October 2015 - mathfun@mailman.xmission.com

[math-fun] Scottish students blocked by croc
by Henry Baker 09 Oct '15

09 Oct '15

FYI -- Crocodile tears ? http://i.kinja-img.com/gawker-media/image/upload/s--h3u6a22J--/146622979031… http://www.bbc.co.uk/news/uk-scotland-34476699 Crocodile maths question 'was challenging'

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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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Re: [math-fun] Dave Dyer's nutty knots & physics idea
by Warren D Smith 09 Oct '15

09 Oct '15

DD: Knots are only possible in three dimensions. That's why space has three dimensions. Particles are knots in the fabric of spacetime. Anti-particles are anti-knots, which self untangle (and release energy) when brought together. -- That did not actually make sense. (And there are plenty of better-sounding reasons why space should be 3D.) First, knot-like objects are possible for k-manifolds embedded within (k+2)-manifolds (the usual case is k=1). Second, "anti-knots" in a string CANNOT annihilate with knots further up same string, which indeed was recently proven allegedly in posts here about "sum" of two knots never equalling the unknot, and anyhow allegedly has been proven many ways by many people. (Er, you could call that the theorem of the "non-existence of forget-me-knots.") But, if you were to postulate our 3-space really is embedded in some 5-space -- then I suppose you could try to postulate particles are the k=3 analogue of knots [call them "3knots"], and you could try to hope the no-cancel theorem is untrue when k=3 despite its truth when k=1. I doubt the latter hope -- Can Mr. Asimov tell us the answer? If indeed 3knots are capable of cancelling and exhibiting interesting group-relations under "summation," that would be very interesting. But I doubt it. Also, I point out that when an electron and positron annihilate, you don't just get nothing. You get two gamma rays (at least most of the time, anyhow). So I think Dyer would need to postulate not "cancellation" but rather a more interesting set of "group relations" among the 3knots. Also, I point out that photons and gravitons can just get generated ad libitum, e.g. from a hot filament. If these truly were altered topologies, wouldn't that be forbidden? Even if photons are their own antiparticles, you'll still have some issues... -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] How many years left for you to live?
by Warren D Smith 08 Oct '15

08 Oct '15

see https://dl.dropboxusercontent.com/u/3507527/LifeExpectByAgeUSA2011.png -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] The fisherman's knot (and things like it)
by James Propp 08 Oct '15

08 Oct '15

There's an old result in knot theory that has a nice proof due to Conway (which I learned about from a Martin Gardner column ages ago): You can't tie nontrivial knots in different parts of a rope and pass them through each other in such a fashion that they cancel. (Can anyone provide a reference to the relevant column?) Proof idea: Create a tube that follows one knot and swallows the other. It occurs to me that this proof does not apply to configurations like the fisherman's knot (see the top picture in https://en.wikipedia.org/wiki/File:Fisherman's_knot.png). Or at least, it's not obvious to me how to apply Conway's trick in this context. (Maybe use a tube that isn't cylindrical, but bifurcates like a pair of pants? I don't see how to make this work.) So now I'm wondering: Is it possible to have something nontrivial like the fisherman's knot in one stretch of a two-rope cable, and something nontrivial in another stretch of the two-rope cable, and cancel them by appropriate manipulations of the cable? Jim Propp

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[math-fun] signature 3 vs signature 2
by Bill Gosper 08 Oct '15

08 Oct '15

Tabulating 2F1s without symbolic parameters nor argument makes about as much sense as trying to build a bestiary with complete genomes. Unless the table is gigantic, the chance of matching your exact animal is minute. Nevertheless, you might enjoy a few closed forms, simultaneously expressed as 2F1[1/3,2/3,1,<algebraic>], Gammas, and 2F1[1/2,1/2,1,<algebraic>] = K(<algebraic>). --rwg gosper.org/sig2sig3.png (Still no luck simplifying that ridiculous expression for 2F1[1/3,2/3,1,-1].)

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[math-fun] buy your very own mini-brain for $1.25
by Warren D Smith 05 Oct '15

05 Oct '15

https://news.brown.edu/articles/2015/10/minibrain Get neurons, grow them in a little spherical mold under the right conditions, and they'll connect themselves to form a "functioning mini-brain."

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[math-fun] curious denesting
by Bill Gosper 04 Oct '15

04 Oct '15

1/3 2 + 2 1 1/3 6 ---------- = - (1 + 2 ) 1/3 9 4 - 3 2 I.e., Out[156]= ((2 + 2^(1/3))/(4 - 3 2^(1/3)))^(1/6) In[157]:= Strad[%] Out[157]= (1 + 2^(1/3) + 2^(2/3))^(1/3) In[158]:= Strad[%] Out[158]= (1 + 2^(1/3))/3^(1/3) --rwg

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[math-fun] This GIF Takes 1,000 Years To Loop
by Henry Baker 03 Oct '15

03 Oct '15

Don't try this at home, millenials: https://www.fastcodesign.com/3051444/yes-this-gif-takes-1000-years-to-loop GIFs are designed to loop short clipsÂ-generally a couple seconds longÂ-over and over until you close your browser window. But As Long As Possible is a different kind of GIF; one rendered on a geologic time scale. Created by Helsinki-based artists Juha van Ingen and Janne SÃ¤rkelÃ¤, it's designed to loop only once every thousand years. "the finished GIF is over 12 gigabytes" (it probably won't work on your 32-bit browser). "made of 48,140,288 numbered frames, that change about every 10 minutes."

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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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