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European Journal of Applied Sciences – Vol. 10, No. 6
Publication Date: December 25, 2022
DOI:10.14738/aivp.106.13388. Pan, H. (2022). The Art of Spirals. European Journal of Applied Sciences, 10(6). 109-120.
Services for Science and Education – United Kingdom
The Art of Spirals
Hongjun Pan
University of North Texas, Denton, TX 76203, USA
ABSTRACT
Interesting graphic patterns can be created by mathematically manipulating the
spirals calculated by the galactic spiral equations. Such mathematical manipulation
is achieved by multiplying the spiral data with a special mathematical function
named as art-pattern function. This method can be extended to parametric curve
equations other than spiral equations and to 3-dimension graphics. The graphic
creativity is unlimited through this method. This method will have wide
applications in many aspects of our daily life.
Keywords: The art of spirals, spirals, galactic spiral equations, art-pattern function,
artwork creation.
INTRODUCTION
The art is a special pattern which can give people aesthetic pleasing. This pattern can be static,
such as paintings, sculptures, architectures, hairstyle, clothes, makeup, tattoos, etc. It can also
be dynamic, such as dance, acrobatics, magic show, sports, music, movies, etc. This pattern may
express a specific meaning (such as the famous painting Uncle Sam poster, labelled with “I want
you”), or it may be just an appreciation of “beauty” and pleasing. The art is also a procedure or
process to make somethings, often seen in the patent descriptions. Spirals are the most popular
geometric patterns in the universe, they appear everywhere in the nature with innumerable
patterns, every spiral pattern in the nature is an artwork, they can be small as DNA helices and
large as spiral galaxies. Many mathematical spiral equations have been invented through the
history of science, like the Archimedean spiral, Euler spiral, Fermat’s spiral, hyperbolic spiral,
logarithmic spiral, Fibonacci spiral, etc. Spirals are among the oldest geometric shapes widely
used in creations of arts and architectures throughout entire human history and can be traced
back to the Stone Age. A very good book “Spirals in nature and art” by Sir Theodore Andrea
Cook gives a comprehensive description in this subject [1]; and graphic patterns created from
the parametric curves have been well developed and described in the book by A. Lastra [2];
such creations become much easier with computer graphics techniques. The author developed
a new set of galactic spiral equations from the ROTASE model to simulate the spiral patterns of
barred galaxies [3], the ROTASE model is the short name of the ROtating Two Arm Sprinkler
Emission model which was proposed to describe the possible mechanism for the formation of
spiral arms of galaxies. Most (if not all) of the spiral patterns of the barred galaxies can be nicely
simulated by the new galactic spiral equations. This paper will demonstrate that graphic
artworks with very interesting patterns can be created by mathematically manipulating the
spirals made with galactic spiral equations.
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THE GALACTIC SPIRAL EQUATIONS
The following are the primary differential galactic spiral equations derived from the ROTASE
model, please refer to the references [3, 4] for the detail of derivation.
⎩
⎪
⎨
⎪
⎧ �� = �! ∗ "
#$!% "! ��
�� = �! ∗ -�(�) − $
#$!% "! 2 ��
(1)
The equations (1) can be solved in a polar coordinate system for three different cases, ρ > 1, ρ
= 1 and ρ < 1, respectively. The Rb is the half length of the galactic bar; the θ is the galactic bar
rotation angle used as time counting in the ROTASE model, and the parameter ρ is the ratio of
the X-matter emission velocity over the flat rotation velocity of the galactic disc, the ρ can
change with time (θ) in any format, the galactic spiral pattern will be decided by the behaviour
of the ρ. For general application of the equations (1) for other than the simulation of spiral
galaxies, the ρ can be treated as a normal parameter which is function of θ, and the Rb is set to
1. A mini computer program can be written to calculate the x and y, and the calculated x and y
must be rotated counter clockwise by the following equations (2) for final spiral plotting, the
counter clockwise rotation is well explained in the reference [4]:
3
�'(�) = �(�) ∗ cos(−�) + �(�) ∗ sin(−�)
�'
(�) = −�(�) ∗ sin(−�) + �(�) ∗ cos (−�)
(2)
Plotting the x’(θ), y’(θ) will produce the calculated spiral, the x’ may be changed to -x’ depending
on the rotation direction of the spirals. If the parameter ρ is less than 1, the spiral will be a
spiral-ring pattern with the radius of the ring defined by the equation (3) as:
� = ("
)*+ (3)
The spiral-ring pattern can only be produced by the new galactic spiral equations at the
moment compared to all other available spiral equations. In this paper, the calculation will be
carried out by setting the rotation angle θ as the following:
�, = � ∗ � (4)
Where k is the natural number (1,2,3,4,5....) and the δ is the selected angle (degree) based on
the pattern to be created.
The various regular spiral patterns can be generated by the galactic spiral equations (1) with
different parameter ρ as shown in references [3, 5]. However, much more interesting patterns
can be generated by manipulating the galactic spirals with the following equations:
3
�(�) = �(�) ∗ �′(�)
�(�) = �(�) ∗ �'
(�)
(5)
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Pan, H. (2022). The Art of Spirals. European Journal of Applied Sciences, 10(6). 109-120.
URL: http://dx.doi.org/10.14738/aivp.106.13388
Where, the f(θ) is a special mathematical function named as the art-pattern function, which can
be any function based on one’s selection, the graphic patterns of plotting (X(θ), Y(θ)) is very
sensitive to the f(θ) and the θ, may be unpredictable with surprises. In this paper, the data
points are plotted in two ways: 1. scattered plot with bright yellow dots and blue background;
2. scattered plot with bright yellow dots, blue background, and straight red lines between the
adjacent data points, this may give additional visual effects.
THE GALACTIC SPIRAL EQUATIONS
This section lists selected patterns created by several art-pattern functions and related
parameters, as demonstration that very rich and appealing spiral artworks can be created in
this way.
Figure 1 shows the patterns of the spirals manipulated by the sin function with different
parameters. The first row of the Figure 1 is the plots of scatted data points, and the second row
is the plots of the scattered data points with red straight lines between the adjacent data points.
The rest of the figures in this paper will be plotted in the same way without additional
statements except for those figures with specific descriptions. It is difficult to predict what will
be seen when selecting an art-pattern function, it is completely a “trying and seeing”. The value
of pi is 3.141592, and the value of the δ(degree) must be converted to radian due to the angular
unit of the trigonometric functions of the computer program used in this calculation.
Unintentional mis-matched unit of angle in the computation can cause a very weird result [6].
Figure 1: Graphic patterns created by the manipulation of the sin functions.
Table 1. Art-pattern functions and parameters for Figure 1
Art-pattern function ρ δ No. of points
Fig. 1a sin(k*pi/100) 2.5 179.98 4000
Fig. 1b sin(k*pi/50) 2.5 89.98 4000
Fig. 1c sin(k*pi/100) 2.5 119.98 4000
Fig. 1d sin(k*pi/200) 2.5 71.99 4000
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Figure 2 shows the patterns of the spirals manipulated by the linearly decreasing function with
different parameters, the parameter ρ is less than 1 which should give a spiral ring pattern, but
the patterns look significantly different from the spiral-ring pattern by the manipulation of the
linearly decreasing function. The Figure 2a looks like a 5-petal flower with partially overlapped
petals. The Figure 2b has a very cute little star in the center.
Figure 2: Graphic patterns created by the manipulation of the linearly decreasing
functions.
Table 2. Art-pattern functions and parameters for Figure 2
Art-pattern function a ρ δ No. of points
Fig. 2a 1 - a * k / 4000 1 0.95 71.98 11-4000
Fig. 2b 1 - a * k / 4000 1 0.95 71.9 4500
Fig. 2c 1 - a * k / 4000 0.95 0.99 138.50776 4500
The Figure 3 shows patterns like 5-petal flowers; specially, the Figure 3c shows a line texture
in the petal which is very similar to the plumbago flowers. Figure 3h is created with Photoshop
to add various colors in different areas for additional artistic effect.
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Pan, H. (2022). The Art of Spirals. European Journal of Applied Sciences, 10(6). 109-120.
URL: http://dx.doi.org/10.14738/aivp.106.13388
Figure 3: Graphic patterns created by the manipulation of the linearly decreasing
functions.
Table 3. Art-pattern functions and parameters for Figure 3
Art-pattern function a ρ δ No. of points
Fig. 3a 1 - a * k / 4000 1 2.5 71.997 4000
Fig. 3b 1 - a * k / 4000 1 2.5 71.98 4000
Fig. 3c 1 - a * k / 4000 1 2.5 71.985 4500
Fig. 3d 1 - a * k / 4000 1 2.5 71.97 4000
Figure 4 shows that if the δ equal to 71.9°, only 0.1° less from 72°, the pattern becomes almost
a round pattern with many textures inside which can be decorated by different colors based on
people’s preference.
Figure 4: Graphic patterns created by the manipulation of the linearly decreasing function.
Table 4. Art-pattern functions and parameters for Figure 4
Art-pattern function a ρ δ No. of points
Fig. 4a 1-a*k/4000 0.9 2.5 71.9 4000
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Figure 5 shows the 4 round patterns with rich textures inside made by linearly decreasing
functions with different parameters. The δ for the Figure 5b, 5c and 5d is vey close to the golden
angle 137.50776°.
Figure 5: Graphic patterns created by the manipulation of the linearly decreasing
functions.
Table 5. Art-pattern functions and parameters for Figure 5
Art-pattern function a ρ δ No. of points
Fig. 5a 1 - a * k / 4000 1 2.5 71 4000
Fig. 5b 1 - a * k / 4000 0.9 2.5 137.60776 4500
Fig. 5c 1 - a * k / 4000 0.9 2.5 137.40776, 4500
Fig. 5d 1 - a * k / 4000 1 2.5 137.00776 4000
Figure 6a and 6b are the patterns made by sin art-pattern function, which look like firework
show. The Figure 6c and 6d are the patterns made by the linearly decreasing art-pattern
function, the Figure 6g looks like an umbrella, and the Figure 6h looks like an old-style oil paper
umbrella.
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Pan, H. (2022). The Art of Spirals. European Journal of Applied Sciences, 10(6). 109-120.
URL: http://dx.doi.org/10.14738/aivp.106.13388
Figure 6: Graphic patterns created by the manipulation of the linearly decreasing
functions and sin functions.
Table 6. Art-pattern functions and parameters for Figure 6
Art-pattern function a ρ δ No. of points
Fig. 6a sin(3 * k * pi / 4000) 2.5 71.99 4000
Fig. 6b sin(k*pi/500) 2.5 71.98 4000
Fig. 6c 1 - a * k / 4000 1 2.5 70.001 4000
Fig. 6d 1 - a * k / 4000 1 2.5 140 4000
Figure 7a shows the pattern made by the art-pattern sin function, the sin function created many
loops with the size gradually increasing, those loops form 4 spirals. The Figure 7c is created
with δ = 1°.
Figure 7: Graphic patterns created by the manipulation of the sin functions.
Table 7. Art-pattern functions and parameters for Figure 7
Art-pattern function ρ δ No. of points
Fig. 7a sin(k*pi/100) 2.5 179 4000
Fig. 7c sin(k*pi*/200) 2.5 1 4000
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The Figure 8 shows the spiral-ring pattern modulated by a sin function, the magnitude of the
modulation is controlled by the parameter b. The change of δ has significant impact on the
morphology also. Figure 8a shows the small waves on the ring (b = 0.05), the Figure 8b shows
the double waves, the Figure 8c shows the weaved ring. The Figure 8d shows a large, weaved
ring due to the large b value (b = 0.2), the dot size is increased to size 8 in PowerPoint to show
better visual effect.
Figure 8: Graphic patterns created by the manipulation of the sin functions.
Table 8. Art-pattern functions and parameters for Figure 8
Art-pattern function b ρ δ No. of points
Fig. 8a 1 + b* sin(k * pi / 100) 0.05 0.99 71.98 4000
Fig. 8b 1 + b* sin(k * pi / 100) 0.05 0.99 71.95 3550
Fig. 8c 1 + b* sin(k * pi / 100) 0.05 0.99 71.9 4000
Fig. 8d 1 + b* sin(k * pi / 100) 0.2 0.99 71.7 4000
Two different art-pattern functions can be combined to produce interesting patterns as shown
in Figure 9. One can compare Figure 9a with Figure 2a, Figure 9b with 3b, and Figure 9c with
Figure 3d respectively to see the combination effect.
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Pan, H. (2022). The Art of Spirals. European Journal of Applied Sciences, 10(6). 109-120.
URL: http://dx.doi.org/10.14738/aivp.106.13388
Figure 9: Graphic patterns created by the manipulation of the combined linearly
decreasing functions and sin functions.
Table 9. Art-pattern functions and parameters for Figure 9
Art-pattern function a b ρ δ No. of points
Fig. 9a (1-a*k/4000)*(1 + b* sin(k * pi
/ 100))
1 0.3 0.95 71.98 146-3900
Fig. 9b (1-a*k/4000)*(1 + b* sin(k * pi
/ 100))
1 0.3 2.5 71.98 4000
Fig. 9c (1-a*k/4000)*(1 + b* sin(k * pi
/ 100))
1 0.3 2.5 71.97 4000
Figure 10 shows 3 additional graphic patterns, each pattern has unique well-arranged texture
showing special aesthetic effect.
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Figure 10: Graphic patterns created by the manipulation of sin functions and the
combined linearly decreasing functions and sin functions.
Table 10 Art-pattern functions and parameters for Figure 10
Art-pattern function ρ δ No. of points
Fig. 10a sin(k*pi*/250) 0.99999 71.3 4000
Fig. 10b (1-1*k/4000)*sin(k*pi/300) 0.99999 71.9 4000
Fig. 10c sin(k*pi/200) 0.99999 71.7 4000
Figures 11, 12 and 13 show the examples of the decoration patterns made with the combination
of selected patterns above, they can be used in wallpaper, gift wrapping paper, textiles, or any
other suitable areas.
Figure 11: Decoration pattern made from Figure 1b.
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Pan, H. (2022). The Art of Spirals. European Journal of Applied Sciences, 10(6). 109-120.
URL: http://dx.doi.org/10.14738/aivp.106.13388
Figure 12: Decoration pattern made from Figure 1b and Figure 8d.
Figure 13: Decoration pattern made from Figure 1b and Figure 9e.
DISCUSSION
One can see that very interesting graphic patterns can be created by manipulating the spirals
through mathematical method, and all types of art-pattern functions can be used with different
parameters, which can create innumerable artwork patterns; therefore, the creativity with such
mathematical method is unlimited. This method can be extended to parametric curve equations
other than spiral equations and to 3-dimension graphics. The mathematical method
demonstrated in this paper will have wide applications in many aspects of our daily life, such
as the artwork creations, decorations, sculptures, architectures, textiles, advertisements,
industrious designs, printing industry, etc. It will be extremely challenge for someone to create
mathematic equations to produce those patterns.
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CONCLUSION
Innumerable artwork patterns can be created by mathematically manipulating the galactic
spirals and other graphics with various art-pattern functions. It will have wide applications in
many aspects of our daily life.
References
Cook, T., ed.: Spirals in nature and art. J. Murray, 1903.
Lastra, A., ed.: Parametric Geometry of Curves and Surfaces. Birkhäuser, 2021. ISBN 978-3-030-81316-1.
Pan, H., Spirals and Rings in Barred Galaxies by the ROTASE Model. 2021. IJP 9(6), p. 286–307.
Pan, H., Pitch Angle Calculation of Spiral Galaxies Based on the ROTASE Model. 2021. IJP 9(2), p. 71–82.
Pan, H., Introduction of New Spiral Formulas from ROTASE Model and Application to Natural Spiral Objects. 2021.
AJAMS 7(2), p. 66–76.
Pan, H., Special spirals are produced by the ROTASE galactic spiral equations with the sequential prime numbers.
2022. AJAMS, 8(4), p. 69–77.