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  • 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 6:00 pm on Sunday, 15 January, 2017 Permalink | Reply
    Tags: , Barkeater Lake, combinatorics, , Dinette Set, Dinosaur Comics, Nancy, Redeye, 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

    • 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?

      Like

      • 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 ;-))

        Like

  • Joseph Nebus 6:00 pm on Sunday, 8 January, 2017 Permalink | Reply
    Tags: , Argyle Sweater, Birdbrains, Boomerangs, 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, , Ripley's Believe It Or Not, , 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, Crisis of Foundations, , , , 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.

     
  • Joseph Nebus 6:00 pm on Sunday, 18 December, 2016 Permalink | Reply
    Tags: , dinosaurs,   

    Reading the Comics, December 17, 2016: Sleepy Week Edition 


    Comic Strip Master Command sent me a slow week in mathematical comics. I suppose they knew I was on somehow a busier schedule than usual and couldn’t spend all the time I wanted just writing. I appreciate that but don’t want to see another of those weeks when nothing qualifies. Just a warning there.

    'Dadburnit! I ain't never gonna git geometry!' 'Bah! Don't fret, Jughaid --- I never understood it neither! But I still manage to work all th' angles!'

    John Rose’s Barney Google and Snuffy Smith for the 12th of December, 2016. I appreciate the desire to pay attention to continuity that makes Rose draw in the coffee cup both panels, but Snuffy Smith has to swap it from one hand to the other to keep it in view there. Not implausible, just kind of busy. Also I can’t fault Jughaid for looking at two pages full of unillustrated text and feeling lost. That’s some Bourbaki-grade geometry going on there.

    John Rose’s Barney Google and Snuffy Smith for the 12th is a bit of mathematical wordplay. It does use geometry as the “hard mathematics we don’t know how to do”. That’s a change from the usual algebra. And that’s odd considering the joke depends on an idiom that is actually used by real people.

    Patrick Roberts’s Todd the Dinosaur for the 12th uses mathematics as the classic impossibly hard subject a seven-year-old can’t be expected to understand. The worry about fractions seems age-appropriate. I don’t know whether it’s fashionable to give elementary school students experience thinking of ‘x’ and ‘y’ as numbers. I remember that as a time when we’d get a square or circle and try to figure what number fits in the gap. It wasn’t a 0 or a square often enough.

    'Teacher! Todd just passed out! But he's waring one of those medic alert bracelets! ... Do not expose the wearer of this bracelet to anything mathematical, especially x's and y's, fractions, or anything that he should remember for a test!' 'Amazing how much writing they were able to fit on a little ol' T-Rex wrist!'

    Patrick Roberts’s Todd the Dinosaur for the 12th of December, 2016. Granting that Todd’s a kid dinosaur and that T-Rexes are not renowned for the hugeness of their arms, wouldn’t that still be enough space for a lot of text to fit around? I would have thought so anyway. I feel like I’m pluralizing ‘T-Rex’ wrong, but what would possibly be right? ‘Ts-rex’? Don’t make me try to spell tyrannosaurus.

    Jef Mallett’s Frazz for the 12th uses one of those great questions I think every child has. And it uses it to question how we can learn things from statistical study. This is circling around the “Bayesian” interpretation of probability, of what odds mean. It’s a big idea and I’m not sure I’m competent to explain it. It amounts to asking what explanations would be plausibly consistent with observations. As we get more data we may be able to rule some cases in or out. It can be unsettling. It demands we accept right up front that we may be wrong. But it lets us find reasonably clean conclusions out of the confusing and muddy world of actual data.

    Sam Hepburn’s Questionable Quotebook for the 14th illustrates an old observation about the hypnotic power of decimal points. I think Hepburn’s gone overboard in this, though: six digits past the decimal in this percentage is too many. It draws attention to the fakeness of the number. One, two, maybe three digits past the decimal would have a more authentic ring to them. I had thought the John Allen Paulos tweet above was about this comic, but it’s mere coincidence. Funny how that happens.

     
  • Joseph Nebus 6:00 pm on Friday, 16 December, 2016 Permalink | Reply
    Tags: , , , , , trees   

    The End 2016 Mathematics A To Z: Tree 


    Graph theory begins with a beautiful legend. I have no reason to suppose it’s false, except my natural suspicion of beautiful legends as origin stories. Its organization as a field is traced to 18th century Köningsburg, where seven bridges connected the banks of a river and a small island in the center. Whether it was possible to cross each bridge exactly once and get back where one started was, they say, a pleasant idle thought to ponder and path to try walking. Then Leonhard Euler solved the problem. It’s impossible.

    Tree.

    Graph theory arises whenever we have a bunch of things that can be connected. We call the things “vertices”, because that’s a good corner-type word. The connections we call “edges”, because that’s a good connection-type word. It’s easy to create graphs that look like the edges of a crystal, especially if you draw edges as straight as much as possible. You don’t have to. You can draw them curved. Then they look like the scary tangles of wire around your wireless router complex.

    Graph theory really got organized in the 19th century, and went crazy in the 20th. It turns out there’s lots of things that connect to other things. Networks, whether computers or social or thematically linked concepts. Anything that has to be delivered from one place to another. All the interesting chemicals. Anything that could be put in a pipe or taken on a road has some graph theory thing applicable to it.

    A lot of graph theory ponders loops. The original problem was about how to use every bridge, every edge, exactly one time. Look at a tangled mass of a graph and it’s hard not to start looking for loops. They’re often interesting. It’s not easy to tell if there’s a loop that lets you get to every vertex exactly once.

    What if there aren’t loops? What if there aren’t any vertices you can step away from and get back to by another route? Well, then you have a tree.

    A tree’s a graph where all the vertices are connected so that there aren’t any closed loops. We normally draw them with straight lines, the better to look like actual trees. We then stop trying to make them look like actual trees by doing stuff like drawing them as a long horizontal spine with a couple branches sticking off above and below, or as * type stars, or H shapes. They still correspond to real-world things. If you’re not sure how consider the layout of one of those long, single-corridor hallways as in a hotel or dormitory. The rooms connect to one another as a tree once again, as long as no room opens to anything but its own closet or bathroom or the central hallway.

    We can talk about the radius of a graph. That’s how many edges away any point can be from the center of the tree. And every tree has a center. Or two centers. If it has two centers they share an edge between the two. And that’s one of the quietly amazing things about trees to me. However complicated and messy the tree might be, we can find its center. How many things allow us that?

    A tree might have some special vertex. That’s called the ‘root’. It’s what the vertices and the connections represent that make a root; it’s not something inherent in the way trees look. We pick one for some special reason and then we highlight it. Maybe put it at the bottom of the drawing, making ‘root’ for once a sensible name for a mathematics thing. Often we put it at the top of the drawing, because I guess we’re just being difficult. Well, we do that because we were modelling stuff where a thing’s properties depend on what it comes from. And that puts us into thoughts of inheritance and of family trees. And weird as it is to put the root of a tree at the top, it’s also weird to put the eldest ancestors at the bottom of a family tree. People do it, but in those illuminated drawings that make a literal tree out of things. You don’t see it in family trees used for actual work, like filling up a couple pages at the start of a king or a queen’s biography.

    Trees give us neat new questions to ponder, like, how many are there? I mean, if you have a certain number of vertices then how many ways are there to arrange them? One or two or three vertices all have just the one way to arrange them. Four vertices can be hooked up a whole two ways. Five vertices offer a whole three different ways to connect them. Six vertices offer six ways to connect and now we’re finally getting something interesting. There’s eleven ways to connect seven vertices, and 23 ways to connect eight vertices. The number keeps on rising, but it doesn’t follow the obvious patterns for growth of this sort of thing.

    And if that’s not enough to idly ponder then think of destroying trees. Draw a tree, any shape you like. Pick one of the vertices. Imagine you obliterate that. How many separate pieces has the tree been broken into? It might be as few as two. It might be as many as the number of remaining vertices. If graph theory took away the pastime of wandering around Köningsburg’s bridges, it has given us this pastime we can create anytime we have pen, paper, and a long meeting.

     
  • Joseph Nebus 6:00 pm on Wednesday, 14 December, 2016 Permalink | Reply
    Tags: , , , ,   

    The End 2016 Mathematics A To Z: Smooth 


    Mathematicians affect a pose of objectivity. We justify this by working on things whose truth we can know, and which must be true whenever we accept certain rules of deduction and certain definitions and axioms. This seems fair. But we choose to pay attention to things that interest us for particular reasons. We study things we like. My A To Z glossary term for today is about one of those things we like.

    Smooth.

    Functions. Not everything mathematicians do is functions. But functions turn up a lot. We need to set some rules. “A function” is so generic a thing we can’t handle it much. Narrow it down. Pick functions with domains that are numbers. Range too. By numbers I mean real numbers, maybe complex numbers. That gives us something.

    There’s functions that are hard to work with. This is almost all of them, so we don’t touch them unless we absolutely must. But they’re functions that aren’t continuous. That means what you imagine. The value of the function at some point is wholly unrelated to its value at some nearby point. It’s hard to work with anything that’s unpredictable like that. Functions as well as people.

    We like functions that are continuous. They’re predictable. We can make approximations. We can estimate the function’s value at some point using its value at some more convenient point. It’s easy to see why that’s useful for numerical mathematics, for calculations to approximate stuff. The dazzling thing is it’s useful analytically. We step into the Platonic-ideal world of pure mathematics. We have tools that let us work as if we had infinitely many digits of precision, for infinitely many numbers at once. And yet we use estimates and approximations and errors. We use them in ways to give us perfect knowledge; we get there by estimates.

    Continuous functions are nice. Well, they’re nicer to us than functions that aren’t continuous. But there are even nicer functions. Functions nicer to us. A continuous function, for example, can have corners; it can change direction suddenly and without warning. A differentiable function is more predictable. It can’t have corners like that. Knowing the function well at one point gives us more information about what it’s like nearby.

    The derivative of a function doesn’t have to be continuous. Grumble. It’s nice when it is, though. It makes the function easier to work with. It’s really nice for us when the derivative itself has a derivative. Nothing guarantees that the derivative of a derivative is continuous. But maybe it is. Maybe the derivative of the derivative has a derivative. That’s a function we can do a lot with.

    A function is “smooth” if it has as many derivatives as we need for whatever it is we’re doing. And if those derivatives are continuous. If this seems loose that’s because it is. A proof for whatever we’re interested in might need only the original function and its first derivative. It might need the original function and its first, second, third, and fourth derivatives. It might need hundreds of derivatives. If we look through the details of the proof we might find exactly how many derivatives we need and how many of them need to be continuous. But that’s tedious. We save ourselves considerable time by saying the function is “smooth”, as in, “smooth enough for what we need”.

    If we do want to specify how many continuous derivatives a function has we call it a “Ck function”. The C here means continuous. The ‘k’ means there are the number ‘k’ continuous derivatives of it. This is completely different from a “Ck function”, which would be one that’s a k-dimensional vector. Whether the “C” is boldface or not is important. A function might have infinitely many continuous derivatives. That we call a “C function”. That’s got wonderful properties, especially if the domain and range are complex-valued numbers. We couldn’t do Complex Analysis without it. Complex Analysis is the course students take after wondering how they’ll ever survive Real Analysis. It’s much easier than Real Analysis. Mathematics can be strange.

     
  • Joseph Nebus 6:00 pm on Tuesday, 13 December, 2016 Permalink | Reply
    Tags: , , , MacArthur Genius Grants, , ,   

    Reading the Comics, December 10, 2016: E = mc^2 Edition 


    And now I can finish off last week’s mathematically-themed comic strips. There’s a strong theme to them, for a refreshing change. It would almost be what we’d call a Comics Synchronicity, on Usenet group rec.arts.comics.strips, had they all appeared the same day. Some folks claiming to be open-minded would allow a Synchronicity for strips appearing on subsequent days or close enough in publication, but I won’t have any of that unless it suits my needs at the time.

    Ernie Bushmiller’s for the 6th would fit thematically better as a Cameo Edition comic. It mentions arithmetic but only because it’s the sort of thing a student might need a cheat sheet on. I can’t fault Sluggo needing help on adding eight or multiplying by six; they’re hard. Not remembering 4 x 2 is unusual. But everybody has their own hangups. The strip originally ran the 6th of December, 1949.

    People contorted to look like a 4, a 2, and a 7 bounce past Dethany's desk. She ponders: 'Performance review time ... when the company reduces people to numbers.' Wendy, previous star of the strip, tells Dethany 'You're next.' Wendy's hair is curled into an 8.

    Bill holbrook’s On The Fastrack for the 7th of December, 2016. Don’t worry about the people in the first three panels; they’re just temps, and weren’t going to appear in the comic again.

    Bill holbrook’s On The Fastrack for the 7th seems like it should be the anthropomorphic numerals joke for this essay. It doesn’t seem to quite fit the definition, but, what the heck.

    Brian Boychuk and Ron Boychuk’s The Chuckle Brothers on the 7th starts off the run of E = mc2 jokes for this essay. This one reminds me of Gary Larson’s Far Side classic with the cleaning woman giving Einstein just that little last bit of inspiration about squaring things away. It shouldn’t surprise anyone that E equalling m times c squared isn’t a matter of what makes an attractive-looking formula. There’s good reasons when one thinks what energy and mass are to realize they’re connected like that. Einstein’s famous, deservedly, for recognizing that link and making it clear.

    Mark Pett’s Lucky Cow rerun for the 7th has Claire try to use Einstein’s famous quote to look like a genius. The mathematical content is accidental. It could be anything profound yet easy to express, and it’s hard to beat the economy of “E = mc2” for both. I’d agree that it suggests Claire doesn’t know statistics well to suppose she could get a MacArthur “Genius” Grant by being overheard by a grant nominator. On the other hand, does anybody have a better idea how to get their attention?

    Harley Schwadron’s 9 to 5 for the 8th completes the “E = mc2” triptych. Calling a tie with the equation on it a power tie elevates the gag for me. I don’t think of “E = mc2” as something that uses powers, even though it literally does. I suppose what gets me is that “c” is a constant number. It’s the speed of light in a vacuum. So “c2” is also a constant number. In form the equation isn’t different from “E = m times seven”, and nobody thinks of seven as a power.

    Morrie Turner’s Wee Pals rerun for the 8th is a bit of mathematics wordplay. It’s also got that weird Morrie Turner thing going on where it feels unquestionably earnest and well-intentioned but prejudiced in that way smart 60s comedies would be.

    Sarge demands to know who left this algebra book on his desk; Zero says not him. Sarge ignores him and asks 'Who's been figuring all over my desk pad?' Zero also unnecessarily denies it. 'Come on, whose is it?!' Zero reflects, 'Gee, he *never* picks on *me*!'

    Mort Walker’s vintage Beetle Bailey for the 18th of May, 1960. Rerun the 9th of December, 2016. For me the really fascinating thing about ancient Beetle Bailey strips is that they could run today with almost no changes and yet they feel like they’re from almost a different cartoon universe from the contemporary comic. I don’t know how that is, or why it is.

    Mort Walker’s Beetle Bailey for the 18th of May, 1960 was reprinted on the 9th. It mentions mathematics — algebra specifically — as the sort of thing intelligent people do. I’m going to take a leap and suppose it’s the sort of algebra done in high school about finding values of ‘x’ rather than the mathematics-major sort of algebra, done with groups and rings and fields. I wonder when holding a mop became the signifier of not just low intelligence but low ambition. It’s subverted in Jef Mallet’s Frazz, the title character of which works as a janitor to support his exercise and music habits. But it is a standard prop to signal something.

     
  • Joseph Nebus 6:00 pm on Monday, 12 December, 2016 Permalink | Reply
    Tags: , , , definite integrals,   

    The End 2016 Mathematics A To Z: Riemann Sum 


    I see for the other A To Z I did this year I did something else named for Riemann. So I did. Bernhard Riemann did a lot of work that’s essential to how we see mathematics today. We name all kinds of things for him, and correctly so. Here’s one of his many essential bits of work.

    Riemann Sum.

    The Riemann Sum is a thing we learn in Intro to Calculus. It’s essential in getting us to definite integrals. We’re introduced to it in functions of a single variable. The functions have a domain that’s an interval of real numbers and a range that’s somewhere in the real numbers. The Riemann Sum — and from it, the integral — is a real number.

    We get this number by following a couple steps. The first is we chop the interval up into a bunch of smaller intervals. That chopping-up we call a “partition” because it’s another of those times mathematicians use a word the way people might use the same word. From each one of those chopped-up pieces we pick a representative point. Now with each piece evaluate what the function is for that representative point. Multiply that by the width of the partition it was in. Then take those products for each of those pieces and add them all together. If you’ve done it right you’ve got a number.

    You need a couple pieces in place to have “the” Riemann Sum for something. You need a function, which is fair enough. And you need a partitioning of the interval. And you need some representative point for each of the partitions. Change any of them — function, partition, or point — and you may change the sum you get. You expect that for changing the function. Changing the partition? That’s less obvious. But draw some wiggly curvy function on a sheet of paper. Draw a couple of partitions of the horizontal axis. (You’ll probably want to use different colors for different partitions.) That should coax you into it. And you’d probably take it on my word that different representative points give you different sums.

    Very different? It’s possible. There’s nothing stopping it from happening. But if the results aren’t very different then we might just have an integrable function. That’s a function that gives us the same Riemann Sum no matter how we pick representative points, as long as we pick partitions that get finer and finer enough. We measure how fine a partition is by how big the widest chopped-up piece is. To be integrable the Riemann Sum for a function has to get to the same number whenever the partition’s size gets small enough and however we pick points inside. We get the lovely quiet paradox in which we add together infinitely many things, each of them infinitesimally tiny, and get a regular old number out of all that work.

    We use the Riemann Sum for what we call numerical quadrature. That’s working out integrals on the computer. Or calculator. Or by hand. When we do it by evaluating numbers instead of using analysis. It’s very easy to program. And we can do some tricks based on the Riemann Sum to make the numerical estimate a closer match to the actual integral.

    And we use the Riemann Sum to learn how the Riemann Integral works. It’s a blessedly straightforward thing. It appeals to intuition well. It lets us draw all sorts of curves with rectangular boxes overlaying them. It’s so easy to work out the area of a rectangular box. We can imagine adding up these areas without being confused.

    We don’t use the Riemann Sum to actually do integrals, though. Numerical approximations to an integral, yes. For the actual integral it’s too hard to use. What makes it hard is you need to evaluate this for every possible partition and every possible pick of representative points. In grad school my analysis professor worked through — once — using this to integrate the number 1. This is the easiest possible thing to integrate and it was barely manageable. He gave a good try at integrating the function ‘f(x) = x’ but admitted he couldn’t do it. None of us could.

    When you see the Riemann Sum in an Introduction to Calculus course you see it in simplified form. You get partitions that are very easy to work with. Like, you break the interval up into some number of equally-sized chunks. You get representative points that follow one of a couple good choices. The left end of the partition. The right end of the partition. The middle of the partition.

    That’s fine, numerically. If the function is integrable it doesn’t matter what partition or representative points we pick. And it’s fine for learning about whether functions are integrable. If it matters whether you pick left or middle or right ends of the partition then the function isn’t integrable. The instructor can give functions that break integrability based on a given partition or endpoint choice or whatever.

    But that isn’t every possible partition and every possible pick of representative points. I suppose it’s possible to work all that out for a couple of really, really simple functions. But it’s so much work. We’re better off using the Riemann Sum to get to formulas about integrals that don’t depend on actually using the Riemann Sum.

    So that is the curious position the Riemann Sum has. It is a fundament of integral calculus. It is the way we first define the definite integral. We rely on it to learn what definite integrals are like. We use it all the time numerically. We never use it analytically. It’s too hard. I hope you appreciate the strange beauty of that.

     
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