Decorative Knotting Using Trigonometric Parametrizations

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Decorative Knotting Using Trigonometric Parametrizations.
By Lindsay D Taylor
Introduction
This document describes the formaƟon of decoraƟve and interesƟng knots using computer 3‐D graphics.
It is wriƩen from the perspecƟve of the pracƟcaliƟes of paƩern design, not mathemaƟcal theory. The
approach uses trigonometric parameterizaƟons for the knots and is based on the use of epitrochoid and
hypotrochoid plane curves interlaced by using one or more sin funcƟons in the 3rd dimension. In its
simplest form they can be represented by the following equaƟons.
|
|
EquaƟons 1
|
for
p, q are non‐zero integers and
loops face inwards.
When
the loops face outwards, while when
the
The form used for the x and y equaƟons was derived from that used by Edwards (1) who produced a
staggering and oŌenƟmes bewildering range of complex and intricate decoraƟve paƩerns for 2‐D
computer graphics in emulaƟon of those produced in the 18th and 19th centuries using the �Geometric
Lathe� and the SpirographŽ toy (2) of the 1960�s.
Figure 1 shows an interlaced compound triple knot formed by the procedures described in this
document.
Figure 1 ‐ Compound Triple Knot
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Methodology
The paƩerns were realized using Blender 3D (3). Blender 3D is a free 3D graphics applicaƟon which has
been released under the terms of the GNU General Public License. It is available for mulƟple operaƟng
systems and contains a feature set comparable with high‐end commercial 3D graphics soŌware. One of
the strengths of Blender is that it comes with an embedded Python interpreter, which allows it to run
scripts wriƩen in that programming language. These scripts can use the Blender/Python applicaƟon
programming interface (API) to expand Blender's funcƟonality
Figure 2 ‐ Surface creaƟon from NURBS curve
The paƩerns were generated as NURBS curves generated by a Python script. A NURBS curve is deﬁned in
terms of a set of �control points� which deﬁne a �control polygon�. The curves were then turned
into virtual 3D objects using an �extrude along path� technique. This consists of creaƟng a surface by
sweeping a separately‐created proﬁle along the curved path by assigning a Bevel Object (BevOb) to the
curve. A visualizaƟon of the process can be seen in Figure 2.
Results and Discussion
Torus Knots using Outward-Looping Curves
The Trefoil Knot
A wide range of torus knots can be prepared using the above‐described simple equaƟons. That is to say,
without the use of the standard equaƟons used to describe a torus knot (4).
The Trefoil Knot, when considered as a simple outward facing loop paƩern (
), requires separate
handling and will be considered ﬁrst. It can be described by the following equaƟons:
Figure 3 shows a Trefoil Knot (
EquaƟons 1. When
and demonstrates the eﬀect of varying the
raƟo in
(in this case 0.5) a cusped paƩern results. Above this value a
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looped paƩern is produced with the inner curves touching when
intersect producing the familiar Trefoil Knot paƩern.
. Above this value the loops
It should be noted that the Torus Knot (and the circulaƟng paƩerns which follow) can also be
represented by the more familiar form of the x, y equaƟons represenƟng a hypocycloid curve. That is to
say,
and the sign change in the x equaƟon as shown below also results in a Trefoil Knot.:
Figure 3 ‐ Trefoil Knot
Torus Knots using � Outer CirculaƟng PaƩerns�
�.
For the following paƩerns the value of p is increased. In the days of the �Mechanical Lathe� the x‐y
plane paƩerns of this type were known as circulaƟng paƩerns because they took more than one cycle to
get back to the starƟng point. When considered in terms of 3‐D knot paƩerns this is the most basic
deﬁniƟon of a torus knot. If
, the equaƟons generate standard torus knots. The requirement
for p and q to be relaƟvely prime sƟll stands.
It is interesƟng that the ﬁrst paƩern formed in this fashion also generate a Trefoil Knot as seen by the
following equaƟons:
With this type of paƩern the number of loops is equal to
and the number of cycles taken to
return to the starƟng point is p. Figure 4 and Figure 5 show groups of torus knots. The equaƟons used for
their producƟon follow:
(2, 7) Knot
xval = cos(2*θ) + 0.2*cos(‐5* θ)
yval = sin(2* θ) + 0.2*sin(‐5* θ)
zval = 0.35*sin(7* θ)
(2,9) Knot
xval = cos(2* θ) + 0.25*cos(‐7* θ)
yval = sin(2* θ) + 0.25*sin(‐7* θ)
zval = 0.35*sin(9* θ)
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Figure 4 ‐ Some Torus Knots
The procedure can produce knots with
as can be seen with the series of knots shown in Figure 5.
Figure 5 ‐ Five Loop Series of Torus Knots
(2, 5) Knot
(3, 5) Knot
(4, 5) Knot
The data presented here show a procedure for generaƟng torus knots without the use of the standard
equaƟons. In this respect, Kauﬀman (5) has previously shown that all Torus Knots can be expressed as
Fourier Knots of the form Fourier‐(1, 3, 3) and Hoste (6) that their simplest representaƟon is as Fourier‐
(1,1,2) knots. Thus, it is not surprising that many Torus Knots, when derived from a diﬀerent viewpoint,
can also be represented as Fourier‐(1, 2, 2) knots as is shown in this document. The equaƟons shown
above have the limitaƟon that they are not universal for all Torus Knots (viz Figure Eight Knot) and are
not as intuiƟve to use as the standard equaƟons for Torus Knots.
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The Figure Eight Knot and other Inner Looping Patterns
The following paƩerns share the feature that
Figure Eight Knot
Figure 6 shows three knots which apparently share a similar underlying structure. The ﬁrst is the Figure
Eight Knot. Since, in this case, the inner loops are required to intersect
. The knot can be
described by the following simple set of equaƟons.
Figure 6a
Figure 6 ‐ Figure Eight and Similar Knots
Figure 6b is an inner circulaƟng paƩern (p = 2) while Figure 6c, although simpler in appearance than
Figure 6b is, in fact, an example of a compound knot. More examples of this type of knot will be given in
the next secƟon.
Figure 6b
Figure 6c
Inner CirculaƟng PaƩerns
Inner circulaƟng paƩerns have the characterisƟc of having
. Figure 7 shows three such paƩerns
with n/m < 1. InspecƟon shows that Figure 7a and Figure 7b have the appearance of decorated torus
knots which return to their starƟng point in two cycles. That this type of equaƟon does represent a torus
knot is easily demonstrated by reducing the n/m raƟo as is shown in Figure 8 where
was reduced to
0.3 in the equaƟons for Figure 7a (and
used in the z equaƟon). It is clear from this and other work
(5), (7) that there are several conceptual ways to deﬁne a Trefoil Knot (and other Torus Knots).
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Figure 7c shows an example where q = 3. It is an interesƟng paƩern as it has three inner and outer loops
and also takes three cycles to return to the starƟng posiƟon.
Figure 7 ‐ Inner CirculaƟng PaƩerns
Figure 8 ‐ Trefoil Knot using "Inner Loops"
Figure 7a
Figure 7b
Figure 7c
Some Compound Patterns
Compound paƩerns are those in which an addiƟonal trig funcƟon has been added to the x and y
equaƟons. Only a single aspect of this broad area will be examined here � paƩerns made from groups
of simple knots.
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Pairs of Knots.
The simplest examples of this are the Granny Knot and Square Knot which are both pairs of 3‐crossing
knots. InspecƟon of Figure 9 demonstrates that these two knots share the same x,y paƩern and merely
diﬀer in their interlacing. Thus, they can be formed using the same x,y equaƟons and merely altering the
z equaƟon. These equaƟons are disƟnct and simpler than those described by Trautwein (7) who used a
more organic and complex 3‐dimensional approach to knot design which used both cos and sin
funcƟons in all three equaƟons.
Figure 9 ‐ The Square and Granny Knots
For the Granny Knot the following z equaƟon is used:
while for the Square Knot the corresponding equaƟon is:
An alternaƟve, simpler, set of equaƟons for the Granny Knot follows. It is based on a four outer loop
paƩern with intersecƟng central curve
Knot pairs need not be limited to 3‐crossing knots. Figure 10 shows two related examples of pairs of
4‐crossing knots. Figure 10b is probably the simplest form for equaƟons describing a pair of 4‐crossing
knots.
Figure 10 ‐ 4‐Crossing Knot Pairs
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Figure 10a
)
Figure 10b
Larger Groupings of Knots
An advantage of the more 2‐dimensional approach to knot design used in this document is that it can be
used to produce larger grouping of knots using relaƟvely minor alteraƟons to the equaƟons. Some
results of this can be seen in Figure 1 and Figure 11 and Figure 12
Figure 11 ‐ A Compound Triple Knot
Figure 11 EquaƟons
The following style of knot (Figure 12) has been created with from between three and six outer knots by
minor alteraƟons to the equaƟons.
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Figure 12 ‐ A Compound Quadruple Knot
Figure 12 EquaƟons
A Note on the Interlacing of Patterns
The approach to interlacing paƩerns used in this document was simple � the use of a sine funcƟon.
Although it can be used to successfully produce fully‐interlaced paƩerns in a variety of paƩerns, it is a
naďve approach. The simplest approach, which is to use a number of sine waves in the z equaƟon which
equal the number of crossings, can fail in even a very simple knot. This occurs most frequently when
there are regions in the paƩern where the crossings are near each other while they are distant in other
regions of the paƩern. In some cases beƩer interlacing can be obtained by the addiƟon of an addiƟonal
sin funcƟon in the z equaƟon to modulate its shape. This approach was used in the producƟon of
several of the knots described in this document. AlternaƟvely, increasing the frequency of the sine
waves can oŌen produce a successfully interlaced paƩern but results in an ugly paƩern. This problem is
illustrated in Figure 13.
Figure 13 ‐ Interlacing problems and a soluƟon
The simple four outer loop paƩern is shown with three interlacing soluƟons. In Figure 13a. the
interlacing fails when
� the number of crossings. In Figure 13b when
was used, a
successfully interlaced paƩern was achieved but also resulted in the presence of unnecessary waves in
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the outer regions of the paƩern.
This diﬃculty can be seen in many of the paƩerns produced by Morita (8) , which used this method of
curve interlacing. It should be noted that in his work the addiƟonal curves were, in some cases, used to
advantage when the individual paƩern was used as a unit of a larger paƩern!
A pragmaƟc soluƟon to many of these problems can be achieved by a minor recoding of the script which
generates the paƩern to have a diﬀerent frequency of the sine wave in diﬀerent regions of the paƩern ‐
see Figure 13c. The following extract of the script code demonstrates this soluƟon by the use of an
if�else condiƟonal statement when calculaƟng the z equaƟon.
xval = cos(pi*u) + 1.5*cos(‐3*pi*u)
yval = sin(pi*u) + 1.5*sin(‐3*pi*u)
if 0.167 < u < 0.333 or 0.666 < u < 0.834 or \
1.167 < u < 1.333 or 1.667 < u < 1.834:
zval = 0.35*sin(16*pi*u)
else:
zval = 0.35*sin(4*pi*u)
A more elegant soluƟon to the problem can be achieved in many cases by creaƟng paƩerns which use
fully‐interlaced torus knots. This procedure is described in the accompanying document.
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Works Cited
1. Edwards, Ross. Microcomputer Art. Australia : PrenƟce‐Hall, 1985. ISBN 0 7248 0795 0.
2. Weisstein, Eric W. Spirograph. From MathWorld‐‐A Wolfram Web Resource.
hƩp://mathworld.wolfram.com/Spirograph.html.
3. Blender 3D. hƩp://www.blender.org/ . [Online] The Blender FoundaƟon.
4. Torus Knot. From Wikipedia, the free encyclopedia. hƩp://en.wikipedia.org/wiki/Torus_knot.
5. Kauﬀman, Louis H. Fourier Knots. hƩp://www.math.uic.edu/~kauﬀman/PJFourier.pdf.
6. Hoste, Jim. Torus Knots are Fourier‐(1, 1, 2) Knots. 2008. arXiv:0708.3590v1.
7. Trautwein, Aaron. Harmonic Knots. Ph. D Thesis. July 1995. Department of MathemaƟcs, University of
Iowa, Iowa City, Iowa, U.S.A.
8. Morita, Katsumi. Shapes of Knot PaƩerns. Forma. 2007, Vol. 22, pp. 75‐91.
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