September 2015 - mathfun@mailman.xmission.com

[math-fun] Computational effort
by James Propp 01 Mar '16

01 Mar '16

I don't know what the relevant units of computation are --- floating-point operations are the ones I've heard about most, but what if you're not using floating point? --- but I'm wondering if anyone has tried to quantify "how much computing" took place in (a) code-breaking at Bletchley Park, (b) the Manhattan Project, and (c) the Apollo space program. It seems likely to me that my laptop, running some silly number-crunching problem in Mathematica and returning an answer after six hours that I could probably have figured out in my head if I'd thought a little bit more before jumping in to write some code, has them all beat. What do you think? (And while we're on the subject: What's the canonical example of wasted computational effort? Isn't there some well-known old story of someone who had an early computer calculate the determinant of a singular matrix?) Jim Propp

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[math-fun] San Jose Math Decagon
by Bill Gosper 17 Feb '16

17 Feb '16

A couple of years ago Julian and I showed some animations to the San Jose Math Circle. gosper.org/decazoom.gif suggests why the area of a circle is the radius times half the circumference, and displays the continued fraction of a very lame approximation to π. --rwg

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[math-fun] xkcd
by rcs＠xmission.com 17 Dec '15

17 Dec '15

Today's XKCD cartoon has a nice collection of approximations. http://xkcd.com/1047/ There's the ancient expression 6 pi^5 for the mass ratio of the proton to the electron, and the more accurate (e^8 - 10)/phi. Robert Munafo & REIS are at least partially responsible. Rich

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[math-fun] Julian pimped my pump
by Bill Gosper 17 Dec '15

17 Dec '15

To run it <http://gosper.org/pumper_2.mp4> continuously, download with Miro, QuickTime-View-Loop. The pumped shapes split lengthwise resemble boats, so a grand entrance to a math museum would be a chain of these boats in an annular canal, passing through four rotors acting as an (approximate) air|water lock, with another four for the exit. Oh, the insurance. --rwg No manual entry for overboard

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[math-fun] A gallery of math art from Bridges 2015
by Alex Bellos 16 Nov '15

16 Nov '15

http://www.theguardian.com/science/alexs-adventures-in-numberland/gallery/2… <http://www.theguardian.com/science/alexs-adventures-in-numberland/gallery/2…>

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[math-fun] Folding a band saw blade
by James Propp 27 Oct '15

27 Oct '15

I took a look at https://www.youtube.com/watch?v=f-7x8X6I3RA because I needed to fold a kid's camping tent --- a task that poses similar differential-geometric challenges (but without the sharp teeth). The solution Paul Sellers advocates uses a table or other external object. Is there an easy-to-describe hands-only two-person solution? Also: Is there a write-up anywhere of the mathematics behind folding a band? I'm guessing that it's the kind of situation where you can get a (2n+1)-to-1 fold for any n. But I don't even see how the 3-to-1 fold works, nor do I see how one can rigorously prove that a 2-to-1 fold is impossible. Come to think of it, I don't even know how to state the problem as a mathematical one. Jim Propp

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[math-fun] Terminology: "Characteristic Generating Function"?
by Bill Gosper 23 Oct '15

23 Oct '15

"Indicator Generating Function"? "Predicate Generating Function"? What is the name for the G.F. that encodes an integer sequence in the exponents with coefficients in {0,1} ? E.g., the OGF for the squares is Sum[n^2*x^n, {n, ∞}] == -((x*(1 + x))/(-1 + x)^3), but the thing I'm looking for is Sum[x^n^2,{n,0,∞}]==(1/2)*(1 + EllipticTheta[3, 0, x]). (Years ago I sent some of you the GF for expressions for k as the sum of n (optionally distinct) members of such a set, in terms of the GF predicating membership in the set. It turned out to be a superspecial case of a formula in Generatingfunctionology that was so general I didn't even recognize it as the same problem.) The reason I ask is that Neil just came up with 1/2 (-1+EllipticTheta[2,0,x]/x^(1/4)+EllipticTheta[3,0,x]) predicating the quartersquares<http://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multip…> A002620 <https://oeis.org/A002620>, from which he got (1/(2 (-1 + x)^2))((-1 + x) x^(3/4) EllipticTheta[2, 0, x] + x (3 - x + (-1 + x) EllipticTheta[3, 0, x])) generating what we think is the number of volumes you can pump with n of the recently discussed rotors, arranged optimally. --rwg

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[math-fun] Add 1 and reverse
by James Propp 13 Oct '15

13 Oct '15

I've been playing with the compound operation on bit-strings of length m in which you (a) add 1 mod 2^m and (b) reverse the order of the bits. Has anyone seen this before? It seems sufficiently simple that I doubt I'm the first person to have played with it. Jim Propp

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[math-fun] Untangling Mechanics & Topology of Overhand Elastic Knots
by Henry Baker 09 Oct '15

09 Oct '15

I finally received a copy from Pedro Reis of his paper; let me know if anyone else needs a copy. Untangling the Mechanics and Topology in the Frictional Response of Long Overhand Elastic Knots M.K. Jawed, 1 P. Dieleman, 2 B. Audoly, 3,* and P.M. Reis 1,2,Â 1 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2 Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 3 Sorbonne UniversitÃ©s, UPMC Univ Paris 06, CNRS, UMR 7190 Institut Jean Le Rond dÂAlembert, F-75005 Paris, France (Received 23 April 2015; revised manuscript received 7 August 2015; published 11 September 2015) We combine experiments and theory to study the mechanics of overhand knots in slender elastic rods under tension. The equilibrium shape of the knot is governed by an interplay between topology, friction, and bending. We use precision model experiments to quantify the dependence of the mechanical response of the knot as a function of the geometry of the self-contacting region, and for different topologies as measured by their crossing number. An analytical model based onthe nonlinear theory of thin elastic rods is then developed to describe how the physical and topological parameters of the knot set the tensile force required for equilibrium. Excellent agreement is found between theory and experiments for overhand knots over a wide range of crossing numbers. DOI: 10.1103/PhysRevLett.115.118302 PACS numbers: 46.25.-y, 02.10.Kn, 46.70.Hg Shoelaces are commonly tied using the reef knot, which comprises two trefoil knots: the first is left handed and the other right handed. Mistakenly tying two consecutive left-handed trefoil knots leads to the mechanically inferior granny knot [1], whose lower performance illustrates the important interplay between topology and mechanics. >From polymer chains [2] to the shipping industry, knots are ubiquitous across length scales [3]. Whereas they can appear spontaneously [4] and are sometimes regarded as a nuisance (e.g., in hair and during knitting), knots as fasteners of filamentary structures have applications in biophysics [5], surgery [6,7], fishing [8], sailing [9], and climbing [10]. Frictional knots have also been added to fibers for increased toughness [11]. Even if the quantitative study of knots has remained primarily in the realm of pure mathematics [12], there have been empirical attempts to characterize their mechanical properties according to strength or robustness [13,14]. However, these metrics rely strongly on material-specific properties and are therefore of limited applicability across different systems and length scales [3]. Recent studies have addressed the mechanics of knots from a more fundamental perspective [15,16]. For example, existing theories on flexible strings (with zero bending stiffness) [17,18] treat friction using the capstan equation [19]. Finite element simulations of knots have also been performed in instances where bending cannot be neglected [20] and friction has been treated perturbatively for trefoil knots tied in elastic rods [21,22]. Still, predictively understanding the mechan- ics of knots remains a challenging endeavor, even for the simplest types of elastic knots, due to the complex coupling of the various physical ingredients at play. Here, we perform a systematic investigation of elastic knots under tension and explore how their mechanical response is influenced by topology. We perform precision model experiments and rationalize the observed behavior through an analysis based on KirchhoffÂs geometrically nonlinear model for slender elastic rods. Our theory takes into account regions of self-contact, where friction dominates. Focus is given to open overhand knots [Figs. 1(a)Â1(d)]. These knots comprise a braid with arc length l, a loop with arc length ?, and two tails onto which a tensile load is applied. The topology of the braid is quantified by the unknotting number n Â¼ Ã°? - 1Ã=2 (number of times the knot must be passed through itself to untie it), where ? is the crossing number (number of apparent crossing nodes). In Fig. 1(e), we plot the traction force F as a function of the end-to-end shortening, e (e Â¼ 0 corresponds to a straight configuration, without a knot) for a variety of knots in the range 1 = n = 10. We find that F depends nonlinearly on e and varies significantly with n. We shall provide an analytical solution for the relation between the knot topology (defined by n) and the braid geometry. We then extend our analysis to identify the underlying physical ingredients and predictively capture the experimental mechanical response.

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[math-fun] Does anyone know if there are an infintude of these
by Bill Gosper 02 Oct '15

02 Oct '15

2F1[rational,rational,1 or 1/2,x] ==algebraic1[x] EllipticK[algebraic2[x]] identities? Or at least any more than a,b,c = 1/2,1/2,1 1/4,1/2,1 1/4,3/4,1 1/4/1/4,1 1/4,1/4,1/2 1/4,3/4,1/2 (sort of) 1/8,5/8,1 1/12,7/12,1 ? This last one features seriously unwieldy cubics. Tnx, --rwg

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