November 2016 - mathfun@mailman.xmission.com

[math-fun] The base 2-i, digits 0,i^0..3 spacefill
by Bill Gosper 24 Nov '16

24 Nov '16

x xor <non dyadic-rational>: Discontinuous at the dyadic rationals. Continuous everywhere else? --rwg Date: 2016-11-23 02:51 From: Joerg Arndt <arndt(a)jjj.de> To: math-fun <math-fun(a)mailman.xmission.com> * Bill Gosper <billgosper(a)gmail.com> [Nov 23. 2016 10:21]: > [...] > He says it's really just prf dressed up. And he makes the remarkable > observation > that the frac4 dragon > gosper.org/4flopfour.png > is the bar graph of x xor 2/3 ! > gosper.org/xor667fade.png > It appears that x xor r makes a fractile for every rational r. --rwg Correct: write 1 in binary as 0.11111..., then observe that (x xor r) + ((1-x) xor r) == 1. Best regards, jj

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Re: [math-fun] The base 2-i, digits 0,i^0..3 spacefill
by Bill Gosper 23 Nov '16

23 Nov '16

On Fri, Nov 11, 2016 at 12:44 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > (whose?) underlying the pretty gosper.org/dana.PNG has only moderately > interesting > Fourier gosper.org/rt5-324.png and polygonal gosper.org/rt5.png > approximations. > --rwg > But we should remember that, for images lacking bilateral symmetry, there are *uncountably many* filled shapes, since you can flip a coin as to whether to mirror image ("flop" --BBM) at each level of recursion. Even unto the lowly dyadic (Heighway) dragon which, if you conjugate at every level, becomes simply a triangular patch of a square grid. Two back-to-back: gosper.org/dragrid199.png --rwg Is there a gallery of variously flopped dragons anywhere? Julian, can they be made with piecewiserecursivefractal?

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[math-fun] holonomicfunctions.m ?
by Bill Gosper 22 Nov '16

22 Nov '16

Mma 10 and 11 can't read the file. http://www.risc.jku.at/research/combinat/software/ergosum/RISC/HolonomicFun… says "Right now you are using Version 1.7.1 released on October 9, 2013. ..." How does it know? ("You are *here*.") "... This version is compatible with Mathematica versions from 5.2 to 9.0." I'd try an older Mma, but then V 11 tends to have major license snits requiring manual reactivation. Before I download and reinstall, is anybody else using Koutschan's package? --rwg

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Re: [math-fun] Space-filling curves again
by Bill Gosper 21 Nov '16

21 Nov '16

On 2016-10-19 10:11, James Propp wrote: > Very pretty. What's the math here? (Apologies if you've already answered > this question.) > > Jim Propp Julian wrote a nifty Fourier expander for recursive Koch polygons. E.g., ptsgnlst2Fouriermat[{0, 1, I^(2/3), 1 + I^(2/3), 2}, {1, -1, 1, -1}] spacefills the triangle joining 0 to 2 via 1+i√3. Actually, it makes an infinite 3x3 matrix product for the coefficients a(k). Then Sum a(k+1/m) exp(2 i π (k+1/m)) repeats the fractal on the sides of an m-gon. The gif just accumulates consecutive harmonics. --rwg > > On Wednesday, October 19, 2016, Bill Gosper <billgosper(a)gmail.com> wrote: > >> With much help from Julian, gosper.org/hellodoily.gif >> --rwg >> >> On Fri, Sep 30, 2016 at 10:13 AM, Bill Gosper <billgosper(a)gmail.com >> <javascript:;>> wrote: >> >> > On Thu, Sep 29, 2016 at 8:33 PM, Bill Gosper <billgosper(a)gmail.com >> <javascript:;>> wrote: >> > >> >> The animation I intended was perhaps too computationally ambitious. >> >> Meanwhile, gosper.org/FHilbert.gif is just conventional plots of the >> >> first 288 >> >> approximations. I dislike such plots due to the finite line thickness, >> >> which someone might imagine contributed to spacefilling. >> >> If you read all the frames into Preview, say, you can see that the >> texture >> >> of frame n repeats at frame 4n. This gives us a way to interpolate >> >> the nonexistent frame n+1/4 from frame 4n+1. >> >> >> >> Also uploaded: gosper.org/outcirc.gif made by flipping a sign in >> >> incirc.gif. --rwg >> >> Stay tuned for a Ptolemaic Hilbert sweep from a multi-hour computation. >> >> >> > >> > Three hours. Foo, the Symbolics machine could do this nearly in real >> time. >> > gosper.org/hilbert286.gif is the 287 rotor sweep. (288 would >> > have been too gross.) It was to have been a single frame of the >> frequency >> > expansion animation, analogous to gosper.org/FHilbert.gif , but the >> > project >> > would not have contributed much insight into the final "catastrophic" >> > dimension bump. >> > >> > I haven't tried very hard to telescope Julian's 3⨉3 matrix product for >> the >> > Fourier coefficients. Success would provide a flood of strange >> identities >> > like (d247) and (d246) in http://www.tweedledum.com/rwg/idents.htm , >> > which come from the family that includes the Sierpinski Curve and >> > Koch's Snowflake (http://gosper.org/fst.pdf pp137, 138) rather than >> > Hilbert's curve. Failure to telescope would encumber the infinite >> > products in the identities with matrices instead of scalars. --rwg >> > The infinite Fourier series faithfully represents Hilbert's function, >> > except it misses the top two corners (the endpoints of Hilbert's curve) >> > and moves them instead to top center (where the animation begins and >> > ends). This is a quadruple point if you think of the Fourier function on >> > [0,1], but only a triple point if you regard it as a periodic function, >> > since >> > a period contains only one of its endpoints. >> > >> >> >> >> On 2016-09-26 18:42, James Propp wrote: >> >> > I hope Bill will write a lively document (or create a lively video) >> that >> >> > explains what's wrong with so many accounts of spacefilling curves. >> >> > >> >> > (I myself wish that more accounts started in a "Not Knot"-ish vein, >> >> > explaining why the "kindergartner's space-filling curve" --- >> scribbling >> >> > one's crayon back and forth until the square is filled --- isn't a >> >> solution >> >> > to the mathematician's problem.) >> >> > >> >> >> >> >> >> > Is there a way to relax an approximation to a space-filling curve >> in >> >> >> > continuous time so that it works out its kinks and regresses to >> >> simpler >> >> >> > approximations? >> >> >> > >> >> >> > (No interim self-intersections please!) >> >> >> > >> >> >> > Jim Propp >> >> >> >> >> >> http://gosper.org/FDrags128.mp4 >> >> >> >> >> >> If people really want to see it, Julian's recent Fourier matrix >> >> product can >> >> >> produce >> >> >> the analogous animation for Hilbert's "curve". >> >> > >> >> > >> >> > I would like to see it. >> >> > >> >> > But it is a seductive thought crime to view a spacefilling function as >> >> some >> >> >> kind of >> >> >> limit of "spacefilling curves"! Those curves Henry sketches are mere >> >> >> schematics, >> >> >> of no mathematical consequence. This leads inevitably to embarrassed >> >> >> hemming >> >> >> and hawing about how the area jumps from 0 to 1 at the very last >> >> moment, >> >> >> when >> >> >> both interior and exterior suddenly disappear and become boundary. >> >> >> Successive partial sums of the Fourier series are even more >> >> seductive. But >> >> >> no matter how many terms you take, you're still infinitely far from >> the >> >> >> end. >> >> >> --rwg >> >> > >> >> > >> >> > That's one way to look at it. But it depends on the notion of >> path-space >> >> > you use and what metric you put on it, doesn't it? In particular, it's >> >> > crucial to look at paths equipped with a parametrization. Then you >> >> really >> >> > can get convergence to a limit. And the limiting object is a >> continuous >> >> > function from a line segment onto a square. If you just look at the >> >> range >> >> > of a path and not the parametrization, you can't describe the square >> as >> >> the >> >> > limit in any meaningful way. Which I gather is part of Bill's point. >> >> > >> >> > Jim Propp

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[math-fun] 700 foot ice wall ?
by Henry Baker 21 Nov '16

21 Nov '16

The TV series 'Game of Thrones' claims a 700' high ice wall. Game of Thrones uses significant amounts of CGI, so the only place this ice wall exists is inside a computer. Nevertheless, could a real ice wall of its dimensions be built and stand? In various scenes, the ice wall is essentially vertical. I haven't seen the more-or-less vertical ice walls created by 'calving' glaciers personally, so I don't know whether any of them attain 700' in height. However, unlike those glacier ice walls, the Game of Thrones ice wall is 2-sided, and is warmer on one side -- enough to 'weep' under the sun. Sooooo... Based on standard Earth gravity and standard water ice, is a 700' ice wall possible? Note that the GoT ice wall is flat, so it doesn't gain any stability from being curved, and as a result, could probably not stand up to the high winds that its mere presence would generate. Ice is only a tad lighter than water, and 700' is approx. 22x the 32' of water that equals 1 atmosphere. I don't think that WWII *steel* submarines could withstand 700' depths of water, so the pressure at the bottom of such a wall would be quite high and would likely require some reinforcement.

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[math-fun] Cellular Automata & "Social Dynamics" !?!
by Henry Baker 21 Nov '16

21 Nov '16

I was following up on a conjecture I heard regarding *immigration* (!) -- that there is a "tipping point for tolerance" to immigrants which is reached at approx. 15% of the population. I thought that this was quite strange -- that there might be a mathematical model in which such a statement could have a numerical answer. The answer appears to be yes! It turns out that these "Social Dynamics" kinds of theories started with this seminal paper: SCHELLING, T. 1971. "Dynamic Models of Segregation". Journal of Mathematical Sociology 1: 143-186. https://www.stat.berkeley.edu/~aldous/157/Papers/Schelling_Seg_Models.pdf There are also continuous (analytic) models (Pollicott, Weiss. "The Dynamics of Schelling-Type Segregation Models and a Nonlinear Graph Laplacian Variational Problem", Adv. in Appl. Math. 27, 1 (July 2001), 17-40.) This paper is a kind of overview of the "Social Dynamics" field: http://jasss.soc.surrey.ac.uk/4/4/6.htmlhttp://jasss.soc.surrey.ac.uk/4/4/6… Do Irregular Grids make a Difference? Relaxing the Spatial Regularity Assumption in Cellular Models of Social Dynamics Journal of Artificial Societies and Social Simulation vol. 4, no. 4 <http://jasss.soc.surrey.ac.uk/4/4/6.html> Three decades of CA-modelling in the social sciences have shown that the cellular automata framework is a useful tool to explore the relationship between micro assumptions and macro outcomes in social dynamics. However, virtually all CA-applications in the social sciences rely on a potentially highly restrictive assumption, a rectangular grid structure. In this paper, we relax this assumption and introduce irregular grids with variation in the structure and size of neighbourhoods between locations in the grid. We test robustness of two applications from our previous work that are representative for two broad classes of CA models, migration dynamics and influence dynamics. We tentatively conclude that both influence dynamics and migration dynamics have important general properties that are robust to variation in the grid structure. In the wake of Schelling's (1971) and Sakoda's (1971) pioneering contributions[1], cellular automata (CA) models are increasingly used to study a variety of social dynamics (for some background cf. Hegselmann 1996, Hegselmann and Flache 1998). The most prominent reason is that CAs can be seen as multi-agent systems based on locality with overlapping interaction structures. In this perspective, CAs are attractive as a modelling framework that may provide a better understanding of micro/macro relations. However, many social scientists reject the CA approach on the grounds that the CA framework is far too simple, or -to put it mildly- far too idealised, to be an appropriate tool for modelling social processes. They argue that standard CA assumptions like discrete time, regular grid structures, finite sets of states etc. may make the approach so overly simplistic that it is questionable whether its results can be generalised beyond the particular framework. While we do not necess arily share this concern, we feel that systematic analysis of the robustness of CA modelling is necessary.

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[math-fun] web site change
by Tom Karzes 20 Nov '16

20 Nov '16

Hi, I was forced to switch to a new web hosting company, and as a consequence I finally registered my own domain name. My old web site (and email through it) will soon (in about a week) cease to work. In the meantime, my old URL redirects to my new one, which is: http://www.karzes.com The reason I'm sending this is because some of you may have bookmarked or linked to my interactive polyhedron page. The new URL for that is: http://www.karzes.com/polyhedra/polyhedron.html Sorry for the nuisance. Tom

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Re: [math-fun] The Hippasus Integers
by Dan Asimov 19 Nov '16

19 Nov '16

Thanks. Yes, I have that book and agree it is a fine book to have. —Dan >From: Eugene Salamin via math-fun <math-fun(a)mailman.xmission.com> >You were right the first time. If h=1 mod 4, the ring is Z[(1 + sqrt(h))/2]. Otherwise, the ring is Z[sqrt(h)]. It is understood that square factors are removed from h, so that h is squarefree. This is all derived in Harvey Cohn, "Advanced Number Theory", Dover, p. 45, a wonderful book.

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Re: [math-fun] The Hippasus Integers
by Dan Asimov 19 Nov '16

19 Nov '16

[By the way, nifty use of the blackboard font ℚ!] I didn't know the discover of the irrationality of sqrt(2) was known by name! I'd like to see a reference for that, since I always heard the history was unclear. ----- The Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67, 163. The ℚ(√−1) numbers are known as Gaussian integers. The ℚ(√−3) numbers are known as Eisenstein integers. The ℚ(√−7) numbers are known as Kleinian integers. It's bugged me for awhile that ℚ(√−2) wasn't named. Today I decided that the obvious name was ... The ℚ(√−2) numbers are now known as Hippasus integers. Hippasus proved √2 was irrational. He was then murdered. Seemed like an apt name to use. I calculated the nine types of Heegner primes and plotted them out. http://community.wolfram.com/groups/-/m/t/965609 <http://community.wolfram.com/groups/-/m/t/965609> Anyone agree / disagree about calling ℚ(√−2) the Hippasus integers? ----- I might be misinterpreting what is meant by "The ℚ(√[whatever]) numbers" are known as [whoever] integers." Usually ℚ(√[whatever]) is labeled an example of a "number field", which I think is any field of algebraic numbers with a finite index over ℚ. But to use the word "integers" implies that they are the roots of monic polynomials, with integer coefficients, that lie in the number field. For the number field ℚ(√-h) where h is a Heegner number the set of algebraic integers is as follows: Let O_h be the ring of algebraic integers of ℚ(√-h), Then: O_1 = Z[i] O_2 = Z[sqrt(-2)] O_h = Z[(1 + sqrt(h))/2] for h = -3, -7, -11, -19, -43, -67, -163. —Dan

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Re: [math-fun] The Hippasus Integers
by Dan Asimov 19 Nov '16

19 Nov '16

As a very kind colleague pointed out: There is still a typo: the complete absence of any correction. As this mundane finder (apply S_6*, S_5 respectively) pointed out: > SHOULD READ: ----- O_h = Z[(1 + sqrt(-h))/2] for -h = -3, -7, -11, -19, -43, -67, -163 ----- ((( And NOT: >> O_h = Z[(1 + sqrt(h))/2] for -h = -3, -7, -11, -19, -43, -67, -163 ))) As my maternal grandmother used to say: Oy!!!!!!!!!!!!!!!!!!!!!!! I can say this: The descent into total senility is at least interesting! —Dan ———— * But notice that Out(S_6) is different. > On 11/18/16, Dan Asimov <asimov(a)msri.org> wrote: >> Correction: >> >>> On Nov 18, 2016, at 10:30 AM, Dan Asimov <asimov(a)msri.org> wrote: >>> >>> Let O_h be the ring of algebraic integers of ℚ(√-h), Then: >>> >>> O_1 = Z[i] >>> >>> O_2 = Z[sqrt(-2)] >>> >>> O_h = Z[(1 + sqrt(h))/2] for h = -3, -7, -11, -19, -43, -67, -163. >> >> >> >> The last line should read: >> >> ----- >> O_h = Z[(1 + sqrt(h))/2] for -h = -3, -7, -11, -19, -43, -67, -163. >> ----- >> >> (and in that e-mail "discover" should be "discoverer". Argggh.

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