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# The Mathematics of Taffy Pullers

Authors:
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