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The Mathematics of Taffy Pullers

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THE MATHEMATICS OF TAFFY PULLERS
JEAN-LUC THIFFEAULT
Abstract. We describe a number of devices for pulling candy, called tay pullers,
that are related to pseudo-Anosov maps of punctured spheres. Though the math-
ematical connection has long been known for the two most common tay puller
models, we unearth a rich variety of early designs from the patent literature, and
introduce a new one.
Introduction
Tay is a type of candy made by ﬁrst heating sugar to a critical temperature,
letting the mixture cool on a slab, then repeatedly ‘pulling’ — stretching and
folding — the resulting mass. The purpose of pulling is to get air bubbles into the
tay, which gives it a nicer texture. Many devices have been built to assist pulling,
and they all consist of a collection of ﬁxed and moving rods, or pins. Figure 1
shows the action of such a tay puller from an old patent. Observe that the tay
(pictured as a dark mass) is stretched and folded on itself repeatedly. As the rods
move, the tay is caught on the rods and its length is forced to grow exponentially.
The eectiveness of a tay puller is directly proportional to this growth, since
more growth implies a more rapid trapping of the air bubbles. Given a pattern of
periodic rod motion, regarded as orbits of points in the plane, the mathematical
challenge is to compute the growth.
We will describe in broad terms how the computation of growth is achieved.
The framework involves the topological dynamics ideas pioneered by William
Thurston, but we will shy away from a complete treatment involving rigorous
deﬁnitions. Instead we will boil down the computation to its essence: the relation-
ship between maps of the torus and those on a punctured sphere. Computations
on the former involve simple linear algebra, and the tay pullers are described by
the latter. We will also show an explicit example that involves surfaces of higher
genus than a torus, which allow us to describe tay pullers with more than three
or four moving rods. Throughout, we will give examples of tay pullers from the
patent literature as well as a newly-invented one. Finally, we answer the question:
which tay puller is the ‘best’ in a mathematical sense?
Supported by NSF grant CMMI-1233935. Contains an extra appendix compared to the version
published in Mathematical Intelligencer.
1
THE MATHEMATICS OF TAFFY PULLERS 2
Figure 1. Action of the tay puller patented by Richards (1905). The
rod motion is equivalent (conjugate) to that of Fig. 6.
Some history
Until the late 19th century, tay was pulled by hand — an arduous task. The
process was ripe for mechanization. The ﬁrst patent for a mechanical tay puller
was by Firchau (1893): his design consisted of two counter-rotating rods on con-
centric circles. This is not a ‘true’ tay puller: a piece of tay wrapped around
the rods will not grow exponentially. Firchau’s device would have been terrible at
pulling tay, but it was likely never built.
In 1900, Herbert M. Dickinson invented the ﬁrst nontrivial tay puller, and
described it in the trade journal The Confectioner. His machine involved a ﬁxed
rod and two rods that move back-and-forth. The moving rods are ‘tripped’ to
THE MATHEMATICS OF TAFFY PULLERS 3
exchange position when they reach the limit of their motion. Dickinson later
patented the machine (Dickinson 1906) and assigned it to Herbert L. Hildreth, the
owner of the Hotel Velvet on Old Orchard Beach, Maine. Tay was especially
popular at beach resorts, in the form of salt water tay (which is not really made
using salt water). Hildreth sold his ‘Hildreth’s Original and Only Velvet Candy’ to
the Maine tourists as well as wholesale, so he needed to make large quantities of
tay. Though he was usually not the inventor, he was the assignee on several tay
puller patents in the early 1900s. In fact several such patents were ﬁled in a span
of a few years by several inventors, which led to lengthy legal wranglings. Some
of these legal issues were resolved by Hildreth buying out the other inventors; for
instance, he acquired one patent for $75,000 (about two million of today’s dollars). Tay was becoming big business. Shockingly, the tay patent wars went all the way to the US Supreme Court. The opinion of the Court was delivered by Chief Justice William Howard Taft. The opinion shows a keen grasp of topological dynamics (Hildreth v. Mastoras, 1921): The machine shown in the Firchau patent [has two pins that] pass each other twice during each revolution [. . . ] and move in concentric circles, but do not have the relative in-and-out motion or Figure 8 movement of the Dickinson machine. With only two hooks there could be no lapping of the candy, because there was no third pin to re-engage the candy while it was held between the other two pins. The movement of the two pins in concentric circles might stretch it somewhat and stir it, but it would not pull it in the sense of the art. The Supreme Court opinion displays the fundamental insight that at least three rods are required to produce some sort of rapid growth. Moreover, the ‘Figure 8’ motion is identiﬁed as key to this growth. We shall have more to say on this rod motion as we examine in turn the dierent design principles. The Dickinson tay puller may have been the ﬁrst, but it was overly complicated and likely never used to make large quantities of candy. A similar rod motion can be obtained by a much simpler mechanism, which was introduced in a patent by Robinson and Deiter (1908) and is still in use today. In this device, two rods move in counter-rotating orbits around a ﬁxed rod (Fig. 2). We call this design the standard 3-rod tay puller. Three-rod taffy pullers Tay pullers involving three rods (some of which may be ﬁxed) are the easiest to describe mathematically. The action of arguably the simplest such puller, from the mathematical standpoint, is depicted in Fig. 3(a). By action, we mean the eect of the puller on a piece of abstract ‘tay.’ For this puller, the ﬁrst and second rods are THE MATHEMATICS OF TAFFY PULLERS 4 (a) (b) Figure 2. (a) Tay puller from the patent of Robinson and Deiter (1908). (b) The motion of the rods. interchanged clockwise, then the second and third are interchanged counterclock- wise. Notice that each rod undergoes a ‘Figure 8’ motion, as shown in Fig. 3(b) for 1 3period. We now demonstrate that such a tay puller motion arises naturally from linear maps on the torus. (We leave out many mathematical details — see for example Farb and Margalit (2011), Fathi, Laundenbach, and Poénaru (1979), and Thurston (1988) for the full story.) We use the standard model of the torus T2as the unit square [0,1]2with opposite edges identiﬁed. Consider the linear map ι:T2T2, deﬁned by ι(x)=xmod 1. The map ιis an involution (ι2=id) with four ﬁxed points on the torus [0,1]2, (1) p0=( 0 0 ) ,p1=(1 20 ) ,p2=(1 2 1 2),p3=( 0 1 2). THE MATHEMATICS OF TAFFY PULLERS 5 (a) (b) (c) Figure 3. (a) The action of a 3-rod tay puller. The ﬁrst and second rods are interchanged clockwise, then the second and third rods are interchanged counterclockwise. (b) Each of the three rods moves in a Figure-8. (c) Tay puller from the patent of Nitz (1918), where rods alternate between the two wheels. Figure 4(a) shows how the dierent sections of T2are mapped to each other under ι; arrows map to each other or are identiﬁed because of periodicity. The quotient space S=T2, depicted in Fig. 4(b), is actually a sphere in the topological sense (it has genus zero). We can see this by ‘gluing’ the identiﬁed edges to obtain Fig. 4(c). The 4 ﬁxed points of ιabove will play a special role, so we puncture the sphere at those points and write S=S0,4, which indicates a surface of genus 0 with 4 punctures. Now let’s take a general linear map φ:T2T2. We write φ(x)=M·xmod 1, with x[0,1]2and Ma matrix in SL2(Z), (2) M= a b c d!,a,b,c,dZ,ad bc =1. This guarantees that φis an orientation-preserving homeomorphism — a continu- ous map of T2with a continuous inverse. The map φﬁxes p0=( 0 0 ) and permutes THE MATHEMATICS OF TAFFY PULLERS 6 A A B B (a) AB p0p1 p3p2 (b) p0p1p2 p3 (c) Figure 4. (a) Identiﬁcation of regions on T2=[0,1]2under the map ι. (b) The surface S=T2, with the four ﬁxed points of ιshown. (c) S is a sphere with four punctures, denoted S0,4. the ordered set (p1,p2,p3). For example, the map (3) φ(x)= 2 1 1 1!·xmod 1 maps (p1,p2,p3) to (p3,p1,p2). This is an Anosov map: it has a real eigenvalue larger than one (in magnitude). We call the spectral radius λof the matrix Mthe dilatation of the map φ. A key fact is that the length of any noncontractible simple closed curve on T2grows as λnas the number of iterates n→ ∞ (Fathi, Laundenbach, and Poénaru 1979). Because ιφ=φι, a linear map such as φon the torus projects nicely to the punctured sphere S0,4=T2. The induced map on S0,4is called pseudo-Anosov rather than Anosov, since the quotient of the torus by ιcreated four singularities. (We shall not need precise deﬁnitions of these terms; here by Anosov map we mean a linear map on the torus with spectral radius larger than 1, and by pseudo-Anosov we mean the same map projected to S0,4.) Let’s see how the action of the map (3) gives the tay puller in Fig. 3. The permuted points (p1,p2,p3) play the role of the rods of the tay puller. Figure 5(a) (left) shows two curves on the torus, which project to curves on the punctured sphere S0,4(right). (Whenever we say curve, we will actually mean an equivalence class of curves under homotopy ﬁxing the punctures.) The blue curve from p2 to p3should be identiﬁed with the piece of tay in Fig. 3(a). Now if we act on the curves with the torus map (3), we obtain the curves in Fig. 5(b) (left). After taking the quotient with ι, the curves project down as in Fig. 5(b) (right). This has the same shape as our tay in Fig. 3(a) (third frame) for 1 3period of the tay puller. THE MATHEMATICS OF TAFFY PULLERS 7 p0p1 p3p2 p0 p3 p0p1p0 p0p1p2p3 (a) p0 0p0 2 p0 3 p0 1 p0 0 p0 0p0 0 p0 2 p0 1 p0 0p0 2 p0 3p0 1 (b) Figure 5. (a) Two curves on the torus T2(left), which project to curves on the punctured sphere S0,4(right). (b) The two curves transformed by the map (3) (left), and projected onto S0,4(right). The transformed blue curve is the same as in the third frame of Fig. 3(a). What we’ve essentially shown is that the tay puller in Fig. 3 can be described by projecting the Anosov map (3) of the torus to a pseudo-Anosov map of S0,4. The growth of the length of tay, under repeated action, will be given by the spectral radius λof the matrix M, here λ=ϕ2with ϕbeing the Golden Ratio 1 2(1+5). This tay puller is a bit peculiar in that it requires rods to move in a Figure-8 motion, as shown in Fig. 3(b). This is challenging to achieve mechanically, but surprisingly such a device was patented by Nitz (1918) (Fig. 3(c)), and then apparently again by Kirsch (1928). The device requires rods to alternately jump between two rotating wheels. All 3-rod devices can be treated in the same manner, including the standard 3-rod tay puller depicted in Fig. 2. We will not give the details here, but it can be shown to arise from the linear map (4) φ(x)= 5 2 2 1!·xmod 1 THE MATHEMATICS OF TAFFY PULLERS 8 (a) (b) Figure 6. (a) Side view of the standard 4-rod tay puller from the patent of Thibodeau (1903), with four rotating rods set on two axles. (b) Rod motion. which has λ=χ2. Here χ=1+2 is the Silver Ratio (Finn and Thieault 2011). Are four rods better than three? As discussed in the previous section, all 3-rod tay pullers arise from Anosov maps of the torus. This is not true in general for more than three rods, but it is true for several speciﬁc devices. Probably the most common device is the standard 4-rod tay puller, which was invented by Thibodeau (1903) and is shown in Fig. 6. It seems to have been rediscovered several times, such as by Hudson (1904). The design of Richards (1905) is a variation that achieves the same eect, and his patent has some of the prettiest diagrams of tay pulling in action (Fig. 1). Mathematically, the 4-rod puller was studied by MacKay (2001) and Halbert and Yorke (2014). The rod motion for the standard 4-rod puller is shown in Fig. 6(b). Observe that the two orbits of smaller radius are not intertwined, so topologically they might as well be ﬁxed rods. This tay puller arises from an Anosov map such as (4), but with all four points (p0,p1,p2,p3) of S0,4identiﬁed with rods. We relabel the four points (p0,p1,p2,p3) as (1,4,3,2), as in Fig. 7(a) (left), which gives the order of the rods on the right in that ﬁgure. The boundary point labeled 0 plays no direct THE MATHEMATICS OF TAFFY PULLERS 9 1 4 23 1 2 141 0102 0102 012 3 4 (a) 1040 2030 10 20 104010 00 200 1 00 200 1 00 1020 3040 (b) Figure 7. (a) Three curves on the torus T2(left), which project to curves on the punctured sphere S0,5(right). (b) The three curves transformed by the map (5) (left), and projected onto S0,5(right). Compare these to the last frame of Fig. 1. role, but prevents us from shortening curves by passing them ‘behind’ the sphere.1 Puncturing at this extra point gives the space S0,5, the sphere with 5 punctures. 1In the 3-rod case, the point labeled 0 in Fig. 5 plays this role. In the 4-rod case, we need to use S0,5in order to allow for this extra point. There are no more ﬁxed points available, since φ(x) in (5) only has 4. However, a period-2 point of φwill do, as long as the two iterates are also mapped to each other by ι. The map (5) actually has 14 orbits of period 2, but only two of those are also invariant under ι: n(1 40 ) ,(3 40 )oand n(1 4 1 2),(3 4 1 2)o. The second choice would put the boundary point being between two rods, so we choose the ﬁrst orbit. The two iterates are labeled 01and 02in Fig. 7(a). They are interchanged in Fig. 7(b) after applying the map (5), but they both map to the same point on the sphere S0,5=(T2)− {0}, since they also satisfy ι(01)=02. THE MATHEMATICS OF TAFFY PULLERS 10 Now act on the curves in Fig. 7(a) with the map (5) φ(x)= 3 2 4 3!·xmod 1. This map ﬁxes each of the points 1,2,3,4, just as the 4-rod tay puller does. Figure 7(b) shows the action of the map on curves anchored on the rods: it acts in exactly the same manner as the standard 4-rod tay puller. In fact, (6) 3 2 4 3!= 1 0 1 1! 5 2 2 1! 1 0 1 1!1 which means that the maps (4) and (5) are conjugate to each other. Conjugate maps have the same dilatation (the trace is invariant), so the standard 3-rod and 4-rod tay pullers arise from essentially the same Anosov map, only interpreted dierently. In other words, at least for the standard 4-rod puller, the addition of a rod does not increase the eectiveness of the device. Anew device All the devices we described so far arise from maps of the torus. Now we give an example of a device that arises from a branched cover of the torus, rather than directly from the torus itself. (A theorem of Franks and Rykken (1999) implies that the dilatation λmust also be quadratic in this case.) Figure 8 shows such a device, designed and built by Alexander Flanagan and the author. It is a simple modiﬁcation of the standard 4-rod design (Fig. 6), except that the two arms are of equal length, and the axles are extended to become ﬁxed rods. There are thus 6 rods in play, and we shall see that this device has a rather large dilatation. The construction of a map describing this 6-rod device uses the two involutions of the closed (unpunctured) genus two surface S2shown in Fig. 9. Imagine that an Anosov map gives the dynamics on the left ‘torus’ of the surface. The involution ι1 extends those dynamics to a genus two surface. The involution ι2is then used to create the quotient surface S0,6=S22. The 6 punctures will correspond to the rods of the tay puller. A bit of experimentation suggests starting from the Anosov map (7) φ(x)= 11 23!·xmod 1. Referring to the points (1), this map ﬁxes p0and p1and interchanges p2and p3. For our purposes, we cut our unit cell for the torus slightly dierently, as shown in Fig. 10(a). In addition to p0and p1, the map has four more ﬁxed points: (8) ( 1 31 3),(1 3 1 3),(1 6 2 3),(1 6 1 3). THE MATHEMATICS OF TAFFY PULLERS 11 (a) (b) Figure 8. (a) A 6-pronged tay puller designed and built by Alexan- der Flanagan and the author. (b) The motion of the rods, with two ﬁxed axles. ι1 01 02 (a) ι2 1 2 3 4 5 6 (b) Figure 9. The two involutions of a genus two surface S2as rotations by π. (a) The involution ι1has two ﬁxed points; (b) ι2has six. To create our branched cover of the torus, we will make a cut from the point 01= (1 31 3) to 02=(1 3 1 3), as shown in Fig. 10(b). We have also labeled by 1–6 the points that will correspond to our rods. The arrows show identiﬁed opposite edges; we have eectively cut a slit in two tori, opened the slits into disks, and glued the tori at those disks to create a genus two surface. The involution ι1from Fig. 9(a) corresponds to translating the top half in Fig. 10(b) down to the bottom half; the only ﬁxed points are then 01and 02. For the involution ι2of Fig. 9(b), ﬁrst divide Fig. 10(b) into four sectors with 2–5 at their center; then rotate each sector THE MATHEMATICS OF TAFFY PULLERS 12 (a) 01 02 1 2 3 4 5 6 (b) 0 0 1 2 3 4 5 6 (c) Figure 10. (a) A dierent unfolding of the torus. The four ﬁxed points of ιare indicated. (b) Two copies of the torus glued together after removing a disk. The points 01,2are at (1 3 1 3). This gives the genus two surface S2. The two tori are mapped to each other by the involution ι1from Fig 9, with ﬁxed points 01,2. The involution ι2 acts on the individual tori with ﬁxed points 1,...,6. (c) The quotient surface S22, which is the punctured sphere S0,6. by πabout its center. This ﬁxes the points 1–6. The quotient surface S22gives the punctured sphere S0,6, shown in Fig. 10(c). The points 01,2are mapped to each other by ι2and so become identiﬁed with the same point 0. In Fig. 11(a) we reproduce the genus two surface, omitting the edge identiﬁca- tions for clarity, and draw some arcs between our rods. Now act on the surface (embedded in the plane) with the map (7). The polygon gets stretched, and we cut and glue pieces following the edge identiﬁcations to bring it back into its initial domain, as in Fig. 11(b). Punctures 2 and 5 are ﬁxed, 1 and 4 are swapped, as are 3 and 6. This is exactly the same as for a half-period of the puller in Fig. 8(b). After acting with the map we form the quotient surface S22=S0,6, as in Fig. 11(c). Now we can carefully trace out the path of each arc, and keep track of which side of the arcs the punctures lie. The paths in Fig. 11(c) are identical to the arcs in Fig. 12, and we conclude that the map (7) is the correct description of the six-rod puller. Its dilatation is thus the largest root of x24x+1, which is 2 +3. THE MATHEMATICS OF TAFFY PULLERS 13 01 02 2 1 13 4 6 6 5 (a) 00 1 00 2 20 40 4060 10 30 30 50 (b) 00 00 10 20 30 40 50 60 (c) Figure 11. (a) The genus two surface from Fig. 10(b), with opposite edges identiﬁed and arcs between the rods. (b) The surface and arcs after applying the map (7) and using the edge identiﬁcations to cut up and rearrange the surface to the same initial domain. (c) The arcs on the punctured sphere S0,6=S22, with edges identiﬁed as in Fig. 10(c). The description of the surface as a polygon in the plane, with edge identiﬁcations via translations and rotations, comes from the theory of ﬂat surfaces (Zorich 2006). In this viewpoint the surface is given a ﬂat metric, and the corners of the polygon correspond to conical singularities with inﬁnite curvature. Here, the two singular- ities 01,2have cone angle 4π, as can be seen by drawing a small circle around the points and following the edge identiﬁcations. The sum of the two singularities is 8π, which equals 2π(4g4) by the Gauss–Bonnet formula, with g=2 the genus. What is the best taffy puller? There are many other tay puller designs found in the patent literature. (See the Appendix for some examples.) A few of these have a quadratic dilation, like the examples we discussed, but many don’t: they involve pseudo-Anosov maps that are more complicated than simple branched covers of the torus. We will not give a detailed construction of the maps, but rather report the polynomial whose largest root is the dilatation and oer some comments. The polynomials were obtained THE MATHEMATICS OF TAFFY PULLERS 14 012 3 45 6 (a) 00 10 20 30 4050 60 (b) Figure 12. (a) A sphere with six punctures (rods) and a seventh puncture at the ﬁxed point 0, with arcs between the punctures, as in Fig. 11(a). (b) The arcs after a half-period of the rod motion in Fig. 8(b). These are identical to the arcs of Fig 11(c). using the computer programs braidlab (Thieault and Budiši´c 2013–2017) and train (Hall 2012). Many tay pullers are planetary devices — these have rods that move on epicy- cles, giving their orbits a ‘spirograph’ appearance. The name comes from Ptole- maic models of the solar system, where planetary motions were apparently well- reproduced using systems of gears. Planetary designs are used in many mixing devices, and are a natural way of creating tay pullers. Kobayashi and Umeda (2007, 2010) and Finn and Thieault (2011) have designed and studied a class of such devices. A typical planetary device, the mixograph, is shown in Fig. 13. The mixograph consists of a small cylindrical vessel with three ﬁxed vertical rods. A lid is lowered onto the base. The lid has two gears each with a pair of rods, and is itself rotating, resulting in a net complex motion as in Fig. 13(c) (top). The mixograph is used to measure properties of bread dough: a piece of dough is placed in the device, and the torque on the rods is recorded on graph paper, in a similar manner to a seismo- graph. An expert on bread dough can then deduce dough-mixing characteristics from the graph (Connelly and Valenti-Jordan 2008). THE MATHEMATICS OF TAFFY PULLERS 15 (a) (b) (c) Figure 13. (a) The mixograph, a planetary rod mixer for bread dough. (b) Top section with four moving rods (above), and bot- tom section with three ﬁxed rods (below). (c) The rod motion is complex (top), but is less so in a rotating frame (bottom). (Courtesy of the Department of Food Science, University of Wisconsin. Photo by the author, from Finn and Thieault (2011).) Clearly, passing to a uniformly-rotating frame does not modify the dilatation. For the mixograph, a co-rotating frame where the ﬁxed rods rotate simpliﬁes the orbits somewhat (Fig. 13(c), bottom). The rod motion of Fig. 13(c) (bottom) must be repeated six times for all the rods to return to their initial position. The dilatation for the co-rotating map is the largest root of x84x7x6+4x4x24x+1, which is approximately 4.1858. The reader might be wondering at this point: which is the best tay puller? Did all these incremental changes and new designs in the patent literature lead to measurable progress in the eectiveness of tay pullers? Table 1 collects the characteristic polynomials and the dilatations (the largest root) for all the tay pullers discussed here and a few others included at the end. The total number of rods is listed (the number in parentheses is the number of ﬁxed rods). The column labeled prequires a bit of explanation. Comparing the dierent tay pullers is not straightforward. To keep things simple, we take the eciency to be the total dilatation for a full period, deﬁned by all the rods returning to their initial position. For example, referring to Table 1, for the Nitz 1918 device THE MATHEMATICS OF TAFFY PULLERS 16 Table 1. Eciency of tay pullers. A number of rods such as 6 (2) indicates 6 total rods, with 2 ﬁxed. The largest root of the polynomial is the dilatation. The dilatation corresponds to a fraction pof a full period, when each rod returns to its initial position. The entropy per period is log(dilatation)/period, which is a crude measure of eciency. Here ϕ=1 2(1 +5) is the Golden Ratio, and χ=1+2 is the Silver Ratio. puller ﬁg. rods polynomial dilatation pentropy/ period standard 3-rod 2 3 (1) x26x+1χ21 1.7627 Nitz (1918) 3 3 x23x+1ϕ21 /32.8873 standard 4-rod 6 4 x26x+1χ21 1.7627 Thibodeau (1904) 14 4 x23x+1ϕ21 /32.8873 6-rod 8 6 (2) x24x+1 2 +31 /22.6339 McCarthy (1916) 15(c) 4 (3) x218x+1ϕ61 2.8873 15(d) 4 (3) x436x3+54x236x+1 34.4634 1 3.5399 mixograph 13(c) 7 x84x7x6+4x4 x24x+14.1858 1 /68.5902 Jenner (1905) 16 5 (3) x48x32x28x+1 (ϕ+ϕ)21 2.1226 Shean (1914) 17 6 x24x+1 2 +31 /22.6339 McCarthy (1915) 18 5 (2) x420x326x220x+1 21.2667 1 3.0571 The McCarthy (1916) device has two conﬁgurations. This is the co-rotating version of the mixograph (Fig. 13(c), bottom). the rods return to the same conﬁguration (as a set) after p=1/3 period. Hence, the dilatation listed, ϕ2, is for 1/3 period. We deﬁne a puller’s eciency as the entropy (logarithm of the dilatation) per period. In this case the eciency is log(ϕ2)/(1/3) =6 log ϕ2.8873. By this measure, the mixograph is the clear winner, with a staggering eciency of 8.5902. Of course, it also has the most rods. The large eciency is mostly due to how long the rods take to return to their initial position. Some general observations can be made regarding practical tay pullers. With a few exceptions, they all give pseudo-Anosov maps. Though we did not deﬁne this term precisely, in this context it implies that any initial piece of tay caught on the rods will grow exponentially. The inventors were thus aware, at least intuitively, that there should be no unnecessary rods. Another observation is that most of the dilatations are quadratic numbers. There are probably a few reasons for this. One is that the polynomial giving the dilatation expresses a recurrence relation REFERENCES 17 that characterizes how the tay’s folds are combined at each period. With a small number of rods, there is a limit to the degree of this recurrence (2n4 for nrods). A second reason is that more rods does not necessarily mean larger dilatation (Finn and Thieault 2011). On the contrary, more rods allows for a smaller dilatation, as observed when ﬁnding the smallest value of the dilatation (Hironaka and Kin 2006; Lanneau and Thieault 2011; Thieault and Finn 2006; Venzke 2008). The collection of tay pullers presented here can be thought of as a battery of examples to illustrate various types of pseudo-Anosov maps. Even though they did not come out of the mathematical literature, they predate by many decades the examples that were later constructed by mathematicians (Binder 2010; Binder and Cox 2008; Boyland, Aref, and Stremler 2000; Boyland and Harrington 2011; Finn and Thieault 2011; Kobayashi and Umeda 2007, 2010; Thieault and Finn 2006). Acknowledgments. The author thanks Alex Flanagan for helping to design and build the 6-rod tay puller, and Phil Boyland and Eiko Kin for their comments on the manuscript. References Binder, B. J. (2010). “Ghost rods adopting the role of withdrawn baes in batch mixer designs.” Phys. Lett. A 374, 3483–3486. Binder, B. J. and S. M. Cox (2008). “A mixer design for the pigtail braid.” Fluid Dyn. Res. 40, 34–44. Boyland, P. L., H. Aref, and M. A. Stremler (2000). “Topological ﬂuid mechanics of stirring.” J. Fluid Mech. 403, 277–304. Boyland, P. L. and J. Harrington (2011). “The entropy eciency of point-push mapping classes on the punctured disk.” Algeb. Geom. Topology 11(4), 2265–2296. Connelly, R. K. and J. Valenti-Jordan (2008). “Mixing analysis of a Newtonian ﬂuid in a 3D planetary pin mixer.” 86(12), 1434–1440. Dickinson, H. M. (1906). “Candy-pulling machine.” Pat. US831501 A. Farb, B. and D. Margalit (2011). A Primer on Mapping Class Groups. Princeton, NJ: Princeton University Press. Fathi, A., F. Laundenbach, and V. Poénaru (1979). “Travaux de Thurston sur les surfaces.” Astérisque 66-67, 1–284. Finn, M. D. and J.-L. Thieault (2011). “Topological optimization of rod-stirring devices.” SIAM Rev. 53(4), 723–743. Firchau, P. J. G. (1893). “Machine for working candy.” Pat. US511011 A. Franks, J. and E. Rykken (1999). “Pseudo-Anosov homeomorphisms with qua- dratic expansion.” Proc. Amer. Math. Soc. 127, 2183–2192. REFERENCES 18 Halbert, J. T. and J. A. Yorke (2014). “Modeling a chaotic machine’s dynamics as a linear map on a “square sphere”.” Topology Proceedings 44, 257–284. Hall, T. (2012). Train: A C++ program for computing train tracks of surface homeomor- phisms.http://www.liv.ac.uk/~tobyhall/T_Hall.html. Hironaka, E. and E. Kin (2006). “A family of pseudo-Anosov braids with small dilatation.” Algebraic & Geometric Topology 6, 699–738. Hudson, W. T. (1904). “Candy-working machine.” Pat. US752226 A. Jenner, E. J. (1905). “Candy-pulling machine.” Pat. US804726 A. Kirsch, E. (1928). “Candy-pulling machine.” Pat. US1656005 A. Kobayashi, T. and S. Umeda (2007). “Realizing pseudo-Anosov egg beaters with simple mecanisms.” In: Proceedings of the International Workshop on Knot Theory for Scientiﬁc Objects, Osaka, Japan. Osaka, Japan: Osaka Municipal Universities Press, pp. 97–109. (2010). “A design for pseudo-Anosov braids using hypotrochoid curves.” Topol- ogy Appl. 157, 280–289. Lanneau, E. and J.-L. Thieault (2011). “On the minimum dilatation of braids on the punctured disc.” Geometriae Dedicata 152(1), 165–182. MacKay, R. S. (2001). “Complicated dynamics from simple topological hypothe- ses.” Phil. Trans. R. Soc. Lond. A 359, 1479–1496. McCarthy, E. F. (1916). “Candy-pulling machine.” Pat. US1182394 A. McCarthy, E. F. and E. W. Wilson (1915). “Candy-pulling machine.” Pat. US1139786 A. Nitz, C. G. W. (1918). “Candy-puller.” Pat. US1278197 A. Richards, F. H. (1905). “Process of making candy.” Pat. US790920 A. Robinson, E. M. and J. H. Deiter (1908). “Candy-pulling machine.” Pat. US881442 A. Russell, R. G. and R. B. Wiley (1951). “Apparatus for mixing viscous liquids.” Pat. US2577920 A. Shean, G. C. C. and L. Schmelz (1914). “Candy-pulling machine.” Pat. US1112569 A. Thibodeau, C. (1903). “Method of pulling candy.” Pat. US736313 A. (1904). “Candy-pulling machine.” Pat. US772442 A. Thieault, J.-L. and M. D. Finn (2006). “Topology, braids, and mixing in ﬂuids.” Phil. Trans. R. Soc. Lond. A 364, 3251–3266. Thieault, J.-L. and M. Budiši´c (2013–2017). Braidlab: A Software Package for Braids and Loops.http://arXiv.org/abs/1410.0849, Version 3.2.2. Thurston, W. P. (1988). “On the geometry and dynamics of dieomorphisms of surfaces.” Bull. Am. Math. Soc. 19, 417–431. Tumasz, S. E. and J.-L. Thieault (2013a). “Estimating Topological Entropy from the Motion of Stirring Rods.” Procedia IUTAM 7, 117–126. (2013b). “Topological entropy and secondary folding.” J. Nonlinear Sci. 13(3), 511–524. REFERENCES 19 Venzke, R. W. (2008). Braid Forcing, Hyperbolic Geometry, and Pseudo-Anosov Se- quences of Low Entropy. PhD thesis. California Institute of Technology. Zorich, A. (2006). “Flat surfaces.” In: Frontiers in Number Theory, Physics, and Geom- etry. Ed. by P. Cartier et al. Vol. 1. Berlin: Springer, pp. 439–586. Appendix: A few more taffy puller designs Because of space constraints, several tay pullers from the patent literature were omitted from the Mathematical Intelligencer version of this article. We include these here for the interested reader. A 4-rod device with Golden ratio dilatation. Thibodeau’s device in Fig. 6 gave an example of a 4-rod tay puller arising directly from an Anosov map. Another example is a later design of Thibodeau shown in Fig. 14. It consists of three rods moving in a circle, and a fourth rod crossing their path back and forth. This tay puller can be shown to come from the same Anosov as gave us the 3-rod puller in Fig. 3, with dilatation equal to the square of the Golden Ratio. Thus, if one is interested in building a device with a Golden Ratio dilatation, the design in Fig. 14(a) is probably far easier to implement than Nitz’s in Fig. 3(c), since Thibodeau’s does not involve rods being exchanged between two gears. A simple planetary design. McCarthy (1916) has an interesting planetary design for a tay puller (Fig. 15). It has two conﬁgurations, with rod motions shown in Fig. 15(b). Its ﬁrst conﬁguration (pictured in Fig. 15(a) with rod motion as in Fig. 15(c)) is a perfect example of a ‘π1-stirring device,’ a device where only a single rod moves around a set of ﬁxed rods. The optimality of such devices was studied by Boyland and Harrington (2011), and McCarthy’s device is one of their optimal examples. The second conﬁguration (not shown) involves replacing the chain in Fig. 15(a) by two gears in direct contact. This gives the motion in Fig. 15(d), which does appear quite dierent from McCarthy’s sketch (Fig. 15(b), bottom) but is topolog- ically identical. McCarthy himself seemed to prefer the ﬁrst conﬁguration, as he noted a bit wordily in his patent: The planetary course described by this pin, when this modiﬁed construction is employed, gives a constant pull to the candy, but does not accomplish as thorough mixing of the same as when said pin describes the planetary course resulting from the construction of the preferred form of my invention, as hereinbefore ﬁrst described. What he meant by ‘a constant pull to the candy’ is probably that in the second conﬁguration the rod moves back and forth in the center of the device, so the tay would sometimes be unstretched. In the ﬁrst conﬁguration the rod resolutely REFERENCES 20 (a) (b) (c) Figure 14. (a) Tay puller from the patent of Thibodeau (1904), with three rotating rods on a wheel and an oscillating arm. (b) Rod motion. (c) The action of the tay puller, as depicted in the patent. traverses the center of the device in a single direction each time, leading to uniform stretching. This is related to motions that remain ‘pulled tight’ as the rods move (Tumasz and Thieault 2013a,b). As far as the less thorough mixing he mentions is concerned, in one turn of the handle the ﬁrst conﬁguration gives a dilatation of 4.2361, while the second has 2.4229. However, the second design has a larger dilatation for a full period of the rod motion, as given in Table 1. This illustrates the diculties involved in comparing the eciency of dierent devices. In its ﬁrst conﬁguration the device has a quadratic dilatation, the largest root of x218x+1. In its second conﬁguration the dilatation is a quartic number, the largest root of x436x3+5x236x+1. A peculiar dilatation. The design of Jenner (1905), shown in Fig. 16, is a fairly straightforward variant of the other devices we’ve seen. From our point of view it has a peculiar property: its dilatation is the largest root of the polynomial x4 REFERENCES 21 (a) (b) (c) (d) Figure 15. (a) Tay puller from the patent of McCarthy (1916). (b) Rod motion for the two conﬁgurations of the device as sketched in the patent. (c) and (d): the actual rod motions. 8x32x28x+1, which is the strange number (ϕ+ϕ)2, where ϕis the Golden Ratio. Interlocking combs. The tay puller of Shean and Schmelz (1914) is shown in Fig. 17. The design is somewhat novel, since it is not based directly on gears. It consists of two interlocking ‘combs’ of three rods each, for a total of six moving rods. Mathematically, this device has exactly the same dilatation as the earlier 6-rod design (Fig. 8). A similar comb design was later used in a device for homogenizing molten glass (Russell and Wiley 1951). A baroque design. We ﬁnish with the intriguing design of McCarthy and Wilson (1915), shown in Fig. 18. This is the most baroque design we’ve encountered: it REFERENCES 22 (a) (b) Figure 16. Tay puller from the patent of Jenner (1905). (b) The motion of the rods, with three ﬁxed rods in gray. (a) (b) Figure 17. (a) Tay puller from the patent of Shean and Schmelz (1914). (b) The rod motion. contains an oscillating arm, rotating rods, and ﬁxed rods. The inventors did seem to know what they were doing with this complexity: its dilatation is enormous at approximately 21.2667, the largest root of x420x326x220x+1. Why so many designs? There are actually quite a few more patents for tay pullers that were not shown here (only U.S. patents were searched). An obvious question is: why so many? Often the answer is that a new patent is created to REFERENCES 23 (a) (b) Figure 18. (a) Tay puller from the patent of McCarthy and Wilson (1915). (b) Rod motion. get around an earlier one, but the very ﬁrst patents had lapsed by the 1920s and yet more designs were introduced, so this is only a partial answer. Perhaps there is a natural response when looking at a tay puller to think that we can design a better one, since the basic idea is so simple. At least mathematics provides a way of making sure that we’ve thoroughly explored all designs, and to gauge the eectiveness of existing ones. Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA E-mail address:jeanluc@math.wisc.edu ... A more whimsical application looks at the evolution of taffy-pulling machines [52]. Taffy is a soft candy made from repeatedly stretching melted sugar; in this process air is folded in, resulting in a softer, more desirable texture. ... ... Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy In a satisfying connection, \phi + \surd \phi is directly related to the braid dilation of a particular historical taffy stretching device (see Table 1 from reference [52], and Figure 16 from the arxiv version of that article). Indeed, a potential mixing device (which we introduce in Figure 20) based on the max TEPO braid, has point movements that are the natural 2D generalization of the rod movement in this taffy stretching device. ... Article Full-text available The deep connections between braids and dynamics by way of the Nielsen--Thurston classification theorem have led to a wide range of practical applications. Braids have been used to detect coherent structures and mixing regions in oceanic flows, drive the design of industrial mixing machines, contextualize the evolution of taffy pullers, and characterize the chaotic motion of topological defects in active nematics. Mixing plays a central role in each of these examples, and the braids naturally associated with each system come equipped with a useful measure of mixing efficiency, the topological entropy per operation (TEPO). This motivates the following questions. What is the maximum mixing efficiency for braids, and what braids realize this? The answer depends on how we define braids. For the standard Artin presentation, well-known braids with mixing efficiencies related to the golden and silver ratios have been proven to be maximal. However, it is fruitful to consider surface braids, a natural generalization of braids, with presentations constructed from Artin-like braid generators on embedded graphs. In this work, we introduce an efficient and elegant algorithm for finding the topological entropy and TEPO of surface braids on any pairing of orientable surface and planar embeddable graph. Of the myriad possible graphs and surfaces, graphs that can be embedded in R2 as a lattice are a simple, highly symmetric choice, and the braids that result more naturally model the motion of points on the plane. We extensively search for a maximum mixing efficiency braid on planar lattice graphs and examine a novel candidate braid, which we conjecture to have this maximal property. ... A more whimsical application looks at the evolution of taffy-pulling machines 17 . Taffy is a soft candy made from repeatedly stretching melted sugar; in this process air is folded in, resulting in a softer, more desirable texture. ... Preprint Full-text available The deep connections between braids and dynamics by way of the Nielsen-Thurston classification theorem have led to a wide range of practical applications. Braids have been used to detect coherent structures and mixing regions in oceanic flows, drive the design of industrial mixing machines, contextualize the evolution of taffy pullers, and characterize the chaotic motion of topological defects in active nematics. Mixing plays a central role in each of these examples, and the braids naturally associated with each system come equipped with a useful measure of mixing efficiency, the topological entropy per operation (TEPO). This motivates the following questions. What is the maximum mixing efficiency for braids, and what braids realize this? The answer depends on how we define braids. For the standard Artin presentation, well-known braids with mixing efficiencies related to the golden and silver ratios have been proven to be maximal. However, it is fruitful to consider surface braids, a natural generalization of braids, with presentations constructed from Artin-like braid generators on embedded graphs. In this work, we introduce an efficient and elegant algorithm for finding the topological entropy and TEPO of surface braids on any pairing of orientable surface and planar embeddable graph. Of the myriad possible graphs and surfaces, graphs that can be embedded in$\mathbb{R}^2as a lattice are a simple, highly symmetric choice, and the braids that result more naturally model the motion of points on the plane. We extensively search for a maximum mixing efficiency braid on planar lattice graphs and examine a novel candidate braid, which we conjecture to have this maximal property. ... is the unique invariant function of Eq. (37) and that any initial function will converge to it. It was noted to us by S. Berman that when p(x) is a cosine, as in Eq. (38), the invariant function P (x) is the Weierstrass function, which was the first published example of a continuous function that is nowhere differentiable. ... Preprint Full-text available Active fluids, composed of individual self-propelled agents, can generate complex large-scale coherent flows. A particularly important laboratory realization of such an active fluid is a system composed of microtubules, aligned in a quasi-two-dimensional (2D) nematic phase, and driven by ATP-fueled kinesin motor proteins. This system exhibits robust chaotic advection and gives rise to a pronounced fractal structure in the nematic contours. We characterize such experimentally derived fractals using the power spectrum and discover that the power spectrum decays ask^{-\beta}$for large wavenumbers$k$. The parameter$\beta$is measured for several experimental realizations. Though$\beta$is effectively constant in time, it does vary with experimental parameters, indicating differences in the scale-free behavior of the microtubule-based active nematic. Though the fractal patterns generated in this active system are reminiscent of passively advected dye in 2D chaotic flows, the underlying mechanism for fractal generation is more subtle. We provide a simple, physically inspired mathematical model of fractal generation in this system that relies on the material being locally compressible, though the total area of the material is conserved globally. The model also requires that large-scale density variations be injected into the material periodically. The model reproduces the power spectrum decay$k^{-\beta}$seen in experiments. Linearizing the model of fractal generation about the equilibrium density, we derive an analytic relationship between$\beta$and a single dimensionless quantity$r\$, which characterizes the compressibility.
... These drawings are reported in Fig. 8.5-b. These machines are also subject of interest in mathematics as in [Thiffeault, 2018], where taffy machines are described as topological toruses and compared to find the "best taffy puller in a mathematical case". ...
Thesis
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