[math-fun] Untangling Mechanics & Topology of Overhand Elastic Knots
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, 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 Universités, 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 ¼ ð? - 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 ¼ 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.
Holy cr@p, I've been tying my shoes wrong all my life, but am only now paying the price! My bootlaces have worn down to hard, slippery cores which, to my complete vexation, came undone before I made it out the door, despite violent tightening. I never realized the bowknot I learned as a child was a granny in a hat. Must've learned from my mother. I still remember my father's stern warning against grannies. --rwg On 2015-09-18 12:57, Henry Baker wrote:
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,†
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 Universités, UPMC Univ Paris 06, CNRS, UMR 7190 Institut Jean Le Rond d’Alembert, 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 Kirchhoff’s 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)–1(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 ¼ ð? - 1Þ=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 ¼ 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.
Tying (ahem) two threads together, the cited paper and the "fisherman's knot" ignore an issue crucial to fishermen. Fishing line (and "leaders") are rated by breaking strength (e.g., 25 lb test). But if you tie the line to anything using an ordinary knot, it will typically break at the knot well short of 25 lbs of tension, because the knot pinches and cuts the line. My father always used a barrel knot. --rwg
Three years ago, when we were both visiting MSRI, mathematician Ben Young showed me how to modify my five-decade-old shoelace-tying technique by doing a double-wrap where I'd normally do a single-wrap. It helps a lot. My kids (when asking me to do it for them in the morning) call it the "mathematician knot". Ben also showed me a lovely way to peel an orange so that the peel comes off as a single piece, reminiscent of an integral sign. But I've forgotten how to do it. Besides, the oranges I can get in the Boston area aren't worth eating. (I miss Berkeley produce!) Jim Propp On Wed, Oct 7, 2015 at 10:23 PM, rwg <rwg@sdf.org> wrote:
Holy cr@p, I've been tying my shoes wrong all my life, but am only now paying the price! My bootlaces have worn down to hard, slippery cores which, to my complete vexation, came undone before I made it out the door, despite violent tightening. I never realized the bowknot I learned as a child was a granny in a hat. Must've learned from my mother. I still remember my father's stern warning against grannies. --rwg
On 2015-09-18 12:57, Henry Baker wrote:
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,†
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 Universités, UPMC Univ Paris 06, CNRS, UMR 7190 Institut Jean Le Rond d’Alembert, 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 Kirchhoff’s 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)–1(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 ¼ ð? - 1Þ=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 ¼ 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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On 2015-10-08 06:28, James Propp wrote:
Three years ago, when we were both visiting MSRI, mathematician Ben Young showed me how to modify my five-decade-old shoelace-tying technique by doing a double-wrap where I'd normally do a single-wrap. It helps a lot. My kids (when asking me to do it for them in the morning) call it the "mathematician knot".
Ben also showed me a lovely way to peel an orange so that the peel comes off as a single piece, reminiscent of an integral sign. But I've forgotten how to do it.
gosper.org/onepiececover.png , gosper.org/tennis2.gif ? How about a loop instead of an integral sign? gosper.org/tennis.gif How about gmo star fruit? gosper.org/tennis3.gif Try this Julianism from 5 yrs ago: Clear[Sphericon]; Sphericon[n_, skip_, rotate_] /; n \[Element] Integers && n >= 3 && skip \[Element] Integers && 1 <= skip < n && rotate \[Element] Integers := Block[{plot, a1, a2, a3, a4}, plot = (a1 = Table[{Cos[2*k*\[Pi]/n], Sin[2*k*\[Pi]/n], 0}, {k, 0, n - 1}]; a2 = Table[Block[{a = a1[[i, 2]] - a1[[Mod[i + skip, n, 1], 2]], b = a1[[Mod[i + skip, n, 1], 2]], v}, v = t*a1[[i]] + (1 - t)*a1[[Mod[i + skip, n, 1]]]; Which[a + b > 0 && b > 0, v, a + b <= 0 && b > 0, v /. t -> (-t*b/a), a + b > 0 && b <= 0, v /. t -> (t + (t - 1)*b/a), True, Sequence[]]], {i, 1, n}]; Print[ParametricPlot[a2[[All, {1, 2}]], {t, 0, 1}]]; Print[ParametricPlot[a2[[All, {1, 2}]], {t, 0, 1/2}]]; a3 = Table[{{1, 0, 0}, {0, Cos[u], -Sin[u]}, {0, Sin[u], Cos[u]}}.v, {v, a2}]; a4 = Table[{{Cos[2*rotate*\[Pi]/n], -Sin[2*rotate*\[Pi]/n], 0}, {Sin[2*rotate*\[Pi]/n], Cos[2*\[Pi]*rotate/n], 0}, {0, 0, 1}}.v, {v, a3}] /. u -> u + \[Pi]; Join[a3, a4]); ParametricPlot3D[plot, {t, 0, 1}, {u, 0, \[Pi]}]]; Sphericon[n_, rotate_] := Sphericon[n, 1, rotate] SetAttributes[Which, SequenceHold]
Besides, the oranges I can get in the Boston area aren't worth eating. (I miss Berkeley produce!)
After 40yrs in CA, I claim Florida oranges put California's to shame. When I was a kid in western New Jersey, there was a refrigerated shed that sold Florida produce trucked up in "reefers". Also, Egan's Market (RiP) in Cambridge sold https://en.wikipedia.org/wiki/Ugli_fruit . Regrettably, the article omits the crate label--a hideous bulldog saying "But the affliction is only skin-deep." --rwg
Jim Propp
On Wed, Oct 7, 2015 at 10:23 PM, rwg <rwg@sdf.org> wrote:
Holy cr@p, I've been tying my shoes wrong all my life, but am only now paying the price! My bootlaces have worn down to hard, slippery cores which, to my complete vexation, came undone before I made it out the door, despite violent tightening. I never realized the bowknot I learned as a child was a granny in a hat. Must've learned from my mother. I still remember my father's stern warning against grannies. --rwg
On 2015-09-18 12:57, Henry Baker wrote:
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,†
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 Universités, UPMC Univ Paris 06, CNRS, UMR 7190 Institut Jean Le Rond d’Alembert, 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 Kirchhoff’s 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)–1(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 ¼ ð? - 1Þ=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 ¼ 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.
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.
Athanasius Kircher said the world is bound with secret knots: and Kelvin thought atoms were knots in the ether (or rather the æther). In college I was hoping that elementary particles would turn out to be little 3-manifolds… why knot? Cris On Oct 8, 2015, at 12:01 PM, Dave Dyer <ddyer@real-me.net> wrote:
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.
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On Oct 8, 2015, at 12:04 PM, Cris Moore <moore@santafe.edu> wrote:
Athanasius Kircher said the world is bound with secret knots: and Kelvin thought atoms were knots in the ether (or rather the æther). In college I was hoping that elementary particles would turn out to be little 3-manifolds… why knot?
Can this æther be used for anæsthesia?
On Oct 8, 2015, at 12:01 PM, Dave Dyer <ddyer@real-me.net> wrote:
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.
This is true for the usual kind of knots: simple closed curves. Indeed they can knot only in three dimensions. But interestingly, there are higher dimensional knots made of k-spheres S^k homeomorphically mapped into R^(k+2).* For example, just as Closure(R^2-R^0) = ([0,oo) x R^0) x S^1 (R^0 is just a single point), we also have Closure(R^3-R^1) = ([0,oo) x R^1) x S^1 and Closure(R^4-R^2) = ([0,oo) x R^2) x S^1. Note that each space above of the form [0,oo) x R^(k-1) is just the closed half-space H^k of R^k. Now let's make an overhand knot K from a *closed interval* in a closed half-space H^3 = ([0,oo) x R^2), so that the endpoints lie on bd(H^3) = R^2. Assume K is smoothly embedded such that its endpoints are arranged to be *perpendicular* to the R^2 = bd(H^3). Because Closure(R^4-R^2) = H^3 x S^1, we have a circle's worth of H^3's. So we can can take K and *spin* it around the R^2 = bd(H^3), keeping its endpoints fixed throughout. Just as the union of longitude lines on a sphere equals the sphere, the union of all these spun images of K forms a S^2 as well. Call the sphere Spun(K). It can be shown that Spun(K) is knotted because the fundamental group of its complement: pi_1(R^4 - Spun(K)) is not isomorphic to pi_1(R^4 - S^2) = Z where S^2 is a standard 2-sphere in R^4. —Dan ________________________________________ * Note: An easy but *uninteresting* way to knot a 2-sphere in R^4 is to just consider an ordinary knot K in R^3 x {0} \sub R^4, and then take the union of all line L(p,q) segments of the form L(p,(0,0,0,1)) and L(p,(0,0,0,-1)) where p ranges over all points p of K. Call this set S(K). This set S(K) will in fact be a topological 2-sphere, but its embedding in R^4 is not what is called "locally flat" at the poles. Instead of being truly knotted globally, it is knotted only because of weirdness at (0,0,0,1) and (0,0,0,-1). This is somewhat related to the reason that the top illustration at https://www.win.tue.nl/~aeb/at/algtop-5.html <https://www.win.tue.nl/~aeb/at/algtop-5.html> is not equivalent to a line segment via any homeomorphism of R^3 to itself.
For hiking boots, I learned at some point to first tie the laces so as to make the ordinary bow, and then take one bow loop in each hand and use them to tie another overhand "knot".* As long as the original bow loops are long enough, this hugely helps to avoid the shoelaces' untying themselves. —Dan ________________ * By which I just mean making an X from the two bow loops and then putting one through the space at the bottom of the X and pulling them tight. Just like the first step in tying a bow.
On Oct 8, 2015, at 6:28 AM, James Propp <jamespropp@gmail.com> wrote:
Three years ago ... mathematician Ben Young showed me how to modify my five-decade-old shoelace-tying technique by doing a double-wrap where I'd normally do a single-wrap. It helps a lot. My kids (when asking me to do it for them in the morning) call it the "mathematician knot".
One piece orange peel, from art: An Old Woman seated sewing 1655, Johannes van der Aack (detail) https://farm3.staticflickr.com/2717/4411307509_afb81d6276_o.jpg On Thu, Oct 8, 2015 at 8:28 AM, James Propp <jamespropp@gmail.com> wrote:
Three years ago, when we were both visiting MSRI, mathematician Ben Young showed me how to modify my five-decade-old shoelace-tying technique by doing a double-wrap where I'd normally do a single-wrap. It helps a lot. My kids (when asking me to do it for them in the morning) call it the "mathematician knot".
Ben also showed me a lovely way to peel an orange so that the peel comes off as a single piece, reminiscent of an integral sign. But I've forgotten how to do it. Besides, the oranges I can get in the Boston area aren't worth eating. (I miss Berkeley produce!)
Jim Propp
On Wed, Oct 7, 2015 at 10:23 PM, rwg <rwg@sdf.org> wrote:
Holy cr@p, I've been tying my shoes wrong all my life, but am only now paying the price! My bootlaces have worn down to hard, slippery cores which, to my complete vexation, came undone before I made it out the door, despite violent tightening. I never realized the bowknot I learned as a child was a granny in a hat. Must've learned from my mother. I still remember my father's stern warning against grannies. --rwg
On 2015-09-18 12:57, Henry Baker wrote:
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,†
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 Universités, UPMC Univ Paris 06, CNRS, UMR 7190 Institut Jean Le Rond d’Alembert, 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 Kirchhoff’s 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)–1(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 ¼ ð? - 1Þ=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 ¼ 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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participants (8)
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Cris Moore -
Dan Asimov -
Dan Asimov -
Dave Dyer -
Henry Baker -
James Buddenhagen -
James Propp -
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