December 2018 - mathfun@mailman.xmission.com

[math-fun] Trisection with folding
by James Propp 17 Dec '18

17 Dec '18

In his video "How to Make a Paper Snowflake" ( https://www.youtube.com/watch?v=BIiEsIDenTk&feature=youtu.be ) Matt Parker shows how to trisect a 180 degree angle by pulling a fold in a constrained fashion until it meets a certain point. It seems to me that this method should allow one to construct some angles that can't be trisected with straightedge and compass. And I suspect it's one of the already-studied construction methods that go beyond straightedge and compass. Do any of you know anything about this? Thanks, Jim Propp

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Re: [math-fun] Egyptian Fractions
by Keith F. Lynch 17 Dec '18

17 Dec '18

Cris Moore <moore(a)santafe.edu> wrote: > But it?s a lovely exercise that the greedy algorithm for Egyptian > fractions ? that is, subtract the largest reciprocal ? terminates > for any rational number between 0 and 1. There?s an inductive proof, > but the nature of the induction is surprising. Found it! The numerator always decreases. For instance for 4/97, whose GEF is 1/25 + 1/809 + 1/980913 + 1/1924379646225, the remainders are 4/97, 3/2425, 2/1961825, and 1/1924379646225. (My first thought was that the denominators increase so quickly that the partial sums are too close to rationals, thus proving that any non-terminating GEF must be irrational. There were just two minor problems. One is that any such proof would probably also prove that all non-terminating GEFs must be transcendental, which is absurd, since all algebraic irrationals have a non-terminating GEF. The other is that the inverse Sylvester sequence (1/A000058) is a non-terminating GEF that converges to 1.) It's also true for any rational number between 0 and 2. For instance for 19/10, whose GEF is 1/1 + 1/2 + 1/3 + 1/15, the remainders are 19/10, 9/10, 2/5, and 1/15. It's not true for most larger rationals, but I think I can extend the proof to cover them anyway. I've read that it's not known whether GEFs constrained to only use odd denominators will terminate for every rational with an odd denominator. (It's easy to see that none will terminate for any rational with (in simplest form) an even denominator.) There's no obvious pattern in the numerators of the remainders. For instance 2/3, whose OGEF is 1/3 + 1/5 + 1/9 + 1/45 has remainders of 2/3, 1/3, 2/15, and 1/45. Apparently nobody has yet found any rational with an odd denominator known not to have a terminating OGEF.

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[math-fun] What's this fractal's name?
by Mike Stay 16 Dec '18

16 Dec '18

I've seen it here on the list multiple times, but can't find it at the moment. https://imgur.com/a/KeO26Sq -- Mike Stay - metaweta(a)gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com

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[math-fun] Draft of December 2018 blog post
by James Propp 16 Dec '18

16 Dec '18

[Comments on the following are welcome.] Here’s something you’ll never see in popular writing about musicians: Music. The mere word conjures up memories of metronomes and endless scales, the music student’s never-ending fear of playing a wrong note (or the right note at the wrong time), and the frowns of music teachers from whom a curt “Good” was the highest expression of praise. And yet there are some people who just can’t get enough of making music; they practice hour after hour, honing their skills, long after the stage of life when there are parents and teachers forcing them to do it. What strange quirk of character compels this behavior? As I said, nobody writes about musicians that way. And yet, when they’re writing about mathematicians, writers sometimes come up with passages like this: You and math – one of the greatest love/hate relationships of all time. What is it about the subject that excites us yet sends a chilling tingle down our spine at the same time? How can it be so precise, yet so fickle? We may never know the answers to these questions, but we do know that math is ubiquitous, though some of us may try to hide from it. This is from the introduction to a 2010 profile of mathematics editor Vickie Kearn <http://blog.press.princeton.edu/2010/08/31/executive-editor-vickie-kearn-an…>, which I recently saw on the blog-site of Princeton University Press, where Ms. Kearn has worked as senior editor for many years. The intro continues in a similar vein. “… While math may sometimes cause us to cry tears of despair, it has caused Vickie to cry tears of joy.” This is an extreme example, I grant you (the author was an undergraduate intern). But the underlying approach is one I’ve seen other journalists use, though usually not so blatantly. I think I get what those journalists are trying to do: they’re reaching out to people who wouldn’t ordinarily read an article about math and saying “Hey, I’m not so different from you; you should really give my article a try.” But how far can one take this device before it becomes demeaning to the subject matter? A related gripe of mine is news editors who, when publishing an article about mathematics, feel they need to dispel the tension conjured up by the mere scent of math by sticking in a jokey headline like “It all adds up for math whiz” or “Math proves to be winning formula for local teen”. (If you’ve seen headlines like these, please submit them in the Comments!) Do these editors know something that I don’t about the cues that cause a reader to stop reading one article and switch to another? Is it the same thing that Stephen Hawking’s editors knew, when they told him (back when he was writing “A Brief History of Time”) that every equation causes half of the readers of a book to stop reading? I’d like to think that these editors are wrong about the world that they and I live in, but part of me is worried that they’re right — in part because I sometimes stop reading an essay or article when I hit a passage that calls for a little more thought than the part that came before. I’m guessing that some of those journalists and editors are mathphobes, while others are not themselves mathphobes but are eager to keep their stories appealing to a wide range of readers, including mathphobes. But whatever their motive may be for reminding math-anxious readers of their math anxiety, the effect is the same, and I worry that the cumulative impact of these messages normalizes or even valorizes that anxiety. Part of what’s going on is a conflation of mathematics with arithmetic, combined with a fear of numbers. Numbers frighten many people, or leave them feeling cold. But what is accomplished when so many articles about math start by reminding readers of this? Question: Is this sort of pandering to mathphobia an American phenomenon, or is it found in other countries? Somehow I don’t imagine French newspapers (for instance) adopting this sort of tactic. The intern who wrote the passage I quoted strikes me as someone who fears math too much to be a good tour-guide on a mathematical journey for non-mathematicians. Someone like me may not be the best tour-guide either, since I don’t know first-hand what it’s like to be a non-mathematician intimidated by mathematics (though I do know what it’s like to be, for instance, a non-algebraic-geometer intimidated by algebraic geometry, which is a vaguely similar situation). Probably the best tour-guides are people who have a dual perspective as insider and outsider. Who are your favorite converts-as-evangelists in the domain of mathematics? Let me know in the Comments! Jim Propp

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[math-fun] What's this fractal's name?
by Bill Gosper 15 Dec '18

15 Dec '18

On 2018-12-10 15:10, Mike Stay wrote: > I've seen it here on the list multiple times, but can't find it at the moment. > https://imgur.com/a/KeO26Sq I just call it (turned CW 90º) the complex base -2 region. (The digits are 0 and the cube roots of 1.) "Proof": Divide it by -2. (= rotate 180º and halve the scale.) Make four copies, displaced by each of the four digits. Form their union, which will be disjoint except at the boundaries. It's ba-a-ack! —rwg

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Re: [math-fun] spacefilling surface
by Bill Gosper 15 Dec '18

15 Dec '18

On 2018-12-13 14:04, Andy Latto wrote: > Aren't there even simpler solutions? If f:R -> R^3 is space-filling, define > > g:R^2 -> R^3 > > by > > g(x,y) = f(x). > Holy cow, doesn't that cover the image a Continuum number of times!? Warren Smith offered: >just let F(t) be a plane filling curve and then the spacefilling surface mapping (s,t) -> (x,y,z) is (s, F(t)). DAWK—Just "cylindrize" the planefill! Double DAWK: All along, gosper.org/base2funnel1.png was sitting in my Math folder. —rwg > > On Thu, Dec 13, 2018 at 4:34 PM Bill Gosper <billgosper(a)gmail.com> wrote: >> >> Rohan requested a continuous map from a 2D surface onto a non-porous patch >> of >> positive 3D volume. There's one under our noses, but maybe not where you >> think. >> We certainly have continuous maps from 1D onto 3D, so it is >> tempting to compose such a volume-filling function with some sort of >> (necessarily >> multivalued) 1D inverse of a planefilling function. But such inverses >> are wildly >> discontinuous, no matter which values you choose as the definition. >> Although >> 1/2+i/2 is "merely" triple point, the Hilbert "curve" is dense with >> quadruple points. >> The preimage of a sufficiently small neighborhood of such a point will >> comprise at >> least four disjoint pieces. Could the 1D to 3D map possibly smooth >> everything out? >> Maybe, but there's a much cleaner solution. SPOILER: gosper.org/3dsno.png >> gives a >> continuous mapping of an equilateral triangle onto a standard 3 simplex. >> —rwg

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Re: [math-fun] High school physics problem
by Henry Baker 15 Dec '18

15 Dec '18

Hmmm... Evolutionary pressures would seem to have optimized *seed shapes* in such a way to *burrow to an optimum level in the ground*. If someone were to do a high school science project, it would be interesting to do some shaking & "artificial weathering" experiments to see how various seed shapes perform in different soils. So the problem isn't so much that Brazil nuts are competing with almonds or peanuts, but that they are competing with bits of dirt & rocks for an optimum space in soil where they can germinate. At 03:53 AM 12/12/2018, Bill Gosper wrote: >[Eavesdroppers: Other respondents offered > >https://en.wikipedia.org/wiki/Granular_convection and "the Brazil nut effect".] > >On 2018-12-10 09:16, Henry Baker wrote: >> Has this problem been studied before? > >In the New Jersey gravel pits. > >Shore dwellers' lots are plain white sand. > >Your cheapest driveway is a truckload of low grade gravel, with granularity ranging from pebbles all the way down to clay. You sink in if you try to drive (or even walk) on it. > >But after a few weeks of weather, and then initially light usage, all the big pebbles are on top. > >Ârwg

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Re: [math-fun] High school physics problem
by Bill Gosper 14 Dec '18

14 Dec '18

[Eavesdroppers: Other respondents offered https://en.wikipedia.org/wiki/Granular_convection and "the Brazil nut effect".] On 2018-12-10 09:16, Henry Baker wrote: > Suppose we have a large container not 100% filled > with ball bearings (all of exactly the same material > & hence density) but with a wide & random variety of > radii. The largest ball bearings are << the smallest > dimension of the container, and the smallest ball > bearings can get into every corner of the container. > > Initially, the bearings of all sizes are equally > distributed w.r.t. location within the filled > portion of the container. > > We're on Earth with its standard gravity g and > standard atmospheric pressure & temperature (if > that matters). > > Now shake the container vigorously for quite > a while (but not so vigorously enough that the > balls become dented or heated!). > > What is the resulting distribution of sizes > after this shaking? > > A. The same as before. > > B. The larger balls become more likely in > the upper layers. > > C. The larger balls become more likely in > the lower layers. > > I don't have a proof, but I have an intuition > about which answer is correct. > > Has this problem been studied before? > In the New Jersey gravel pits. Shore dwellers' lots are plain white sand. Your cheapest driveway is a truckload of low grade gravel, with granularity ranging from pebbles all the way down to clay. You sink in if you try to drive (or even walk) on it. But after a few weeks of weather, and then initially light usage, all the big pebbles are on top. —rwg

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[math-fun] Fun series
by Dan Asimov 14 Dec '18

14 Dec '18

Pick an integer p in {2, 3, 4, ...} = Z+ - {1}. If you add up the curves in the complex plane C given by f_n(t) = (1/p^n) exp(p^n * it), (so we don't have to worry about convergence), we get the closed plane curve f(t) = Sum_{0 <= n < oo} (1/p^n) exp(p^n * it), 0 <= t <= 2π in C. From plotting this, I know there are a few parts of the curve where f is not one-to-one. It is also clear that the curve contains a *simple* closed curve that bounds a unique minimal open set of positive area. Some questions: --------------- 1. Is the image of f locally simply connected anywhere at all?* 2. Does the image of f contain a differentiable arc anywhere at all? 3. Do the answers depend on the choice of integer p >= 2 ? —Dan ————— * A space X is locally simply connected if for each x in X, every neighborhood of x contains an neighborhood U of x such that U is simply connected: All closed loops in U can be continuously shrunk to a point in U.

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[math-fun] Messages in pi
by James Propp 14 Dec '18

14 Dec '18

Keith Lynch writes: “Carl Sagan’s 1985 novel _Contact_ is a classic, though not very original, SETI novel. Unoriginal except for a subplot about the aliens having discovered messages hidden in the base-eleven digits of pi, which was unfortunately left out of the 1997 film of the same name. I find it an interesting question whether evidence for such a message in pi can ever overcome the possibilities that it's either an astonishing coincidence or someone hacking the computer that's doing the calculation. Presumably if there really was such a message, it would have to be from God, rather than from advanced aliens.” I’m not sure if we’ve discussed this before on math-fun, but I’ve never found this aspect of Sagan’s novel compelling or even coherent. I can imagine a God who intervenes in ways that locally violate the laws of physics, or a God who chooses the laws of physics, but not a God who chooses the laws of mathematics. Is there anything that this might mean? My guess is that, even though Sagan acknowledges in the novel that pi is a mathematical constant rather than a physical one, he didn’t really understand what this entails. Hiding a message in the digits of the base ten expansion of pi is no different from hiding a message in the digits of the base ten expansion of seventeen. (“Mathematicians will tell you that it’s all zeros after the decimal point, but how far out how they really checked?”) I know that the mathematician and science-fiction novelist Greg Egan plays thought experiments with the mutability of math, but I always get the sense with him that at least a fraction of his tongue is in his cheek when he does this, whereas I think Sagan was invoking mathematical ideas in an attempt to convey a sense of transcendence, much as Arthur C. Clarke had done in _2001: A Space Odyssey_ (when the newly transhuman Bowman chides his recently human self for failing to imagine that the numerical pattern governing the proportions of the monolith were limited to three dimensions). Trying to convey something beyond human comprehension is a tricky business; it’s akin to designing good technobabble, but harder. Jim Propp

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