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THE MATHEMATICS OF TAFFY PULLERS

JEAN-LUC THIFFEAULT

Abstract. We describe a number of devices for pulling candy, called taﬀy 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 taﬀy puller

models, we unearth a rich variety of early designs from the patent literature, and

introduce a new one.

Introduction

Taﬀy 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

taﬀy, 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 taﬀy puller from an old patent. Observe that the taﬀy

(pictured as a dark mass) is stretched and folded on itself repeatedly. As the rods

move, the taﬀy is caught on the rods and its length is forced to grow exponentially.

The eﬀectiveness of a taﬀy 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 taﬀy 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 taﬀy pullers with more than three

or four moving rods. Throughout, we will give examples of taﬀy pullers from the

patent literature as well as a newly-invented one. Finally, we answer the question:

which taﬀy 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 taﬀy puller patented by Richards (1905). The

rod motion is equivalent (conjugate) to that of Fig. 6.

Some history

Until the late 19th century, taﬀy was pulled by hand — an arduous task. The

process was ripe for mechanization. The ﬁrst patent for a mechanical taﬀy puller

was by Firchau (1893): his design consisted of two counter-rotating rods on con-

centric circles. This is not a ‘true’ taﬀy puller: a piece of taﬀy wrapped around

the rods will not grow exponentially. Firchau’s device would have been terrible at

pulling taﬀy, but it was likely never built.

In 1900, Herbert M. Dickinson invented the ﬁrst nontrivial taﬀy 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. Taﬀy was especially

popular at beach resorts, in the form of salt water taﬀy (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

taﬀy. Though he was usually not the inventor, he was the assignee on several taﬀy

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).

Taﬀy was becoming big business.

Shockingly, the taﬀy 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 diﬀerent design principles.

The Dickinson taﬀy 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 taﬀy puller.

Three-rod taffy pullers

Taﬀy 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 eﬀect of

the puller on a piece of abstract ‘taﬀy.’ For this puller, the ﬁrst and second rods are

THE MATHEMATICS OF TAFFY PULLERS 4

(a) (b)

Figure 2. (a) Taﬀy 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 taﬀy 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:T2→T2,

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 taﬀy 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) Taﬀy puller from the patent of Nitz (1918), where rods

alternate between the two wheels.

Figure 4(a) shows how the diﬀerent 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/ι, 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:T2→T2. 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,d∈Z,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/ι, 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 ι◦φ=φ◦ι, a linear map such as φon the torus projects nicely to the

punctured sphere S0,4=T2/ι. 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 taﬀy puller in Fig. 3. The

permuted points (p1,p2,p3) play the role of the rods of the taﬀy 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 taﬀy 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 taﬀy in Fig. 3(a) (third frame) for 1

3period of the taﬀy 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 taﬀy 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 taﬀy, 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

taﬀy 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 taﬀy 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 taﬀy 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 Thiﬀeault 2011).

Are four rods better than three?

As discussed in the previous section, all 3-rod taﬀy 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 taﬀy 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 eﬀect,

and his patent has some of the prettiest diagrams of taﬀy 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 taﬀy 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/ι)− {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 taﬀy 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 taﬀy 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 taﬀy pullers arise from essentially the same Anosov map, only interpreted

diﬀerently. In other words, at least for the standard 4-rod puller, the addition of a

rod does not increase the eﬀectiveness 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=S2/ι2. The 6 punctures will correspond to the

rods of the taﬀy puller.

A bit of experimentation suggests starting from the Anosov map

(7) φ(x)= −1−1

−2−3!·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 diﬀerently, as shown in

Fig. 10(a). In addition to p0and p1, the map has four more ﬁxed points:

(8) ( −1

3−1

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 taﬀy 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

3−1

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 eﬀectively 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 diﬀerent 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 S2/ι2, which is the punctured sphere S0,6.

by πabout its center. This ﬁxes the points 1–6. The quotient surface S2/ι2gives

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 S2/ι2=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 x2−4x+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=S2/ι2, 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(4g−4) by the Gauss–Bonnet formula, with g=2 the genus.

What is the best taffy puller?

There are many other taﬀy 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 oﬀer 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 (Thiﬀeault and Budiši´c 2013–2017) and

train (Hall 2012).

Many taﬀy 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 taﬀy pullers. Kobayashi and Umeda

(2007, 2010) and Finn and Thiﬀeault (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 Thiﬀeault (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 x8−4x7−x6+4x4−x2−4x+1, which

is approximately 4.1858.

The reader might be wondering at this point: which is the best taﬀy puller?

Did all these incremental changes and new designs in the patent literature lead

to measurable progress in the eﬀectiveness of taﬀy pullers? Table 1 collects the

characteristic polynomials and the dilatations (the largest root) for all the taﬀy

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 diﬀerent

taﬀy pullers is not straightforward. To keep things simple, we take the eﬃciency

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. Eﬃciency of taﬀy 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

eﬃciency. 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) x2−6x+1χ21 1.7627

Nitz (1918) 3 3 x2−3x+1ϕ21

/32.8873

standard 4-rod 6 4 x2−6x+1χ21 1.7627

Thibodeau (1904) 14 4 x2−3x+1ϕ21

/32.8873

6-rod 8 6 (2) x2−4x+1 2 +√31

/22.6339

McCarthy (1916) †15(c) 4 (3) x2−18x+1ϕ61 2.8873

15(d) 4 (3) x4−36x3+54x2−36x+1 34.4634 1 3.5399

mixograph ‡13(c) 7 x8−4x7−x6+4x4

−x2−4x+14.1858 1

/68.5902

Jenner (1905) 16 5 (3) x4−8x3−2x2−8x+1 (ϕ+√ϕ)21 2.1226

Shean (1914) 17 6 x2−4x+1 2 +√31

/22.6339

McCarthy (1915) 18 5 (2) x4−20x3−26x2−20x+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 eﬃciency as the

entropy (logarithm of the dilatation) per period. In this case the eﬃciency is

log(ϕ2)/(1/3) =6 log ϕ≈2.8873. By this measure, the mixograph is the clear

winner, with a staggering eﬃciency of 8.5902. Of course, it also has the most rods.

The large eﬃciency is mostly due to how long the rods take to return to their initial

position.

Some general observations can be made regarding practical taﬀy 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 taﬀy 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 taﬀy’s folds are combined at each period. With a small

number of rods, there is a limit to the degree of this recurrence (2n−4 for nrods). A

second reason is that more rods does not necessarily mean larger dilatation (Finn

and Thiﬀeault 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 Thiﬀeault 2011; Thiﬀeault and Finn 2006; Venzke 2008).

The collection of taﬀy 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 Thiﬀeault 2011; Kobayashi and Umeda 2007, 2010; Thiﬀeault and Finn

2006).

Acknowledgments. The author thanks Alex Flanagan for helping to design and

build the 6-rod taﬀy puller, and Phil Boyland and Eiko Kin for their comments on

the manuscript.

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Appendix: A few more taffy puller designs

Because of space constraints, several taﬀy 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 taﬀy 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

taﬀy 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 taﬀy 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 diﬀerent 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

taﬀy would sometimes be unstretched. In the ﬁrst conﬁguration the rod resolutely

REFERENCES 20

(a)

(b) (c)

Figure 14. (a) Taﬀy 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 taﬀy 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 Thiﬀeault 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 diﬃculties involved in comparing the eﬃciency of diﬀerent devices. In its ﬁrst

conﬁguration the device has a quadratic dilatation, the largest root of x2−18x+1.

In its second conﬁguration the dilatation is a quartic number, the largest root

of x4−36x3+5x2−36x+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) Taﬀy 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.

8x3−2x2−8x+1, which is the strange number (ϕ+√ϕ)2, where ϕis the Golden

Ratio.

Interlocking combs. The taﬀy 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. Taﬀy puller from the patent of Jenner (1905). (b) The

motion of the rods, with three ﬁxed rods in gray.

(a) (b)

Figure 17. (a) Taﬀy 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 x4−20x3−26x2−20x+1.

Why so many designs? There are actually quite a few more patents for taﬀy

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) Taﬀy 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 taﬀy 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

eﬀectiveness of existing ones.

Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA

E-mail address:jeanluc@math.wisc.edu

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