-Fraenkel Set Theory is known as the Axiom of Choice. It says if we have a collection of mutually disjoint sets, each with at least one thing in them, then it’s possible to pick exactly one item from each of the sets.

    It’s impossible to dispute this is what we have axioms for. It’s about something that feels like it should be obvious: we can always pick something from a set. How could this not be true?

    If it is true, though, we get some unsavory conclusions. For example, it becomes possible to take a ball the size of an orange and slice it up. We slice using mathematical blades. They’re not halted by something as petty as the desire not to slice atoms down the middle. We can reassemble the pieces. Into two balls. And worse, it doesn’t require we do something like cut the orange into infinitely many pieces. We expect crazy things to happen when we let infinities get involved. No, though, we can do this cut-and-duplicate thing by cutting the orange into five pieces. When you hear that it’s hard to know whether to point to the big, distracting thing and run away. If we dump the Axiom of Choice we don’t have that problem. But can we do anything useful without the ability to make a choice like that?

    And we’ve learned that we can. If we want to use the Zermelo-Fraenkel Set Theory with the Axiom of Choice we say we were working in “ZFC”, Zermelo-Fraenkel-with-Choice. We don’t have to. If we don’t want to make any assumption about choices we say we’re working in “ZF”. Which to use depends on what one wants to use.

    Either way Zermelo and Fraenkel and Skolem established set theory on the foundation we use to this day. We’re not required to use them, no; there’s a construction called von Neumann-Bernays-Gödel Set Theory that’s supposed to be more elegant. They didn’t mention it in my logic classes that I remember, though.

    And still there’s important stuff we would like to know which even ZFC can’t answer. The most famous of these is the continuum hypothesis. Everyone knows — excuse me. That’s wrong. Everyone who would be reading a pop mathematics blog knows there are different-sized infinitely-large sets. And knows that the set of integers is smaller than the set of real numbers. The question is: is there a set bigger than the integers yet smaller than the real numbers? The Continuum Hypothesis says there is not.

    Zermelo-Fraenkel Set Theory, even though it’s all about the properties of sets, can’t tell us if the Continuum Hypothesis is true. But that’s all right; it can’t tell us if it’s false, either. Whether the Continuum Hypothesis is true or false stands independent of the rest of the theory. We can assume whichever state is more useful for our work.

    Back to the ideals of mathematics. One question that produced the Crisis of Foundations was consistency. How do we know our axioms don’t contain a contradiction? It’s hard to say. Typically a set of axioms we can prove consistent are also a set too boring to do anything useful in. Zermelo-Fraenkel Set Theory, with or without the Axiom of Choice, has a lot of interesting results. Do we know the axioms are consistent?

    No, not yet. We know some of the axioms are mutually consistent, at least. And we have some results which, if true, would prove the axioms to be consistent. We don’t know if they’re true. Mathematicians are generally confident that these axioms are consistent. Mostly on the grounds that if there were a problem something would have turned up by now. It’s withstood all the obvious faults. But the universe is vaster than we imagine. We could be wrong.

    It’s hard to live up to our ideals. After a generation of valiant struggling we settle into hoping we’re doing good enough. And waiting for some brilliant mind that can get us a bit closer to what we ought to be.

     
    • elkement (Elke Stangl) 10:42 am on Sunday, 1 January, 2017 Permalink | Reply

      Very interesting – as usual! I was also subjected to the New Math in elementary school – the upside was that you got a lot of nice toys for free, as ‘add-ons’ to school books ( … plastic cubes and other toy blocks that should represents members of sets …). Not sure if it prepared one better to understand Russell’s paradox later ;-)

      Liked by 1 person

      • elkement (Elke Stangl) 10:43 am on Sunday, 1 January, 2017 Permalink | Reply

        … and I wish you a Happy New Year and more A-Zs in 2017 :-)

        Liked by 1 person

        • Joseph Nebus 5:34 am on Thursday, 5 January, 2017 Permalink | Reply

          Thanks kindly. I am going to do a fresh A-to-Z, although I don’t know just when. Not in January; haven’t got the energy for it right away.

          Liked by 1 person

      • Joseph Nebus 5:34 am on Thursday, 5 January, 2017 Permalink | Reply

        Oh, now, the toys were fantastic. I suppose it’s a fair guess whether the people who got something out of the New Math got it because they understood fundamentals better in that form or whether it was just that the toys and games made the subject more engaging.

        I am, I admit, a fan of the New Math, but that may just be because it’s the way I learned mathematics, and the way you did something as a kid is always the one natural way to do it.

        Liked by 1 person

  • Joseph Nebus 6:00 pm on Thursday, 29 December, 2016 Permalink | Reply
    Tags: , China, , , , , Mersenne numbers, , ,   

    The End 2016 Mathematics A To Z: Yang Hui’s Triangle 


    Today’s is another request from gaurish and another I’m glad to have as it let me learn things too. That’s a particularly fun kind of essay to have here.

    Yang Hui’s Triangle.

    It’s a triangle. Not because we’re interested in triangles, but because it’s a particularly good way to organize what we’re doing and show why we do that. We’re making an arrangement of numbers. First we need cells to put the numbers in.

    Start with a single cell in what’ll be the top middle of the triangle. It spreads out in rows beneath that. The rows are staggered. The second row has two cells, each one-half width to the side of the starting one. The third row has three cells, each one-half width to the sides of the row above, so that its center cell is directly under the original one. The fourth row has four cells, two of which are exactly underneath the cells of the second row. The fifth row has five cells, three of them directly underneath the third row’s cells. And so on. You know the pattern. It’s the one that pins in a plinko board take. Just trimmed down to a triangle. Make as many rows as you find interesting. You can always add more later.

    In the top cell goes the number ‘1’. There’s also a ‘1’ in the leftmost cell of each row, and a ‘1’ in the rightmost cell of each row.

    What of interior cells? The number for those we work out by looking to the row above. Take the cells to the immediate left and right of it. Add the values of those together. So for example the center cell in the third row will be ‘1’ plus ‘1’, commonly regarded as ‘2’. In the third row the leftmost cell is ‘1’; it always is. The next cell over will be ‘1’ plus ‘2’, from the row above. That’s ‘3’. The cell next to that will be ‘2’ plus ‘1’, a subtly different ‘3’. And the last cell in the row is ‘1’ because it always is. In the fourth row we get, starting from the left, ‘1’, ‘4’, ‘6’, ‘4’, and ‘1’. And so on.

    It’s a neat little arithmetic project. It has useful application beyond the joy of making something neat. Many neat little arithmetic projects don’t have that. But the numbers in each row give us binomial coefficients, which we often want to know. That is, if we wanted to work out (a + b) to, say, the third power, we would know what it looks like from looking at the fourth row of Yanghui’s Triangle. It will be 1\cdot a^4 + 4\cdot a^3 \cdot b^1 + 6\cdot a^2\cdot b^2 + 4\cdot a^1\cdot b^3 + 1\cdot b^4 . This turns up in polynomials all the time.

    Look at diagonals. By diagonal here I mean a line parallel to the line of ‘1’s. Left side or right side; it doesn’t matter. Yang Hui’s triangle is bilaterally symmetric around its center. The first diagonal under the edges is a bit boring but familiar enough: 1-2-3-4-5-6-7-et cetera. The second diagonal is more curious: 1-3-6-10-15-21-28 and so on. You’ve seen those numbers before. They’re called the triangular numbers. They’re the number of dots you need to make a uniformly spaced, staggered-row triangle. Doodle a bit and you’ll see. Or play with coins or pool balls.

    The third diagonal looks more arbitrary yet: 1-4-10-20-35-56-84 and on. But these are something too. They’re the tetrahedronal numbers. They’re the number of things you need to make a tetrahedron. Try it out with a couple of balls. Oranges if you’re bored at the grocer’s. Four, ten, twenty, these make a nice stack. The fourth diagonal is a bunch of numbers I never paid attention to before. 1-5-15-35-70-126-210 and so on. This is — well. We just did tetrahedrons, the triangular arrangement of three-dimensional balls. Before that we did triangles, the triangular arrangement of two-dimensional discs. Do you want to put in a guess what these “pentatope numbers” are about? Sure, but you hardly need to. If we’ve got a bunch of four-dimensional hyperspheres and want to stack them in a neat triangular pile we need one, or five, or fifteen, or so on to make the pile come out neat. You can guess what might be in the fifth diagonal. I don’t want to think too hard about making triangular heaps of five-dimensional hyperspheres.

    There’s more stuff lurking in here, waiting to be decoded. Add the numbers of, say, row four up and you get two raised to the third power. Add the numbers of row ten up and you get two raised to the ninth power. You see the pattern. Add everything in, say, the top five rows together and you get the fifth Mersenne number, two raised to the fifth power (32) minus one (31, when we’re done). Add everything in the top ten rows together and you get the tenth Mersenne number, two raised to the tenth power (1024) minus one (1023).

    Or add together things on “shallow diagonals”. Start from a ‘1’ on the outer edge. I’m going to suppose you started on the left edge, but remember symmetry; it’ll be fine if you go from the right instead. Add to that ‘1’ the number you get by moving one cell to the right and going up-and-right. And then again, go one cell to the right and then one cell up-and-right. And again and again, until you run out of cells. You get the Fibonacci sequence, 1-1-2-3-5-8-13-21-and so on.

    We can even make an astounding picture from this. Take the cells of Yang Hui’s triangle. Color them in. One shade if the cell has an odd number, another if the cell has an even number. It will create a pattern we know as the Sierpiński Triangle. (Wacław Sierpiński is proving to be the surprise special guest star in many of this A To Z sequence’s essays.) That’s the fractal of a triangle subdivided into four triangles with the center one knocked out, and the remaining triangles them subdivided into four triangles with the center knocked out, and on and on.

    By now I imagine even my most skeptical readers agree this is an interesting, useful mathematical construct. Also that they’re wondering why I haven’t said the name “Blaise Pascal”. The Western mathematical tradition knows of this from Pascal’s work, particularly his 1653 Traité du triangle arithmétique. But mathematicians like to say their work is universal, and independent of the mere human beings who find it. Constructions like this triangle give support to this. Yang lived in China, in the 12th century. I imagine it possible Pascal had hard of his work or been influenced by it, by some chain, but I know of no evidence that he did.

    And even if he had, there are other apparently independent inventions. The Avanti Indian astronomer-mathematician-astrologer Varāhamihira described the addition rule which makes the triangle work in commentaries written around the year 500. Omar Khayyám, who keeps appearing in the history of science and mathematics, wrote about the triangle in his 1070 Treatise on Demonstration of Problems of Algebra. Again so far as I am aware there’s not a direct link between any of these discoveries. They are things different people in different traditions found because the tools — arithmetic and aesthetically-pleasing orders of things — were ready for them.

    Yang Hui wrote about his triangle in the 1261 book Xiangjie Jiuzhang Suanfa. In it he credits the use of the triangle (for finding roots) was invented around 1100 by mathematician Jia Xian. This reminds us that it is not merely mathematical discoveries that are found by many peoples at many times and places. So is Boyer’s Law, discovered by Hubert Kennedy.

     
    • gaurish 6:46 pm on Thursday, 29 December, 2016 Permalink | Reply

      This is first time that I have read an article about Pascal triangle without a picture of it in front of me and could still imagine it in my mind. :)

      Like

      • Joseph Nebus 5:22 am on Thursday, 5 January, 2017 Permalink | Reply

        Thank you; I’m glad you like it. I did spend a good bit of time before writing the essay thinking about why it is a triangle that we use for this figure, and that helped me think about how things are organized and why. (The one thing I didn’t get into was identifying the top row, the single cell, as row zero. Computers may index things starting from zero and there may be fair reasons to do it, but that is always going to be a weird choice for humans.)

        Liked by 1 person

  • Joseph Nebus 6:00 pm on Tuesday, 27 December, 2016 Permalink | Reply
    Tags: , , , Riemann hypothesis,   

    The End 2016 Mathematics A To Z: Xi Function 


    I have today another request from gaurish, who’s also been good enough to give me requests for ‘Y’ and ‘Z’. I apologize for coming to this a day late. But it was Christmas and many things demanded my attention.

    Xi Function.

    We start with complex-valued numbers. People discovered them because they were useful tools to solve polynomials. They turned out to be more than useful fictions, if numbers are anything more than useful fictions. We can add and subtract them easily. Multiply and divide them less easily. We can even raise them to powers, or raise numbers to them.

    If you become a mathematics major then somewhere in Intro to Complex Analysis you’re introduced to an exotic, infinitely large sum. It’s spoken of reverently as the Riemann Zeta Function, and it connects to something named the Riemann Hypothesis. Then you remember that you’ve heard of this, because if you’re willing to become a mathematics major you’ve read mathematics popularizations. And you know the Riemann Hypothesis is an unsolved problem. It proposes something that might be true or might be false. Either way has astounding implications for the way numbers fit together.

    Riemann here is Bernard Riemann, who’s turned up often in these A To Z sequences. We saw him in spheres and in sums, leading to integrals. We’ll see him again. Riemann just covered so much of 19th century mathematics; we can’t talk about calculus without him. Zeta, Xi, and later on, Gamma are the famous Greek letters. Mathematicians fall back on them because the Roman alphabet just hasn’t got enough letters for our needs. I’m writing them out as English words instead because if you aren’t familiar with them they look like an indistinct set of squiggles. Even if you are familiar, sometimes. I got confused in researching this some because I did slip between a lowercase-xi and a lowercase-zeta in my mind. All I can plead is it’s been a hard week.

    Riemann’s Zeta function is famous. It’s easy to approach. You can write it as a sum. An infinite sum, but still, those are easy to understand. Pick a complex-valued number. I’ll call it ‘s’ because that’s the standard. Next take each of the counting numbers: 1, 2, 3, and so on. Raise each of them to the power ‘s’. And take the reciprocal, one divided by those numbers. Add all that together. You’ll get something. Might be real. Might be complex-valued. Might be zero. We know many values of ‘s’ what would give us a zero. The Riemann Hypothesis is about characterizing all the possible values of ‘s’ that give us a zero. We know some of them, so boring we call them trivial: -2, -4, -6, -8, and so on. (This looks crazy. There’s another way of writing the Riemann Zeta function which makes it obvious instead.) The Riemann Hypothesis is about whether all the proper, that is, non-boring values of ‘s’ that give us a zero are 1/2 plus some imaginary number.

    It’s a rare thing mathematicians have only one way of writing. If something’s been known and studied for a long time there are usually variations. We find different ways to write the problem. Or we find different problems which, if solved, would solve the original problem. The Riemann Xi function is an example of this.

    I’m going to spare you the formula for it. That’s in self-defense. I haven’t found an expression of the Xi function that isn’t a mess. The normal ways to write it themselves call on the Zeta function, as well as the Gamma function. The Gamma function looks like factorials, for the counting numbers. It does its own thing for other complex-valued numbers.

    That said, I’m not sure what the advantages are in looking at the Xi function. The one that people talk about is its symmetry. Its value at a particular complex-valued number ‘s’ is the same as its value at the number ‘1 – s’. This may not seem like much. But it gives us this way of rewriting the Riemann Hypothesis. Imagine all the complex-valued numbers with the same imaginary part. That is, all the numbers that we could write as, say, ‘x + 4i’, where ‘x’ is some real number. If the size of the value of Xi, evaluated at ‘x + 4i’, always increases as ‘x’ starts out equal to 1/2 and increases, then the Riemann hypothesis is true. (This has to be true not just for ‘x + 4i’, but for all possible imaginary numbers. So, ‘x + 5i’, and ‘x + 6i’, and even ‘x + 4.1 i’ and so on. But it’s easier to start with a single example.)

    Or another way to write it. Suppose the size of the value of Xi, evaluated at ‘x + 4i’ (or whatever), always gets smaller as ‘x’ starts out at a negative infinitely large number and keeps increasing all the way to 1/2. If that’s true, and true for every imaginary number, including ‘x – i’, then the Riemann hypothesis is true.

    And it turns out if the Riemann hypothesis is true we can prove the two cases above. We’d write the theorem about this in our papers with the start ‘The Following Are Equivalent’. In our notes we’d write ‘TFAE’, which is just as good. Then we’d take which ever of them seemed easiest to prove and find out it isn’t that easy after all. But if we do get through we declare ourselves fortunate, sit back feeling triumphant, and consider going out somewhere to celebrate. But we haven’t got any of these alternatives solved yet. None of the equivalent ways to write it has helped so far.

    We know some some things. For example, we know there are infinitely many roots for the Xi function with a real part that’s 1/2. This is what we’d need for the Riemann hypothesis to be true. But we don’t know that all of them are.

    The Xi function isn’t entirely about what it can tell us for the Zeta function. The Xi function has its own exotic and wonderful properties. In a 2009 paper on arxiv.org, for example, Drs Yang-Hui He, Vishnu Jejjala, and Djordje Minic describe how if the zeroes of the Xi function are all exactly where we expect them to be then we learn something about a particular kind of string theory. I admit not knowing just what to say about a genus-one free energy of the topological string past what I have read in this paper. In another paper they write of how the zeroes of the Xi function correspond to the description of the behavior for a quantum-mechanical operator that I just can’t find a way to describe clearly in under three thousand words.

    But mathematicians often speak of the strangeness that mathematical constructs can match reality so well. And here is surely a powerful one. We learned of the Riemann Hypothesis originally by studying how many prime numbers there are compared to the counting numbers. If it’s true, then the physics of the universe may be set up one particular way. Is that not astounding?

     
    • gaurish 5:34 am on Wednesday, 28 December, 2016 Permalink | Reply

      Yes it’s astounding. You have a very nice talent of talking about mathematical quantities without showing formulas :)

      Liked by 1 person

      • Joseph Nebus 5:15 am on Thursday, 5 January, 2017 Permalink | Reply

        You’re most kind, thank you. I’ve probably gone overboard in avoiding formulas lately though.

        Like

  • Joseph Nebus 6:00 pm on Sunday, 25 December, 2016 Permalink | Reply  

    Reading the Comics, December 23, 2016: Weak Pretexts Edition 


    I’ve set the cutoff for the strips this week at Friday because you know how busy the day before Christmas is. If you don’t, then I ask that you trust me: it’s busy. If Comic Strip Master Command sent me anything worthy of comment I’ll report on it next year. I had thought this week’s set of mathematically-themed comics were a weak bunch that I had to strain to justify covering. But on looking at the whole essay … mm. I’m comfortable with it.

    Bill Amend’s FoxTrot for the 18th has two mathematical references. Writing “three kings” as “square root of nine kings” is an old sort of joke at least in mathematical circles. It’s about writing a number using some elaborate but valid expression. It’s good fun. A few years back I had a calendar with a mathematics puzzle of the day. And that was fun up until I noticed the correct answer was always the day of the month so that, say, today’s would have an answer of 25. This collapsed the calendar from being about solving problems to just verifying the solution. It’s a little change but it is one that spoiled it for me.

    And a few years back an aunt and uncle gave me an “Irrational Watch”, with the dial marked only by irrational numbers — π, for example, a little bit clockwise of where ‘3’ ought to go. The one flaw: it used the Euler-Mascheroni constant, a number that’s about 0.57, to designate that time a little past 12:30. The Euler-Mascheroni constant isn’t actually known to be irrational. It’s the way to bet, but it might just be a rational number after all.

    A binary tree, mentioned in the bottom row, is a tree for which every vertex is connected to at most three others. We’ve seen trees recently. And there’s a specially designated vertex known as the root. Each vertex (except the root) is connected to exactly one vertex that’s closer to the root. The structure looks like either the branches or the roots of a tree, depending whether the root is put at the top or bottom. And if you accept a strikingly minimal, Mid-Century Modern style drawing of a (natural) tree.

    Quincy, in monologue: 'I don't understand this math lesson. What I need most is some excuse for not doing my homework. And all I can think of right now is a city-wide blackout.'

    Ted Shearer’s Quincy for the 26th of October, 1977. Reprinted the 20th of December, 2016. There’s much that’s expressive here, although what most catches my eye is the swoopy curves of the soda straw.

    Ted Shearer’s Quincy for the 26th of October, 1977 mentions mathematics. So I’m using that as an excuse to include it. Mostly I like it for the artwork. But the mention of mathematics was strikingly arbitrary; the joke would be the same were it his English homework or Geography or anything else. I suppose mathematics got the nod because it can be written with so few letters. (Art is even more compact, but it would probably distract the reader trying to think of what would be hard to understand about an Art project homework. Difficult, if it were painting or crafting something, sure, but that’s not a challenge in understanding.)

    I don’t know what Marty Links’s Emmy Lou for the 20th was getting at. The strip originally ran the 15th of September, 1964. It seems to be referring to some ephemeral trivia passed around the summer of that year. My guess is it refers to some estimation of the unattached male and female populations of some age set and finding that there were very different numbers. That sort of result is typically done by clever definition, sometimes assisted by double-counting and other sleights of hand. It’s legitimate if you accept the definition. But before reacting too much to any surprising claim one should know just what the claim is, and why it’s that.

    Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is mathematics wordplay. Simpson’s Approximation, mentioned here, is a calculus thing. It’s about integrals. We use it to estimate the value of an integral. We find a parabola that resembles the original function. And then the integral of the parabola should be close to the integral of the original function. The advantage of using a parabola is that we know exactly how to integrate that. You may have noticed a lot of calculus is built on finding a problem that we can do that looks enough like the one we want to do. There’s also a Simpson’s 3/8th Approximation. It uses a cubic polynomial instead of a parabola for the approximation. We can integrate cubics exactly too. It’s called the 3/8 Approximation, or the 3/8 Rule, because the formula for it starts off with a 3/8. So now and then a mathematics thing is named appropriately. Simpson’s Approximation is named for Thomas Simpson, an 18th century mathematician and textbook writer who did show the world the 3/8 Approximation. But other people are known to have used Simpson’s non-3/8 Approximation a century or more before Simpson was born .

    Jason Poland’s Robbie and Bobby seeks much but slight attention from me this week. The first, from the 21st, riffs on the “randomness” of the random acts of kindness. Robbie strives for a truly random act. Randomness is tricky. We’re pretty sure we know what it ought to look like, but it’s so very hard to ever be sure we have randomness. We have a set of possible outcomes of whatever we’re doing; but, should every one of those outcomes be equally likely? Should some be more likely than others? Should some be almost inevitable but a few have long-shot chances? I suspect that when we say “truly random” we are thinking of a uniform distribution, with every different outcome being equally likely. That isn’t always what fits the situation.

    And then on the 23rd the strip names “Zeno’s Paradoxical Pasta”. There are several paradoxes; this surely refers to the one about not being able to cross a distance because one must always get halfway across the distance first. It’s a reliably funny idea. It’s not a paradox by itself, though. What makes the paradox is that Zeno presents several scenarios which ask that we decide whether space and time and movement can be infinitely subdivided or not, and either decision brings up new difficulties.

     
  • Joseph Nebus 6:00 pm on Friday, 23 December, 2016 Permalink | Reply
    Tags: , , , , ,   

    The End 2016 Mathematics A To Z: Weierstrass Function 


    I’ve teased this one before.

    Weierstrass Function.

    So you know how the Earth is a sphere, but from our normal vantage point right up close to its surface it looks flat? That happens with functions too. Here I mean the normal kinds of functions we deal with, ones with domains that are the real numbers or a Euclidean space. And ranges that are real numbers. The functions you can draw on a sheet of paper with some wiggly bits. Let the function wiggle as much as you want. Pick a part of it and zoom in close. That zoomed-in part will look straight. If it doesn’t look straight, zoom in closer.

    We rely on this. Functions that are straight, or at least straight enough, are easy to work with. We can do calculus on them. We can do analysis on them. Functions with plots that look like straight lines are easy to work with. Often the best approach to working with the function you’re interested in is to approximate it with an easy-to-work-with function. I bet it’ll be a polynomial. That serves us well. Polynomials are these continuous functions. They’re differentiable. They’re smooth.

    That thing about the Earth looking flat, though? That’s a lie. I’ve never been to any of the really great cuts in the Earth’s surface, but I have been to some decent gorges. I went to grad school in the Hudson River Valley. I’ve driven I-80 over Pennsylvania’s scariest bridges. There’s points where the surface of the Earth just drops a great distance between your one footstep and your last.

    Functions do that too. We can have points where a function isn’t differentiable, where it’s impossible to define the direction it’s headed. We can have points where a function isn’t continuous, where it jumps from one region of values to another region. Everyone knows this. We can’t dismiss those as abberations not worthy of the name “function”; too many of them are too useful. Typically we handle this by admitting there’s points that aren’t continuous and we chop the function up. We make it into a couple of functions, each stretching from discontinuity to discontinuity. Between them we have continuous region and we can go about our business as before.

    Then came the 19th century when things got crazy. This particular craziness we credit to Karl Weierstrass. Weierstrass’s name is all over 19th century analysis. He had that talent for probing the limits of our intuition about basic mathematical ideas. We have a calculus that is logically rigorous because he found great counterexamples to what we had assumed without proving.

    The Weierstrass function challenges this idea that any function is going to eventually level out. Or that we can even smooth a function out into basically straight, predictable chunks in-between sudden changes of direction. The function is continuous everywhere; you can draw it perfectly without lifting your pen from paper. But it always looks like a zig-zag pattern, jumping around like it was always randomly deciding whether to go up or down next. Zoom in on any patch and it still jumps around, zig-zagging up and down. There’s never an interval where it’s always moving up, or always moving down, or even just staying constant.

    Despite being continuous it’s not differentiable. I’ve described that casually as it being impossible to predict where the function is going. That’s an abuse of words, yes. The function is defined. Its value at a point isn’t any more random than the value of “x2” is for any particular x. The unpredictability I’m talking about here is a side effect of ignorance. Imagine I showed you a plot of “x2” with a part of it concealed and asked you to fill in the gap. You’d probably do pretty well estimating it. The Weierstrass function, though? No; your guess would be lousy. My guess would be lousy too.

    That’s a weird thing to have happen. A century and a half later it’s still weird. It gets weirder. The Weierstrass function isn’t differentiable generally. But there are exceptions. There are little dots of differentiability, where the rate at which the function changes is known. Not intervals, though. Single points. This is crazy. Derivatives are about how a function changes. We work out what they should even mean by thinking of a function’s value on strips of the domain. Those strips are small, but they’re still, you know, strips. But on almost all of that strip the derivative isn’t defined. It’s only at isolated points, a set with measure zero, that this derivative even exists. It evokes the medieval Mysteries, of how we are supposed to try, even though we know we shall fail, to understand how God can have contradictory properties.

    It’s not quite that Mysterious here. Properties like this challenge our intuition, if we’ve gotten any. Once we’ve laid out good definitions for ideas like “derivative” and “continuous” and “limit” and “function” we can work out whether results like this make sense. And they — well, they follow. We can avoid weird conclusions like this, but at the cost of messing up our definitions for what a “function” and other things are. Making those useless. For the mathematical world to make sense, we have to change our idea of what quite makes sense.

    That’s all right. When we look close we realize the Earth around us is never flat. Even reasonably flat areas have slight rises and falls. The ends of properties are marked with curbs or ditches, and bordered by streets that rise to a center. Look closely even at the dirt and we notice that as level as it gets there are still rocks and scratches in the ground, clumps of dirt an infinitesimal bit higher here and lower there. The flatness of the Earth around us is a useful tool, but we miss a lot by pretending it’s everything. The Weierstrass function is one of the ways a student mathematician learns that while smooth, predictable functions are essential, there is much more out there.

     
  • Joseph Nebus 6:00 pm on Wednesday, 21 December, 2016 Permalink | Reply
    Tags: , compression, , Markov Chains, , ,   

    The End 2016 Mathematics A To Z: Voronoi Diagram 


    This is one I never heard of before grad school. And not my first year in grad school either; I was pretty well past the point I should’ve been out of grad school before I remember hearing of it, somehow. I can’t explain that.

    Voronoi Diagram.

    Take a sheet of paper. Draw two dots on it. Anywhere you like. It’s your paper. But here’s the obvious thing: you can divide the paper into the parts of it that are nearer to the first, or that are nearer to the second. Yes, yes, I see you saying there’s also a line that’s exactly the same distance between the two and shouldn’t that be a third part? Fine, go ahead. We’ll be drawing that in anyway. But here we’ve got a piece of paper and two dots and this line dividing it into two chunks.

    Now drop in a third point. Now every point on your paper might be closer to the first, or closer to the second, or closer to the third. Or, yeah, it might be on an edge equidistant between two of those points. Maybe even equidistant to all three points. It’s not guaranteed there is such a “triple point”, but if you weren’t picking points to cause trouble there probably is. You get the page divided up into three regions that you say are coming together in a triangle before realizing that no, it’s a Y intersection. Or else the regions are three strips and they don’t come together at all.

    What if you have four points … You should get four regions. They might all come together in one grand intersection. Or they might come together at weird angles, two and three regions touching each other. You might get a weird one where there’s a triangle in the center and three regions that go off to the edge of the paper. Or all sorts of fun little abstract flag icons, maybe. It’s hard to say. If we had, say, 26 points all sorts of weird things could happen.

    These weird things are Voronoi Diagrams. They’re a partition of some surface. Usually it’s a plane or some well-behaved subset of the plane like a sheet of paper. The partitioning is into polygons. Exactly one of the points you start with is inside each of the polygons. And everything else inside that polygon is nearer to its one contained starting point than it is any other point. All you need for the diagram are your original points and the edges dividing spots between them. But the thing begs to be colored. Give in to it and you have your own, abstract, stained-glass window pattern. So I’m glad to give you some useful mathematics to play with.

    Voronoi diagrams turn up naturally whenever you want to divide up space by the shortest route to get something. Sometimes this is literally so. For example, a radio picking up two FM signals will switch to the stronger of the two. That’s what the superheterodyne does. If the two signals are transmitted with equal strength, then the receiver will pick up on whichever the nearer signal is. And unless the other mathematicians who’ve talked about this were just as misinformed, cell phones pick which signal tower to communicate with by which one has the stronger signal. If you could look at what tower your cell phone communicates with as you move around, you would produce a Voronoi diagram of cell phone towers in your area.

    Mathematicians hoping to get credit for a good thing may also bring up Dr John Snow’s famous halting of an 1854 cholera epidemic in London. He did this by tracking cholera outbreaks and measuring their proximity to public water pumps. He shut down the water pump at the center of the severest outbreak and the epidemic soon stopped. One could claim this as a triumph for Voronoi diagrams, although Snow can not have had this tool in mind. Georgy Voronoy (yes, the spelling isn’t consistent. Fashions in transliterating Eastern European names — Voronoy was Ukrainian and worked in Warsaw when Poland was part of the Russian Empire — have changed over the years) wasn’t even born until 1868. And it doesn’t require great mathematical insight to look for the things an infected population has in common. But mathematicians need some tales of heroism too. And it isn’t as though we’ve run out of epidemics with sources that need tracking down.

    Voronoi diagrams turned out to be useful in my own meager research. I needed to model the flow of a fluid over a whole planet, but could only do so with a modest number of points to represent the whole thing. Scattering points over the planet was easy enough. To represent the fluid over the whole planet as a collection of single values at a couple hundred points required this Voronoi-diagram type division. … Well, it used them anyway. I suppose there might have been other ways. But I’d just learned about them and was happy to find a reason to use them. Anyway, this is the sort of technique often used to turn information about a single point into approximate information about a region.

    (And I discover some amusing connections here. Voronoy’s thesis advisor was Andrey Markov, who’s the person being named by “Markov Chains”. You know those as those predictive-word things that are kind of amusing for a while. Markov Chains were part of the tool I used to scatter points over the whole planet. Also, Voronoy’s thesis was On A Generalization Of A Continuous Fraction, so, hi, Gaurish! … And one of Voronoy’s doctoral students was Wacław Sierpiński, famous for fractals and normal numbers.)

    Voronoi diagrams have a lot of beauty to them. Some of it is subtle. Take a point inside its polygon and look to a neighboring polygon. Where is the representative point inside that neighbor polygon? … There’s only one place it can be. It’s got to be exactly as far as the original point is from the edge between them, and it’s got to be on the direction perpendicular to the edge between them. It’s where you’d see the reflection of the original point if the border between them were a mirror. And that has to apply to all the polygons and their neighbors.

    From there it’s a short step to wondering: imagine you knew the edges. The mirrors. But you don’t know the original points. Could you figure out where the representative points must be to fit that diagram? … Or at least some points where they may be? This is the inverse problem, and it’s how I first encountered them. This inverse problem allows nice stuff like algorithm compression. Remember my description of the result of a Voronoi diagram being a stained glass window image? There’s no reason a stained glass image can’t be quite good, if we have enough points and enough gradations of color. And storing a bunch of points and the color for the region is probably less demanding than storing the color information for every point in the original image.

    If we want images. Many kinds of data turn out to work pretty much like pictures, set up right.

     
    • gaurish 5:10 am on Thursday, 22 December, 2016 Permalink | Reply

      I didn’t know that Voronoy’s thesis was on continued fractions :) Few months ago, I was delighted to see the application of Voronoi Diagram to find answer to this simple geometry problem about maximization: http://math.stackexchange.com/a/1812338/214604

      Like

      • Joseph Nebus 5:11 am on Thursday, 5 January, 2017 Permalink | Reply

        I did not know either, until I started writing the essay. I’m glad for the side bits of information I get in writing this sort of thing.

        And I’m delighted to see the problem. I didn’t think of Voronoi diagrams as a way to study maximization problems but obviously, yeah, they would be.

        Like

    • gaurish 5:04 am on Tuesday, 3 January, 2017 Permalink | Reply

      • Joseph Nebus 5:40 am on Thursday, 5 January, 2017 Permalink | Reply

        You are quite right; I do like that. And it even has a loose connection as it is to my original thesis and its work; part of the problem was getting points spread out uniformly on a plane without them spreading out infinitely far, that is, getting them to cluster according to some imposed preference. It wasn’t artistic except in the way abstract mathematics is a bit artistic.

        Liked by 1 person

  • Joseph Nebus 6:00 pm on Monday, 19 December, 2016 Permalink | Reply
    Tags: , , , links,   

    The End 2016 Mathematics A To Z: Unlink 


    This is going to be a fun one. It lets me get into knot theory again.

    Unlink.

    An unlink is what knot theorists call that heap of loose rubber bands in that one drawer compartment.

    The longer way around. It starts with knots. I love knots. If I were stronger on abstract reasoning and weaker on computation I’d have been a knot theorist. At least graph theory anyway. The mathematical idea of a knot is inspired by a string tied together. In making it a mathematical idea we perfect the string. It becomes as thin as a line, though it curves as much as we want. It can stretch out or squash down as much as we want. It slides frictionlessly against itself. Gravity doesn’t make it drop any. This removes the hassles of real-world objects from it. It also means actual strings or yarns or whatever can’t be knots anymore. Only something that’s a loop which closes back on itself can be a knot. The knot you might make in a shoelace, to use an example, could be undone by pushing the tip back through the ‘knot’. Since our mathematical string is frictionless we can do that, effortlessly. We’re left with nothing.

    But you can create a pretty good approximation to a mathematical knot if you have some kind of cable that can be connected to its own end. Loop the thing around as you like, connect end to end, and you’ve got it. I recommend the glow sticks sold for people to take to parties or raves or the like. They’re fun. If you tie it up so that the string (rope, glow stick, whatever) can’t spread out into a simple O shape no matter how you shake it up (short of breaking the cable) then you have a knot. There are many of them. Trefoil knots are probably the easiest to get, but if you’re short on inspiration try looking at Celtic knot patterns.

    But if the string can be shaken out until it’s a simple O shape, the sort of thing you can place flat on a table, then you have an unknot. Just from the vocabulary this you see why I like the subject so. Since this hasn’t quite got silly enough, let me assure you that an unknot is itself a kind of knot; we call it the trivial knot. It’s the knot that’s too simple to be a knot. I’m sure you were worried about that. I only hear people call it an unknot, but maybe there are heritages that prefer “trivial knot”.

    So that’s knots. What happens if you have more than one thing, though? What if you have a couple of string-loops? Several cables. We know these things can happen in the real world, since we’ve looked behind the TV set or the wireless router and we know there’s somehow more cables there than there are even things to connect.

    Even mathematicians wouldn’t want to ignore something that caught up with real world implications. And we don’t. We get to them after we’re pretty comfortable working with knots. Describing them, working out the theoretical tools we’d use to un-knot a proper knot (spoiler: we cut things), coming up with polynomials that describe them, that sort of thing. When we’re ready for a new trick there we consider what happens if we have several knots. We call this bundle of knots a “link”. Well, what would you call it?

    A link is a collection of knots. By talking about a link we expect that at least some of the knots are going to loop around each other. This covers a lot of possibilities. We could picture one of those construction-paper chains, made of intertwined loops, that are good for elementary school craft projects to be a link. We can picture a keychain with a bunch of keys dangling from it to be a link. (Imagine each key is a knot, just made of a very fat, metal “string”. C’mon, you can give me that.) The mass of cables hiding behind the TV stand is not properly a link, since it’s not properly made out of knots. But if you can imagine taking the ends of each of those wires and looping them back to the origins, then the somehow vaster mess you get from that would be a link again.

    And then we come to an “unlink”. This has two pieces. The first is that it’s a collection of knots, yes, but knots that don’t interlink. We can pull them apart without any of them tugging the others along. The second piece is that each of the knots is itself an unknot. Trivial knots. Whichever you like to call them.

    The “unlink” also gets called the “trivial link”, since it’s as boring a link as you can imagine. Manifested in the real world, well, an unbroken rubber band is a pretty good unknot. And a pile of unbroken rubber bands will therefore be an unlink.

    If you get into knot theory you end up trying to prove stuff about complicated knots, and complicated links. Often these are easiest to prove by chopping up the knot or the link into something simpler. Maybe you chop those smaller pieces up again. And you can’t get simpler than an unlink. If you can prove whatever you want to show for that then you’ve got a step done toward proving your whole actually interesting thing. This is why we see unknots and unlinks enough to give them names and attention.

     
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