I was reading a biography of Donald Coxeter, one of the most important geometers of the 20th century, and it mentioned in passing something Coxeter referred to as Morley’s Miracle Theorem. The theorem was proved in 1899 by Frank Morley, who taught at Haverford College (if that sounds vaguely familiar that’s because you remember it’s where Dave Barry went) and then Johns Hopkins (which may be familiar on the strength of its lacrosse team), and published this in the first issue of the Transactions of the American Mathematical Society. And, yes, perhaps it isn’t actually important, but the result is so unexpected and surprising that I wanted to share it with you. The biography also includes a proof Coxeter wrote for the theorem, one that’s admirably straightforward, but let me show the result without the proof so you can wonder about it.
First, start by drawing a triangle. It doesn’t have to have any particular interesting properties other than existing. I’ve drawn an example one.
The next step is to cut into three equal pieces each of the interior angles of the triangle, and draw those lines. I’m doing that in separate diagrams for each of the triangle’s three original angles because I want to better suggest the process.
I should point out, this trisection of the angles can be done however you like, which is probably going to be by measuring the angles with a protractor and dividing the angle by three. I made these diagrams just by sketching them out, so they aren’t perfect in their measure, but if you were doing the diagram yourself on a sheet of scratch paper you wouldn’t bother getting the protractor out either. (And, famously, you can’t trisect an angle if you’re using just compass and straightedge to draw things, but you don’t have to restrict yourself to compass and straightedge for this.)
Now the next bit is to take the points where adjacent angle trisectors intersect — that is, for example, where the lower red line crosses the lower green line; where the upper red line crosses the left blue line; and where the right blue line crosses the upper green line. Draw lines connecting these points together and …
This new triangle, drawn in purple on my sketch, is an equilateral triangle!
(It may look a little off, but that’s because I didn’t measure the trisectors when I drew them in and just eyeballed it. If I had measured the angles and drawn the new ones in carefully, it would have been perfect.)
I’ve been thinking back on this and grinning ever since reading it. I certainly didn’t see that punch line coming.
Must be the start of school or something. In today’s roundup of mathematically-themed comics there are a couple of strips that I think touch on the question of defining just what the problem is: what are you trying to measure, what are you trying to calculate, what are the rules of this sort of calculation? That’s a lot of what’s really interesting about mathematics, which is how I’m able to say something about a rerun Archie comic. It’s not easy work but that’s why I get that big math-blogger paycheck.
I’d have thought the universe to be at least three-dimensional.
John Hambrock’s The Brilliant Mind of Edison Lee (September 2) talks about the shape of the universe. Measuring the world, or the universe, is certainly one of the older influences on mathematical thought. From a handful of observations and some careful reasoning, for example, one can understand how large the Earth is, and how far away the Moon and the Sun must be, without going past the kinds of reasoning or calculations that a middle school student would probably be able to follow.
There is something deeper to consider about the shape of space, though: the geometry of the universe affects what things can happen in them, and can even be seen in the kinds of physics that happen. A famous, and astounding, result by the mathematical physicist Emmy Noether shows that symmetries in space correspond to conservation laws. That the universe is, apparently, rotationally symmetric — everything would look the same if the whole universe were picked up and rotated (say) 80 degrees along one axis — means that there is such a thing as the conservation of angular momentum. That the universe is time-symmetric — the universe would look the same if it had got started five hours later (please pretend that’s a statement that can have any coherent meaning) — means that energy is conserved. And so on. It may seem, superficially, like a cosmologist is engaged in some almost ancient-Greek-style abstract reasoning to wonder what shapes the universe could have and what it does, but (putting aside that it gets hard to divide mathematics, physics, and philosophy in this kind of field) we can imagine observable, testable consequences of the answer.
Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 5) tells a joke starting with “two perfectly rational perfectly informed individuals walk into a bar”, along the way to a joke about economists. The idea of “perfectly rational perfectly informed” people is part of the mathematical modeling that’s become a popular strain of economic thought in recent decades. It’s a model, and like many models, is properly speaking wrong, but it allows one to describe interesting behavior — in this case, how people will make decisions — without complications you either can’t handle or aren’t interested in. The joke goes on to the idea that one can assign costs and benefits to continuing in the joke. The idea that one can quantify preferences and pleasures and happiness I think of as being made concrete by Jeremy Bentham and the utilitarian philosophers, although trying to find ways to measure things has been a streak in Western thought for close to a thousand years now, and rather fruitfully so. But I wouldn’t have much to do with protagonists who can’t stay around through the whole joke either.
Marc Anderson’s Andertoons (September 6) was probably composed in the spirit of joking, but it does hit something that I understand baffles kids learning it every year: that subtracting a negative number does the same thing as adding a positive number. To be fair to kids who need a couple months to feel quite confident in what they’re doing, mathematicians needed a couple generations to get the hang of it too. We have now a pretty sound set of rules for how to work with negative numbers, that’s nice and logically tested and very successful at representing things we want to know, but there seems to be a strong intuition that says “subtracting a negative three” and “adding a positive three” might just be different somehow, and we won’t really know negative numbers until that sense of something being awry is resolved.
Andertoons pops up again the next day (September 7) with a completely different drawing of a chalkboard and this time a scientist and a rabbit standing in front of it. The rabbit’s shown to be able to do more than multiply and, indeed, the mathematics is correct. Cosines and sines have a rather famous link to exponentiation and to imaginary- and complex-valued numbers, and it can be useful to change an ordinary cosine or sine into this exponentiation of a complex-valued number. Why? Mostly, because exponentiation tends to be pretty nice, analytically: you can multiply and divide terms pretty easily, you can take derivatives and integrals almost effortlessly, and then if you need a cosine or a sine you can get that out at the end again. It’s a good trick to know how to do.
How far back is this rerun from if Moose got lunch for two for $8.95?
Craig Boldman and Henry Scarpelli’s Archie (September 8, rerun) goes back to one of arithmetic’s traditional comic strip applications, that of working out the tip. Poor Moose is driving himself crazy trying to work out 15 percent of $8.95, probably from a quiz-inspired fear that if he doesn’t get it correct to the penny he’s completely wrong. Being able to do a calculation precisely is useful, certainly, but he’s forgetting that in tis real-world application he gets some flexibility in what has to be calculated. He’d save some effort if he realized the tip for $8.95 is probably close enough to the tip for $9.00 that he could afford the difference, most obviously, and (if his budget allows) that he could just as well work out one-sixth the bill instead of fifteen percent, and give up that workload in exchange for sixteen cents.
Well, I did say we were getting to the end of summer. It’s taken only a couple days to get a fresh batch of enough mathematics-themed comics to include here, although the majority of them are about mathematics in ways that we’ve seen before, sometimes many times. I suppose that’s fair; it’s hard to keep thinking of wholly original mathematics jokes, after all. When you’ve had one killer gag about “537”, it’s tough to move on to “539” and have it still feel fresh.
Tom Toles’s Randolph Itch, 2 am (August 27, rerun) presents Randolph suffering the nightmare of contracting a case of entropy. Entropy might be the 19th-century mathematical concept that’s most achieved popular recognition: everyone knows it as some kind of measure of how disorganized things are, and that it’s going to ever increase, and if pressed there’s maybe something about milk being stirred into coffee that’s linked with it. The mathematical definition of entropy is tied to the probability one will find whatever one is looking at in a given state. Work out the probability of finding a system in a particular state — having particles in these positions, with these speeds, maybe these bits of magnetism, whatever — and multiply that by the logarithm of that probability. Work out that product for all the possible ways the system could possibly be configured, however likely or however improbable, just so long as they’re not impossible states. Then add together all those products over all possible states. (This is when you become grateful for learning calculus, since that makes it imaginable to do all these multiplications and additions.) That’s the entropy of the system. And it applies to things with stunning universality: it can be meaningfully measured for the stirring of milk into coffee, to heat flowing through an engine, to a body falling apart, to messages sent over the Internet, all the way to the outcomes of sports brackets. It isn’t just body parts falling off.
Randy Glasbergen’s _The Better Half_ For the 28th of August, 2014.
Randy Glasbergen’s The Better Half (August 28) does the old joke about not giving up on algebra someday being useful. Do teachers in other subjects get this? “Don’t worry, someday your knowledge of the Panic of 1819 will be useful to you!” “Never fear, someday they’ll all look up to you for being able to diagram a sentence!” “Keep the faith: you will eventually need to tell someone who only speaks French that the notebook of your uncle is on the table of your aunt!”
Eric the Circle (August 28, by “Gilly” this time) sneaks into my pages again by bringing a famous mathematical symbol into things. I’d like to make a mention of the links between mathematics and music which go back at minimum as far as the Ancient Greeks and the observation that a lyre string twice as long produced the same note one octave lower, but lyres and strings don’t fit the reference Gilly was going for here. Too bad.
Zach Weinersmith’s Saturday Morning Breakfast Cereal (August 28) is another strip to use a “blackboard full of mathematical symbols” as visual shorthand for “is incredibly smart stuff going on”. The symbols look to me like they at least started out as being meaningful — they’re the kinds of symbols I expect in describing the curvature of space, and which you can find by opening up a book about general relativity — though I’m not sure they actually stay sensible. (It’s not the kind of mathematics I’ve really studied.) However, work in progress tends to be sloppy, the rough sketch of an idea which can hopefully be made sound.
I had thought the folks at Comic Strip Master Command got most of their mathematics-themed comics cleaned out ahead of the end of the school year (United States time zones) by last week, and then over the course of the weekend they went and published about a hundred million of them, so let me try catching up on that before the long dry spell of summer sets in. (And yet none of them mentioned monkeys writing Shakespeare; go figure.) I’m kind of expecting an all-mathematics-strips series tomorrow morning.
Jason Chatfield’s Ginger Meggs (June 12) puns a bit on negative numbers as also meaning downbeat or pessimistic ones. Negative numbers tend to make people uneasy, when they’re first encountered. It took western mathematics several centuries to be quite fully comfortable with them and that even with the good example of debts serving as a mental model of what negative numbers might mean. Descartes, for example, apparently used four separate quadrants, giving points their positions to the right and up, to the left and up, to the left and down, or to the right and down, from the origin point, rather than deal with negative numbers; and the Fahrenheit temperature scale was pretty much designed around the constraint that Daniel Fahrenheit shouldn’t have to deal with negative numbers in measuring the temperature in his hometown of the Netherlands. I have seen references to Immanuel Kant writing about the theoretical foundation of negative numbers, but not a clear explanation of just what he did, alas. And skepticism of exotic number constructs would last; they’re not called imaginary numbers because people appreciated the imaginative leaps that working with the square roots of negative numbers inspired.
Steve Breen and Mike Thompson’s Grand Avenue (June 12) served notice that, just like last summer, Grandma is going to make sure the kids experience mathematics as a series of chores they have to endure through an otherwise pleasant summer break.
Mike Twohy’s That’s Life (June 12) might be a marginal inclusion here, but it does refer to a lab mouse that’s gone from merely counting food pellets to cost-averaging them. The mathematics abilities of animals are pretty amazing things, certainly, and I’d also be impressed by an animal that was so skilled in abstract mathematics that it was aware “how much does a thing cost?” is a pretty tricky question when you look hard at it.
Jim Scancarelli’s Gasoline Alley (June 13) features a punch line that’s familiar to me — it’s what you get by putting a parrot and the subject of geometry together — although the setup seems clumsy to me. I think that’s because the kid has to bring up geometry out of nowhere in the first panel. Usually the setup as I see it is more along the lines of “what geometric figure is drawn by a parrot that then leaves the room”, which I suppose also brings geometry up out of nowhere to start off, really. I guess the setup feels clumsy to me because I’m trying to imagine the dialogue as following right after the previous day’s, so the flow of the conversation feels odd.
Eric the Circle (June 14), this one signed “andel”, riffs on the popular bit of mathematics trivia that in a randomly selected group of 22 people there’s about a fifty percent chance that some pair of them will share a birthday; that there’s a coincidental use for 22 in estimating π is, believe it or not, something I hadn’t noticed before.
Pab Sungenis’s New Adventures of Queen Victoria (June 14) plays with infinities, and whether the phrase “forever and a day” could actually mean anything, or at least anything more than “forever” does. This requires having a clear idea what you mean by “forever” and, for that matter, by “more”. Normally we compare infinitely large sets by working out whether it’s possible to form pairs which match one element of the first set to one element of the second, and seeing whether elements from either set have to be left out. That sort of work lets us realize that there are just as many prime numbers as there are counting numbers, and just as many counting numbers as there are rational numbers (positive and negative), but that there are more irrational numbers than there are rational numbers. And, yes, “forever and a day” would be the same length of time as “forever”, but I suppose the Innamorati (I tried to find his character’s name, but I can’t, so, Pab Sungenis can come in and correct me) wouldn’t do very well if he promised love for the “power set of forever”, which would be a bigger infinity than “forever”.
Mark Anderson’s Andertoons (June 15) is actually roughly the same joke as the Ginger Meggs from the 12th, students mourning their grades with what’s really a correct and appropriate use of mathematics-mentioning terminology.
Keith Knight’s The Knight Life (June 16) introduces a “personal statistician”, which is probably inspired by the measuring of just everything possible that modern sports has gotten around to doing. But the notion of keeping track of just what one is doing, and how effectively, is old and, at least in principle, sensible. It’s implicit in budgeting (time, money, or other resources) that you are going to study what you do, and what you want to do, and what’s required by what you want to do, and what you can do. And careful tracking of what one’s doing leads to what’s got to be a version of the paradox of Achilles and the tortoise, in which the time (and money) spent on recording the fact of one’s recordings starts to spin out of control. I’m looking forward to that. Don’t read the comments.
Max Garcia’s Sunny Street (June 16) shows what happens when anthropomorphized numerals don’t appear in Scott Hilburn’s The Argyle Sweater for too long a time.
I’d like to close out the month by pointing to 4D Visualization, a web site set up by … well, I’m not sure the person, but the contact e-mail address is 4d ( at ) eusebeia.dyndns.org for whatever that’s worth. (Worse, I can not remember what site led me to it; if you’re out there, referent, please say so so I can thank you properly. In the meantime, thank you.) The author takes eleven chapters to discuss ways to visualize four-dimensional structures, and does quite a nice job at it. The ways we visualize three-dimensional structures are used heavily for analogies, and the illustrations — static and animated — build what feels like an intuitive bridge to me, at least.
Eusebeia (if I may use that as a name) goes through cross-sections, which are generally simple to render but which tax the imagination to put together1, and projections, and the subtleties in rendering two-dimensional images of three-dimensional projections of four-dimensional structures so that they’re sensible. It’s all quite good and I’m just sorry that my belief in the promise “More chapters coming soon!” clashes with the notice, “Last updated 13 Oct 2008”.
The main page is still being updated regularly, including a Polytope Of The Month feature. A polytope is what people call a polygon or polyhedron if they don’t want their discussion to carry the connotation of being about a two- or three-dimensional figure. It’s kind of the way someone in celestial mechanics talking about the orbit of an object around another might say periapsis and apoapsis, instead of perigee and apogee or perihelion and aphelion, although as far as I can tell people in celestial mechanics are only that precise if they suspect someone pedantic is watching them. I’m not well-versed enough to say how much polytope is used compared to polyhedron.
Anyway, for those looking for the chance to poke around higher dimensions, consider giving this a try; it’s a good read.
[1: I knew that a three-dimensional cube has, on the right slice, a hexagonal cross-section. It’s something I discovered while fiddling around with the problem of charged particles on a conductive-particule sphere, believe it or not. ]
(I’m putting this little post out because I want to do something more impressive, and I’ll need this lurking in the background. If it seems unmotivated, then, please treat it as a lemma of an essay.)
Most of us have a fairly decent idea of “equals”; at least, we’re fairly sure we know what it means to say “x is equal to y” and can draw from that other conclusions, such as, the controversial “y is equal to x”. Usually we get the idea early in learning arithmetic, and get used to it in working with numbers, and maybe stretch the idea (within the bounds of mathematics) to include things like two angles being equal or two shapes being equal.
Equivalence is a kind of generalizing of this equality idea: we’ll take a bit of what’s interesting about the idea of two things being equal, and use it in a new context. In this new context two things that might not be equal are still similar in some way that’s of interest for whatever we’re working on right now.
To write that “x is equal to y” efficiently we call on the equals sign and just put “x = y”. “x is equivalent to y” also begs for a shorthand notation, at least if you’re doing a lot of work with the idea, and the easiest way to type that is probably just to use a tilde: “x ~ y”, though I admit I prefer using a double tilde, “x ≈ y”, which isn’t too hard to do in HTML but is more work.
For two things to be equivalent you need to say equivalent with respect to what property. Sugar, sand, and salt are pretty much the same if all you’re interested in is how heaps of small-grained particles move; they’re not at all equivalent if you’re baking; and they’re only sort-of equivalent if you’re trying to melt sidewalk ice. You also need to say what set of things you’re drawing from; it’s very hard to answer whether sugar is equivalent to birds if you thought the discussion was about real numbers. Usually in practice the relationship — called the equivalence relation — carries with it an explicit statement of what the set of things is, unless it’s just blisteringly obvious from context.
To say that something is an equivalence relation means that it has to obey three rules, ones that look make it look a lot like ordinary old equality. The first is called reflexivity: any thing in the set is equivalent to itself. Any number equals itself; any article of clothing has the same color as itself; any person has the same gender as herself. Sounds like an unavoidably true property? Consider, for real numbers, the relationship “is less than”; there’s no number that is less than itself. “Is less than” can’t be an equivalence relationship.
The second is called symmetry: if one thing is equivalent to another, then, that other thing is equivalent to the first. If the number we’ve given the name “height” is equal to the number we’ve given the name “length”, so to does the number we’ve given the name “length” equal the number we’ve given the name “height”, and similarly good results can be found with shirts and people’s genders. For numbers, “Is less than” is ruled out right away; but the initially promising “Is less than or equal to”, which satisfies reflexivity, can flop on symmetry: 4 is less than or equal to 12, certainly, but not the other way around.
And the last is called transitivity: if one thing is equivalent to a second, and a second thing is equivalent to a third, then, the first thing has to be equivalent to the third. Ordinary old numbers being equal to one another are still transitive, and those shirts having the same colors work out too, and the same with people sharing a gender. Interestingly, both “Is less than” and “Is less than or equal to” are transitive, but since those fail on reflexivity or on transitivity they’re not equivalence relationships anyway.
There are a lot of equivalences out there, such as two geometric shapes being congruent, or for that matter just being similar (having the same shape but different sizes), or whole numbers having the same remainder when divided by, say, two (which is a fussy way of saying numbers are odd or are even), or two objects having the same temperature, or the like.
Seyma Erbas had a post recently that I quite liked. It’s a nearly visual proof of the irrationality of the square root of two. Proving that the square root of two is irrational isn’t by itself a great trick: either that or the proof there are infinitely many prime numbers is probably the simplest interesting proof-by-contradiction someone could do. The Pythagoreans certainly knew of it, and being the Pythagoreans, inspired confusing legends about just what they did about this irrationality.
Anyway, in the reblogged post here, a proof (by contradiction) that the square root of two can’t be rational is done nearly entirely in pictures. The paper which Seyma Erbas cites, Steven J Miller and David Montague’s “Irrationality From The Book”, also includes similar visual proofs of the irrationality of the square roots of three, five, and six, and if the pictures don’t inspire you to higher mathematics they might at least give you ideas for retiling the kitchen. Miller and Montague talk about the generalization problem — making similar diagrams for larger and larger numbers, such as ten — and where their generalization stops working.
Yesterday I came a across a new (new to me, that is) proof of the irrationality of . I found it in the paper “Irrationality From The Book,” by Steven J. Miller, David Montague, which was recently posted to arXiv.org.
Apparently the proof was discovered by Stanley Tennenbaum in the 1950′s but was made widely known by John Conway around 1990. The proof appeared in Conway’s chapter “The Power of Mathematics” of the book Power, which was edited by Alan F. Blackwell, David MacKay (2005).
Recently my dear love, the professional philosopher, got to thinking about a plane that just touches a sphere, and wondered: where does the plane just touch the sphere? I, the mathematician, knew just what to call that: it’s the “point of tangency”, or if you want a phrasing that’s a little less Law French, the “tangent point”. The tangent to a curve is a flat surface, of one lower dimension than the space has — on the two-dimensional plane the tangent’s a line; in three-dimensional space the tangent’s a plane; in four-dimensional space the tangent’s a pain to quite visualize perfectly — and, ordinarily, it touches the original curve at just the one point, locally anyway.
But, and this is a good philosophical objection, is a “point” really anywhere? A single point has no breadth, no width, it occupies no volume. Mathematically we’d say it has measure zero. If you had a glass filled to the brim and dropped a point into it, it wouldn’t overflow. If you tried to point at the tangent point, you’d miss it. If you tried to highlight the spot with a magic marker, you couldn’t draw a mark centered on that point; the best you could do is draw out a swath that, presumably, has the point, somewhere within it, somewhere.
This feels somehow like one of Zeno’s Paradoxes, although it’s not one of the paradoxes to have come down to us, at least so far as I understand them. Those are all about the problem that there seem to be conclusions, contrary to intuition, that result from supposing that space (and time) can be infinitely divided; but, there are at least as great problems from supposing that they can’t. I’m a bit surprised by that, since it’s so easy to visualize a sphere and a plane — it almost leaps into the mind as soon as you have a fruit and a table — but perhaps we just don’t happen to have records of the Ancients discussing it.
We can work out a good deal of information about the tangent point, and staying on firm ground all the way to the end. For example: imagine the sphere sliced into a big and a small half by a plane. Imagine moving the plane in the direction of the smaller slice; this produces a smaller slice yet. Keep repeating this ad infinitum and you’d have a smaller slice, volume approaching zero, and a plane that’s approaching tangency to the sphere. But then there is that slice that’s so close to the edge of the sphere that the sphere isn’t cut at all, and there is something curious about that point.
One of my mathematics-trivia-of-the-day Twitter feeds mentioned that Saturday was the birthday of Thomas Hobbes (5 April 1588 to 4 December 1679), and yes, that Hobbes. I was surprised; I knew Hobbes had written Leviathan and was famous for philosophical works that I hadn’t read either. I had no idea that he’d done anything important mathematically, but then, the generic biography for a mathematician of the 16th or 17th century is “philosopher/theologian who advanced mathematics in order to further his astronomical research”, so, it wouldn’t be strange.
The MacTutor History of Mathematics archive’s biography explains that he actually came to discover mathematics relatively late in life. They quote John Aubrey’s A Brief Life Of Thomas Hobbes for a scene that’s just beautiful:
He was forty years old before he looked on geometry; which happened accidentally. Being in a gentleman’s library Euclid’s Elements lay open, and ’twas the forty-seventh proposition in the first book. He read the proposition. “By God,” said he, “this is impossible!” So he reads the demonstration of it, which referred him back to such a proof; which referred him back to another, which he also read. … at last he was demonstratively convinced of that truth. This made him in love with geometry.
And so began a love of mathematics that, if MacTutor is right, lasted the half-century he had left to his life. His mathematics work would not displace his place as a philosopher, but then, his accomplishments …
Well, that’s the less cheerful part of it. For example, says MacTutor, shortly before his 1655 publication of De Corpore (On The Body) Hobbes worked out a method of squaring the circle, using straightedge and compass alone. It’s impossible to do this (though that it is impossible would take two more centuries to prove), and Hobbes realized his demonstration was wrong shortly before publication. Rather than remove the proof he added text to explain that it was a false proof.
False proofs can be solid teaching tools: just working out where a proof goes wrong is a good exercise in testing one’s knowledge of concepts and how they relate, and whether a concept is actually well-defined yet. And it’s not like attempting to square the circle is by itself ridiculous. I suspect most mathematicians even today give it a try, at least before they can study the proofs that it’s impossible and they can go on to trying to do Fermat’s Last Theorem.
([Edited 31 May 2017 to change from “you can go on to trying” because I finally noticed the shift in pronouns was weird.])
But Hobbes also included a second attempted proof which he again realized was at best “an approximate quadrature”, and tried a third which he realized was wrong while the book was being printed, so he added a note that it was meant as a problem for the reader. Hobbes was sure he was close, and would keep on trying to prove he had squared the circle to the end of his life. These circle-squarings set off a long-running feud with John Wallis, a pioneer in algebra and calculus (and the person who introduced the ∞ symbol to mathematics), who attacked Hobbes’s mistakes and faulty claims.
Hobbes also refused to have anything to do with the algebra and calculus and the symbolic operations which were revolutionizing mathematics at the time; he wanted geometry and nothing but. MacTutor quotes him as insulting — and here we’re reminded that the 17th century was a golden age of academic insulting — Wallis’s Algebra as “a scab of symbols [ which disfigured the page ] as if a hen had been scraping there”.
The best that MacTutor can say about Hobbes’s mathematics is that while he claimed to do a lot of truly impressive work, none of the things which would have been substantial advances in mathematics were correct. And there is something sad that a person of great intellectual power could be so in love with mathematics and find that love wasn’t reciprocated. He wrote near the end of his life a list of seven problems “sought in vain by the diligent scrutiny of the greatest geometers since the very beginnings of geometry” that he concluded he’d solved; and, to be kind, he’s not renowned as the person who found the center of gravity of the quadrant of a circle.
But that sadness is taking an unfair view of the value of doing mathematics. So Hobbes spent a half-century playing with plane figures without finding something true that future generations would regard as novel — how is that a failing? How many professional mathematicians will do something that’s of any general interest, and won’t even write a classic on social contract theory that people will think they probably should’ve read at some point? He found in geometry something which brought him a sense of wonder, and which was delightful enough to keep him going through long and bitter academic feuds (I grant it’s possible Hobbes enjoyed the feuds; some people do), and without apparently losing his enthusiasm. That’s wonderful, regardless of whether his work found anything original.
So how does the first month of 2014 compare to the last month of 2013, in terms of popularity? The raw numbers are looking up: I went from 176 unique visitors looking at 352 pages in December up to 283 unique visitors looking at 498 pages. If WordPress’s statistics are to be believed that’s my greatest number of page views since June of 2013, and the greatest number of visitors since February. This hurt the ratio of views per visitor a little, which dropped from 2.00 to 1.76, but we can’t have everything unless I write stuff that lots of people want to read and they figure they want to read a lot more based on that, which is just crazy talk. The most popular articles, though, were:
How Many Trapezoids I Can Draw, with my best guess for how many different kinds of trapezoids there are (and despite its popularity I haven’t seen a kind not listed here, which surprises me).
Factor Finding, linking over to IvaSallay’s quite interesting blog with a great recreational mathematics puzzle (or educational puzzle, depending on how you came into it) that drove me and a friend crazy with this week’s puzzles.
What’s The Worst Way To Pack? in which I go looking for the least-efficient packing of spheres and show off these neat Mystery Science Theater 3000 foam balls I got.
The countries sending me readers the most often were the United States (281), Canada (52), the United Kingdom (25), and Austria (23). Sending me just a single reader each this past month were a pretty good list:
Bulgaria, France, Greece, Israel, Morocco, the Netherlands, Norway, Portugal, Romania, Russia, Serbia, Singapore, South Korea, Spain, and Viet Nam. Returning on that list from last month are Norway, Romania, Spain, and Viet Nam, and none of those were single-country viewers back in November 2013.
While reading that biography of Donald Coxeter that brought up that lovely triangle theorem, I ran across some mentions of the sphere-packing problem. That’s the treatment of a problem anyone who’s had a stack of oranges or golf balls has independently discovered: how can you arrange balls, all the same size (oranges are near enough), so as to have the least amount of wasted space between balls? It’s a mathematics problem with a lot of applications, both the obvious ones of arranging orange or golf-ball shipments, and less obvious ones such as sending error-free messages. You can recast the problem of sending a message so it’s understood even despite errors in coding, transmitting, receiving, or decoding, as one of packing equal-size balls around one another.
The “packing density” is the term used to say how much of a volume of space can be filled with balls of equal size using some pattern or other. Patterns called the cubic close packing or the hexagonal close packing are the best that can be done with periodic packings, ones that repeat some base pattern over and over; they fill a touch over 74 percent of the available space with balls. If you don’t want to follow the Mathworld links before, just get a tub of balls, or crate of oranges, or some foam Mystery Science Theater 3000 logo balls as packing materials when you order the new DVD set, and play around with a while and you’ll likely rediscover them. If you’re willing to give up that repetition you can get up to nearly 78 percent. Finding these efficient packings is known as the Kepler conjecture, and yes, it’s that Kepler, and it did take a couple centuries to show that these were the most efficient packings.
While thinking about that I wondered: what’s the least efficient way to pack balls? The obvious answer is to start with a container the size of the universe, and then put no balls in it, for a packing fraction of zero percent. This seems to fall outside the spirit of the question, though; it’s at least implicit in wondering the least efficient way to pack balls to suppose that there’s at least one ball that exists.
I was reading a biography of Donald Coxeter, one of the most important geometers of the 20th century, and it mentioned in passing something Coxeter referred to as Morley’s Miracle Theorem. The theorem was proved in 1899 by Frank Morley, who taught at Haverford College (if that sounds vaguely familiar that’s because you remember it’s where Dave Barry went) and then Johns Hopkins (which may be familiar on the strength of its lacrosse team), and published this in the first issue of the Transactions of the American Mathematical Society. And, yes, perhaps it isn’t actually important, but the result is so unexpected and surprising that I wanted to share it with you. The biography also includes a proof Coxeter wrote for the theorem, one that’s admirably straightforward, but let me show the result without the proof so you can wonder about it.
First, start by drawing a triangle. It doesn’t have to have any particular interesting properties other than existing. I’ve drawn an example one.
The next step is to cut into three equal pieces each of the interior angles of the triangle, and draw those lines. I’m doing that in separate diagrams for each of the triangle’s three original angles because I want to better suggest the process.
I should point out, this trisection of the angles can be done however you like, which is probably going to be by measuring the angles with a protractor and dividing the angle by three. I made these diagrams just by sketching them out, so they aren’t perfect in their measure, but if you were doing the diagram yourself on a sheet of scratch paper you wouldn’t bother getting the protractor out either. (And, famously, you can’t trisect an angle if you’re using just compass and straightedge to draw things, but you don’t have to restrict yourself to compass and straightedge for this.)
Now the next bit is to take the points where adjacent angle trisectors intersect — that is, for example, where the lower red line crosses the lower green line; where the upper red line crosses the left blue line; and where the right blue line crosses the upper green line. Draw lines connecting these points together and …
This new triangle, drawn in purple on my sketch, is an equilateral triangle!
(It may look a little off, but that’s because I didn’t measure the trisectors when I drew them in and just eyeballed it. If I had measured the angles and drawn the new ones in carefully, it would have been perfect.)
I’ve been thinking back on this and grinning ever since reading it. I certainly didn’t see that punch line coming.
Through the MacTutor archive I learn that today’s the birthday of Émile Michel Hyacinthe Lemoine (1840 – 1912), a mathematician I admit I don’t remember hearing of before. His particular mathematical interests were primarily in geometry (though MacTutor notes professionally he became a civil engineer responsible for Paris’s gas supply).
What interests me is that Lemoine looked into the problem of how complicated a proof is, and in just the sort of thing designed to endear him to my heart, he tried to give a concrete measurement of, at least, how involved a geometric construction was. He identified the classic steps in compass-and-straightedge constructions and classified proofs by how many steps these took. MacTutor cites him as showing that the usual solution to the Apollonius problem — construct a circle tangent to three given circles — required over four hundred operations, but that he was able to squeeze that down to 199.
However, well, nobody seems to have been very interested in this classification. That’s probably because the length doesn’t really measure how complicated a proof (or a construction) is: proofs can have a narrative flow, and a proof that involves many steps each of which seems to flow obviously (or which look like the steps in another, already-familiar proof) is going to be easier to read and to understand than one that involves fewer but more obscure steps. This is the sort of thing that challenges attempts to measure how difficult a proof is, even though it’d be interesting to know.
Here’s one of the things that would be served by being able to measure just how long a proof is: a lot of numerical mathematics depends on having sequences of randomly generated numbers, but, showing that you actually have a random sequence of numbers is a deeply hard problem. If you can specify how you get a particular digit … well, they’re not random, then, are they? Unless it’s “use this digit from this randomly generated sequence”. If you could show there’s no way to produce a particular sequence of numbers in any way more efficiently than just reading them off this list of numbers you’d at least have a fair chance at saying this is a truly unpredictable sequence. But, showing that you have found the most efficient algorithm for producing something is … well, it’s difficult to even start measuring that sort of thing, and while Lemoine didn’t produce a very good measure of algorithmic complexity, he did have an idea, and it’s difficult to see how one could get a good measure if one didn’t start with trying not-very-good ones.
Jason Chatfield’s Ginger Meggs (October 14) does the usual confused-student joke. It’s a little unusual in having the subject be different ways to plot data, though, with line graphs, bar graphs, and scatter graphs being shown off. I think remarkable about this is that line graphs and bar graphs were both — well, if not invented, then at least popularized — by one person, William Playfair, who’s also to be credited for making pie charts a popular tool. Playfair, an engineer and economist of the late 18th and early 19th century, and I do admire him for developing not just one but multiple techniques for making complicated information easier to see.
Eric the Circle (October 16) breaks through my usual reluctance to include it — just having a circle doesn’t seem like it’s enough — because it does a neat bit of mathematical joking, in which a cube looks “my dual” in an octahedron. Duals are one of the ways mathematicians transform one problem into another, that usually turns out to be equivalent; what’s surprising is that often a problem that’s difficult for the original is easy, or at least easier, for the dual.
I haven’t forgot my little problem about working out where the apparent edge of the world was, from my visit to the Sleeping Bear Dunes in northern (lower) Michigan. What I have been is stuck on a way to do all the calculations in a way that’s clear and that avoids confusion. I realized the calculations were reasonably clear to me but were hard to describe because I could put into similar-looking symbols a bunch of things I wanted to describe.
So I’ve resolved that the best thing I can do is take some time to describe the things I mean, and why they’ll get the symbols that they do. The first part of this is drawing a slightly more mathematical representation of the situation of standing on top of the dune and looking out at the water, and seeing the apparent edge of the dune as something very much closer than the water is. This is what’s behind my new picture, a cross-section of the dune and a person looking out at its edge.
The slightly dirty secret, though, is that it isn’t. It’s built around logical arguments, certainly, and the more rigorous the argument the better-proven a thing is usually considered to be. But you don’t get results proven with perfectly rigorously airtight deductive reasoning, at least not in the journals and monographs that report interesting new results, because it turns out this requires so much work that it takes forever. What you typically see is enough of an argument to be convincing that anything elided over could be filled in, if required. This is part of why huge results professing major new accomplishments, like a proof of Goldbach’s Conjecture, take time to verify: not only is there a lot that’s there, but suddenly the question of whether the elided steps really are secure has to be filled in.
Most of the big gaps-to-be-filled in basic mathematics were filled in a century ago. Pasch was among the people who found some points in Euclidean geometry where physical intuition about real-world things was assumed into mathematical arguments without it being explicitly stated. This didn’t mean any geometric results were wrong or counterintuitive or anything; just that there were assumptions in the system that Euclid — and everybody else — had made without saying they were making them, which is pretty impressive considering that Euclid thought to mention that he was assuming all right angles were congruent.
One of those discovered spots gets called now Pasch’s Axion, and it gives a good example of the kind of thing which can go centuries being assumed without drawing attention to itself: suppose you have a triangle connecting the points we label A, B, and C. And suppose you have a line which enters the triangle through the leg connecting points A and B, and which doesn’t pass through the point C. Then the line exits the triangle either through the leg between points B and C or through the leg between points C and A.
Obvious? Perhaps, but not more obvious than the axiom that a line segment can be drawn between any two different points, and it’s a special insight to notice these things are assumptions.
I may need to revise my seven-or-so-comic standard for hosting one of these roundups of mathematics-themed comic strips, at least during the summer vacation. We’ll see how they go as the school year picks up and cartoonists return to the traditional jokes of students not caring about algebra and kids giving wiseacre responses to word problems.
Jan Eliot’s Stone Soup began a sequence on the 26th of August in which Holly, the teenager, has to do flash cards to improve her memorization of the multiplication tables. It’s a baffling sequence to me, at least, since I can’t figure why a high schooler needs to study the times tables (on the 27th, Grandmom says it’s because it will make mathematics easier the more arithmetic she can do in her head). It’s also a bit infuriating because I can’t see a way to make sure Holly sees mathematics as tedious drudge work more than getting drilled by flash cards through summer vacation, particularly as she’s at an age where she ought to be doing algebra or trigonometry or geometry.
Steve Moore’s In The Bleachers (September 1) uses a bit of mathematics as a throwaway “something complicated to be thinking about” bit. I do like that the mathematics shown at least parses. I’m not sure offhand what problem the pitcher is trying to solve, that is, but the steps in it are done correctly, and even show off a nice bit of implicit differentiation. That’s a bit of differential calculus where you’ll find the rate of change of one variable with respect to another depends on the value of the variable, which isn’t actually hard to do if you follow the rules correctly but which, as I remember it, produces a vague sense of unease at its introduction. Probably it feels vaguely illicit to have a function defined in, in parts, in terms of itself.
So I want to understand the illusion of being at the edge of the world at the Sleeping Bear Dunes in northern Michigan. Since I like doing mathematics I think of this as a mathematics problem; so, I figure, I need to put together some equations. Before I do that, I need to think of what I want the equations to represent, which is the part of the problem where I build a model of the dunes. In the process I should get at least a qualitative idea of the effect; later, I should be able to quantify that.
What’s a dune? Well, it’s a great whomping big pile of sand right next to the water. There’s more to a dune than that, but since all I’m interested in is how the dune looks, I don’t need to think about much more than what the dune’s shape is, and how it compares to the water beside it. If I wanted to understand the ecology of a dune, or fascinating things like how it moves, then I’d have to model it in greater detail, but for now I’m going to try out this incredibly simple model and see what it gets me.
I got to visit the Sleeping Bear Dunes National Lakeshore earlier this month. I thought I knew what dunes were, from the little piles of sand that accumulated on the Jersey Shore sometimes, but, no. These dunes, at the northern end of Michigan’s lower peninsula, look out on Lake Michigan from, at one spot on the Pierce Stocking Scenic Drive, about 450 feet above sea level. That’s a staggering height, that awed me, particularly as we approached it from the drive, and so had a previously quite nice enough drive through lovely forests come to a sudden, almost explosive, panorama of sand towering far above the ocean.
Maybe a quick working definition of a “scenic overlook” is a spot of land from which you can look noticeably down and see birds flying. This is vey scenic: had there been a Saturn V moon rocket, on the launchpad, at sea level, we would have been looking noticeably down at its escape rocket. For that matter, if there had been a Saturn V moon rocket, on top of which was somehow perched a Gemini-Titan rocket, we’d still be … well, we’d have to look up to see the crew in the Gemini capsule, but we would be about eye level with the top of the Titan booster, anyway.
Something that I imagine no picture except a three-dimensional one is going to capture, though, is the sense that one is standing at the edge of the world. From the top of the dune, the end of the sand seems to be very nearby, maybe a couple dozen feet off, and the water below is so clearly distant that it feels impossibly far away. Walking toward that edge makes the edge of the world recede, of course, but it never quite loses that sense of being on the precipice until quite far along.
Some mad souls do follow a trail all the way down to the beachfront of Lake Michigan; I wasn’t among them. The difficulty in walking back up — all on sand, on a pretty significant slope — from just walking a little near the edge and maybe dropping twenty feet or so in altitude convinced me not to carry on. I didn’t know it was a full 450 feet up, but it was obviously far enough.
The geometry of all this, though, has captivated me, and I hope to spend a couple essays here working out such questions as just how the optical illusion of this edge of the world worked, and how its recession works.
WordPress says that in July 2013 I had 341 pages read, which is down rather catastrophically from the June score of 713. The number of distinct visitors also dropped, though less alarmingly, from 246 down to 156; this also implies the number of pages each visitor viewed dropped from 2.90 down to 2.19. That’s still the second-highest number of pages-per-visitor that I’ve had recorded since WordPress started sharing that information with me, so, I’m going to suppose that the combination of school letting out (so fewer people are looking for help about trapezoids) and my relatively fewer posts this month hit me. There are presently 215 people following the blog, if my Twitter followers are counted among them. They hear about new posts, anyway.
My most popular posts over the past 30 days have been:
Counting From 52 To 11,108, some further work from Professor Inder J Taneja on a lovely bit of recreational mathematics. (Professor Taneja even pops in for the comments.)
Geometry The Old-Fashioned Way, pointing to a fun little web page in which you can work out geometric constructions using straightedge and compass live and direct on the web.
And The $64 Question Was, in which I learned something about a classic game show and started to think about how it might be used educationally.
My all-time most popular post remains How Many Trapezoids I Can Draw, because I think there are people out there who worry about how many different kinds of trapezoids there are. I hope I can bring a little peace to their minds. (I make the answer out at six.)
The countries sending me the most viewers the past month have been the United States (165), then Denmark (32), Australia (24), India (18), and the United Kingdom and Brazil (12 each). Sorry, Canada (11). Sending me a single viewer each were Estonia, Slovenia, South Africa, the Netherlands, Argentina, Pakistan, Angola, France, and Switzerland. Argentina and Slovenia did the same for me last month too.
Ruben Boiling’s Tom the Dancing Bug (July 12) features one of his irresistible (to me) “Super-Fun-Pak Comix”, among them, A Voice From Another Dimension, which is a neat bit of Flatland-inspired fun between points in space. Edwin Abbot Abbot’s Flatland is one of those rare advanced-mathematical concepts that got firmly lodged into the pop culture, probably because it is a supremely accessible introduction to the concept of multidimensional space. People love learning about things which go against their everyday intuition, and the book (eventually) made the new notions of general relativity feel like they could be understood by anyone.
There’s a package of forty challenges offered, things like drawing squares or making rosettes of circles or the like, and if that’s not enough there’s the challenge in beating a par for the number of moves required. Meanwhile I’m gratified to learn that, years after I had to do this stuff for school, I’ve still remembered how to do bisections of lines and angles, and how to drop perpendiculars to a point.
(I haven’t figured yet how to draw a circle to an arbitrary point — sometimes you don’t need to connect anywhere particular — but imagine if I read the instructions maybe that would be obvious or something.)
I’m surprised to discover it’s been over a month since I had a roster of mathematics-themed comic strips to share, but that’s how things happen to happen. It’s also been a month with repeated references to “finding square roots”, I suppose because that sounds like a really math-y thing to do. It’s certainly computationally challenging; the task of finding such is even a (very minor) moment in Isaac Asimov’s magnificent short story about arithmetic, “The Feeling Of Power”. I remember reading the procedure for finding them when I was a kid, and finding that with considerable effort, I was able to, though I’d probably refuse to do more than give a rough estimate of such a root nowadays.
Bill Watterson’s Calvin and Hobbes (June 4, rerun) is another entry in the long string of jokes about “why bother studying mathematics”, but Watterson’s craft lifts it above average. Admire that fourth panel: that’s every resistant student in one pose.
I apologize for being so quiet the past few days. I haven’t had the chance to write what I mean to. To make it up to you please let me reblog this charming tesselation from RobertLovesPi. Tesselations are wonderful sections of mathematics because they lend themselves to stunning pictures and thoughts of impractical ways to redo the kitchen floor. They also depend on symmetries and rotations to work, which is a hallmark of group theory, which starts out by looking at things which look like addition and multiplication and which ends up in things like predicting how many different kinds of subatomic particles there ought to be. (I haven’t gone that far in studying group theory so I’d have to trust other people to fill in some of the gaps here.)
The flow of mathematics-themed comic strips almost dried up in April. I’m going to assume this reflects the kids of the cartoonists being on Spring Break, and teachers not placing exams immediately after the exam, in early to mid-March, and that we were just seeing the lag from that. I’m joking a little bit, but surely there’s some explanation for the rash of did-you-get-your-taxes-done comics appearing two weeks after April 15, and I’m fairly sure it isn’t the high regard United States newspaper syndicates have for their Canadian readership.
Dave Whamond’s Reality Check (April 8) uses the “infinity” symbol and tossed pizza dough together. The ∞ symbol, I understand, is credited to the English mathematician John Wallis, who introduced it in the Treatise on the Conic Sections, a text that made clearer how conic sections could be described algebraically. Wikipedia claims that Wallis had the idea that negative numbers were, rather than less than zero, actually greater than infinity, which we’d regard as a quirky interpretation, but (if I can verify it) it makes for an interesting point in figuring out how long people took to understand negative numbers like we believe we do today.
Jonathan Lemon’s Rabbits Against Magic (April 9) does a playing-the-odds joke, in this case in the efficiency of alligator repellent. The joke in this sort of thing comes to the assumption of independence of events — whether the chance that a thing works this time is the same as the chance of it working last time — and a bit of the idea that you find the probability of something working by trying it many times and counting the successes. Trusting in the Law of Large Numbers (and the independence of the events), this empirically-generated probability can be expected to match the actual probability, once you spend enough time thinking about what you mean by a term like “the actual probability”.
I do want to work out the solution by calculus methods, though, partly because that was actually easier for me, and partly to see whether my audience will put up with such. I’m trying to figure out how to present a more complicated subject which sure looks like it needs calculus to explain, and I’d like to have some sense whether I can write coherently on that topic so.
To set the stage: the problem was about where to stand, behind a tall obscuring fence, so as to see the greatest view of a building hidden behind the fence. To make for simple enough numbers, the viewer is assumed to have eyes six feet off the ground, the fence is eight feet tall, and the building, four feet beyond the fence, is twelve feet tall. Trusting that the ground is level — the reality isn’t quite, as it is at an amusement park — and that you can get as near or as far from the fence as you like, when does the angle between the top of the building and the top of the fence get its biggest?
To get to my next point about Arthur Christmas I needed to know how fast an arbitrary point on the Earth is moving, as the Earth rotates. This required me getting out a sheet of paper and doing some sketches, so, I figured it’s worth a side article to explain what I was doing.
The first thing was that I simplified stuff. In particular, I decided the Earth is near enough a sphere that I’m not bothering with the fact that it isn’t. The difference between an actual sphere and the geoid is not worth bothering with unless you’re timing the retrofire for a ballistically-reentering space capsule. That’s … actually fairly close to the problem I want, about how long it might take the reindeer and sleigh to get back to Arthur Christmas and Grand-Santa, but that’s also too much work for the improvement in the answer I’d get.
I left Arthur Christmas and Grand-Santa in a hypothetical puzzle, inspired by the movie, with them stranded on a tiny island while their team of flying reindeer and sleigh carried on in a straight line without them. I am assuming for the sake of an interesting problem that this means the reindeer are carrying on the Great Circle route, favored by airplanes and satellites, and that the reindeer are in an orbit more like the satellite’s than the reindeers — that is, they keep to a circle in a plane which isn’t rotating while the Earth does, since otherwise, Arthur and Grand-Santa have to wait only for the reindeer to finish one lap around the planet and somehow get up to flying altitude to be picked up. If the reindeer aren’t rotating the with the Earth, then, when the reindeer finish one circuit our heroes are going to be … well, maybe east, maybe west, of the reindeer; the problem is, they’re going to be away.
As promised, since I’ve got the chance, I want to return to the question of the reindeer behavior as shown in the Aardman movie Arthur Christmas, and what would ultimately happen to them if the reindeer carry on as Grand-Santa claims they will. (Again, this does require spoiling a plot point of the film and so I tuck the rest behind a cut.)
I follow several Mathematics twitter accounts, mostly so that I can run across some interesting points I didn’t know about and feel a little dumber the rest of the day (oh, good grief, of course if f is a quasi-convex function and y a convex combination of x and z then f(y) is less than or equal to the maximum of f(x) and f(z)). Mostly they’re little “huh” bits. Unfortunately I’ve lost which one I found this item from originally, but it was just a link to an interesting puzzle result: how to cut a cake into four equal pieces using a single slice.
It’s been long enough since my last roundup of mathematics-themed comics to host a new one. I’m also getting stirred to try tracking how many of these turn up per day, because they certainly feel like they run in a feast-or-famine pattern. There’d be no point to it, besides satisfying my vague feelings that everything can be tracked, but there’s data laying there all ready to be measured, isn’t there?
As before, this is going to be the comics other than those run through King Features Syndicate, since I haven’t found a solution I like for presenting their mathematics-themed comic strips for discussion. But there haven’t been many this month that I’ve seen either, so I can stick with gocomics.com strips for today at least. (I’m also a little irked that Comics Kingdom’s archives are being shut down — it’s their right, of course, but I don’t like having so many dead links in my old articles.) But on with the strips I have got.
I suppose it’s been long enough to resume the review of math-themed comic strips. I admit there are weeks I don’t have much chance to write regular articles and then I feel embarrassed that I post only comic strips links, but I do enjoy the comics and the comic strip reviews. This one gets slightly truncated because King Features Syndicate has indeed locked down their Comics Kingdom archives of its strips, making it blasted inconvenient to read and nearly impossible to link to them in any practical, archivable way. They do offer a service, DailyInk.com, with their comic strips, but I can hardly expect every reader of mine to pay up over there just for the odd day when Mandrake the Magician mentions something I can build a math problem from. Until I work out an acceptable-to-me resolution, then, I’ll be dropping to gocomics.com and a few oddball strips that the Houston Chronicle carries.
The Math Less Traveled over here shows off a lovely way of visualizing the factoring of integers by putting them into patterns inspired by the regular polygons. Some numbers factor into wonderfully obvious patterns; some turn into muddles of dots because integers just work that way. They’re all attractive ways to look at numbers, though.
In an idle moment a while ago I wrote a program to generate "factorization diagrams". Here’s 700:
It’s easy to see (I hope), just by looking at the arrangement of dots, that there are $latex 7 \times 5 \times 5 \times 2 \times 2$ in total.
Here’s how I did it. First, a few imports: a function to do factorization of integers, and a library to draw pictures (yes, this is the library I wrote myself; I promise to write more about it soon!).
The primeLayout function takes an integer n (assumed to be a prime number) and some sort of picture, and symmetrically arranges n copies of the picture.
. I found it in the paper “Irrationality From The Book,” by Steven J. Miller, David Montague, which was recently posted to arXiv.org.
nebusresearch 