3. 1 Introduction
In this paper, I wanted to look at the relationship between Art and Math. Some of the greatest
minds have been both painters and mathematicians. Math was used as a means of achieving a sense
of depth in art. From this the beginnings of art and math becoming one, are rooted in geometry
and the science of optics. By studying geometry and optics, Artist were able to determine how to
create visually pleasing art.
With investigations into geometry and optics, a sense of what we think of as ”aesthetically”
pleasing began being questioned. From these investigations , it was theorized that our sense of what
is visually pleasing, is derived from our innate understanding of the Golden Ratio. By studying the
Golden Ratio patterns were noticed in art and architecture that implied that we have an innate
understanding of what the golden ratio is, although it may have not been defined in explicit terms,
those who designed the pyramids, ancient Greek architects, had an innate understanding of the
golden ratio.
Even in architecture today, we can still see the golden ratio being used, shapes of buildings take
on the proportions of the golden ratio, even some computer monitors are designed with the ratio
in mind.
From the golden ratio, we move onto other aspects of mathematics, the Fibonacci numbers,
which predict patter ens in nature, are related to the golden ratio. We examine the function that
defines the Fibonacci numbers and how they relate to the golden ratio.
We will also look at how mathematicians view math as an aesthetically pleasing art. The logical
rigidity that math follows, allows us to view that which is the truth, in simplest terms. The method
of proof is seen as a form of art. Some math equations to are viewed as art, in terms of elegance,
the power of the equation, and its applications and power.
From methods of proof and equations, we look at a relatively new branch of Mathematics,
Chaos. The term “Chaos” is not used in our normal understanding of the word. In math Chaos
theory is about finding order in random data. The ordering of this data is best done with graphing
software, something that has not’t become readily available until just recently. Powerful computers
were needed to graph the data. The difficulty in creating the drawings of Fractals is that, they are
an iterated function that takes millions of iterations to begin seeing patterns that emerge in the
data.
We will mainly concentrate on the different graphing methods used in creating Fractals. Frac-
tal art can create aesthetically pleasing pictures, that one can appreciate without knowing the
underlying math or techniques that were used to create the image. [2]
Geometry is also used to create art. The obvious branch of geometry that one would normally
look at is that of Euclidean-Geometry, we will instead take a cursory glance at Hyperbolic Geometry.
We will focus on the works of one artist here, the art of M.S. Escher. He used the methods of
hyperbolic geometry to create drawings that trick the eye and also seem to faithfully reproduce
things that we see, or rather, how we seem to view them.
2 How is math Art?
So, how is math art and how is art math? Mathematicians view their subject as the simplest way of
doing something complex. This seems like a contradiction, in’t math in itself complicated? While
some of the theorems and techniques that are used in math seem to be unnecessarily complicated,
they are in fact, the simplest way of achieving that desired goal. Mathematicians view math as
not only truth, a goal that philosophers and mathematicians have chased since the inception of the
3
4. word, but also as supreme beauty. Math is an Art of the mind,the methods of proof that are used
in math are the most pleasing aspects of mathematics.
The power in mathematical thinking is the concept of “abstracting”. Early Greek sculptures
such as Phidias, used the abstract notion of “ideal type” that standardized our conception of what
was aesthetically pleasing. This standardization extended to to human body and numerical ratios
were discovered that produced sculptures and architecture that were consistently pleasing to the
eye. The idea was that perfect art must follow the ideal proportions, giving rise to the concept of
the Golden ratio. The architecture of the Greeks emphasized the ratio. [2] A proof uses a minimum
number of assumptions or previous results to achieve truth. By the nature of proof, mathematicians
seek to find the shortest path possible to the truth, with the least amount of theorems, lemmas or
often times, words. The beauty of proofs are their generality, since proofs are so generalized, we
can use the techniques in the proof to solve a number of concrete examples. Often times, there
are many different ways to prove the same thing. In calculus, there are often times many different
methods to arrive at a solution for a single problem.
3 The Golden Ratio
What exactly is the Golden Ratio and where does it come from? The Golden ratio dates from
as far back as 500 B.C., the artist Phidias, seemed to create statues that embodied the golden
ratio. Some argue that our sense of aesthetics is derived from the Golden Ratio, architects, artists,
book designers and many others are encourage to use the ratio in dimensional relationships in
their works. Many ancient cultures have used and studied the golden ratio, some even used the
ratio without realizing they were using it. The golden ratio can be seen in Art, Architecture, even
swimming pools are designed using the golden ratio.
The Golden ratio appears frequently in the geometry of pentagrams and pentagons, this gives
the number an air of mysticism. The Greeks actually accredit the discovery of the golden ratio to
Pythagoras and his followers.
The golden ratio is defined as follows: The two quantities are in the golden ratio if the ratio
between the sum of those quantities and the larger one is the ratio between the larger one and the
smaller. The golden ratio is a mathematical constant that is approximately equal to, 1.618 or for
an exact number,
1 +
√
5
2
The equation for the golden ratio is as follows:
a + b
a
=
a
b
In 1876, Gustav Fechner polled the responses of a group of people to determind what shapes
they found most pleasing, from his research, he concluded that people prefer rectangular shapes
that are approximations of Golden Rectangles.
The Golden Ratio can be found in the measurement of ancient buildings such as the parthenon
on the Acropolis in Athens. In art the “perfect” human face can be viewed as a sequence of Golden
Ratios. [1]
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5. 4 Fibonacci Numbers
Next we examine the Fibonacci numbers. The Fibonacci numbers are a sequence of numbers named
after Leonardo or Pisa, known as Fibonacci. Fibonacci introduced the sequence in his book Liber
Abaci. Although the number had already been discovered in Indian mathematics, Fibonacci is
accredited with the sequence.
The Fibonacci sequence is defined as follows. The first number of the sequence is 0, the second
is 1. With the first two numbers of the sequence defined, we add the first two numbers together
and add the result to our sequence. Then we take the second and third numbers and add those
together. This is a recursive sequence that continues infinitely. In mathematics we define the
recurrence relation as follows.
Fn =
o if n=0
1 if n=1
Fn−2 + Fn−1 if n > 1
The sequence of numbers that appears is,
0, 1, 1, 2, 3, 5, 8, 13, ...
5 The Relationship between Fibonacci and The Golden Ratio
An interesting connection that the Fibonacci numbers and the Golden ratio have in common is the
equivalence of the solutions to each equation. The Fibonacci sequence has a closed form solution
as follows,
F(n) =
φn − (1 − φ)n
√
5
where the Golden ratio, interpreted as a solution to the polynomial equation:
x2
− x − 1 = 0
has solution,
1 +
√
5
2
Lets look at the sequence in another way. If we take the ratio of the two successive Fibonacci
numbers and we divide each by the number before it, we get the following series of numbers,
1
1
= 1,
2
1
= 2,
3
2
= 1.5,
5
3
= 1.666...,
8
5
= 1.6,
13
8
= 1.625,
21
13
= 1.161538
we can see with successive iterations that the terms of our new sequence begin approaching the
value of the golden ratio.
We can now view a different representation of the Fibonacci sequence. We will make a drawing
of squares. We start with two squares of size one next to each other, on top of these two squares
we draw a square of size 2. Now this gives a side that is 3 units long. we then continue adding
squares around the previous squares. With each new square having sides as long as the sum of the
previous two sides. This is what we call the Fibonacci rectangles.
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6. Figure 1: Golden Rectangles
From here, we draw a quarter of circle in each square. This looks like a spiral but in fact does
not fit the mathematical criteria of a spiral. Although it is a very good approximation of one.
These spirals are seen in the shells of snails and sea shells and many other naturally occurring
phenomena. The interesting part of these picture, is that the spiral in the squares makes a line
form the center of the spiral and increase by a factor of the golden number in each square. So the
points of the spiral are 1.618 times as far from the center after a quarter turn.
Figure 2: The Golden Spiral
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7. 6 Chaos
The name Chaos theory comes from the fact that the systems that describe the theory seem to be
in disorder.[4] But Chaos theory is about finding order in these systems. A condition common in
Chaos theory is the sensitive dependence on initial conditions. A small change in the starting point
of a system can vastly change the outcome of the experiment. The changes in the conditions can
be viewed as background noise, experimental error, or the inaccuracy of the equipment.
The area of Chaos will be the final result of calculations. This is viewed no as an from of art.
An important part of the argument for fractals is this, We can say that a ball appears to be a
point when viewed far away, it has 0-dimension, if we move in close enough to the ball, we can see
that it has 3-dimensions, if we move in closer to the ball, say a ball of twine, then we can see that
the individual strands appear to be 1-dimensional. An object hose irregularity is constant over
different scales is said to be self-similar, we call this a fractal.
6.1 Fractal Art
There are four types of fractals that we will look at, escape time fractals, Lindenmayer systems,
stochastic synthesis, and iterated function systems. The best way to view fractals is through the
use of computer software. Fractals are generated by a function being iterated a large number of
times on the complex plane. Hand drawing fractals is almost impossible, we will see a couple of
examples that show the basic nature of a fractal, but then move on to more complicated systems
that could not be drawn by hand. [5]
6.1.1 Escape Time Fractals
Escape time fractals are defined by a recurrence relation at each point in a space. These fractals
are manipulated by the choice of parameters before the function is iterated. The choice of which
parameters are iterated and how they are mapped to an image, along with the color from the
numbers produced through the calculations, give us a representative image of what is happening
in the function. The easiest way to generate fractal images is with programs such as Ultra fractal,
ChaosPro and Fractint. Examples of this type of fractal are, The Mandolbrot Set, the Newton
Fractal and the burning ship fractal.
7
8. Figure 3: An escape time fractal
6.1.2 Newton Fractal
Here we have an example of the complexity that arises with the complex plane. We characterize
this method on the complex plan by newtons method of approximating the zeros of a polynomial
function. In this representation on the complex plane we choose a starting point that is associated
with a root of an equation, after iterating the function, we can see when the numbers around that
point, converge, are bounded by a circle, or break the orbit and go to infinity.
Figure 4: Newtons Fractal
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9. 6.1.3 Lindenmayer Systems
The design of a lindenmayr system relies on simple geometry and being able to predict the shapes
that result from a system of predetermined rules. Examples of this method of generating fractals
are: The Koch snowflake and the Sierpinski triangle.
The Koch snowflake is constructed by starting with the line segments forming an equulateral
triangle. With these line segments we divide each into thress equal sections, we replace the middle
section of each segment by two additional sections. Each iteration of the procedure adds more
triangles to each section, generating the Koch snowflake. [1]
Figure 5: The Koch snowflake
The construction of the Sierpinski triangle is done in a similar fashion, instead of working
with a one dimensional object such as the Koch snowflake, we use a two dimensional object, a
triangle. We begin with an equilateral triangle, then we draw another equilateral triangle inside
our original triangle. This divides our original triangle into 3 smaller triangles, we repeat this
process indefinitely, each time shading in the triangle that we have drawn. [1]
9
10. Figure 6: The Sierpinski triangle
6.1.4 Iterated function Systems
Iterated function systems and variants thereof have a fixed geometric replacement rule. An example
is the fractal flame. Shapes and colors are determined by easily understood transformations of
shrunk copies of the whole pattern, and since the transformation matrices and deformations have
no particular significance, they are usually input in fractal software visually and often with a real
time preview. Another trend is manual editing, starting from a random fractal (the arbitrary
parameters are many and mostly independent). Apophysis is a popular and very sophisticated
example of this category.
10
11. 6.1.5 Stochastic synthesis
Stochastic synthesis of fractal noise (typically fractal landscapes) is controlled through a few simple
high level parameters and by trying different pseudo random number generator seeds.
Figure 7: Southern Alps
While these pictures are indeed impressive, they have been touched up in image processing pro-
grams. The programs that generate the landscapes do not fill in the shading of the ground or sky.
The program only draws the shapes of the landscape, after this is done, a graphic artist will work
with the resulting image and produce these high quality images.
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12. Figure 8: Mars
7 Hyperbolic Geometry
In math, hyperbolic geometry is branch of geometry that takes Euclid’s parallel postulate and says
that the postulate is false. Of the postulates in Euclidean geometry, assuming this one to be false
gives us another branch of geometry that is a better model of the physical world.
7.1 Tessellations
A subsection of hyperbolic geometry that we will look at is that of Tessellations. A tessellation
is a collection of plane figures that overlaps and has no gaps. We can generalize this to higher
dimensions, this is were hyperbolic geometry comes into play. Tessellations appeared frequently in
the art of M.C. Escher. Escher was inspired by the ornamental art of the Moors, he created designs
that integrated both math and art. [1]
7.2 Wallpaper Groups
An simple version of tessellations are 2-dimensional wallpaper groups. This is a classification of a
two dimensional repetitive pattern, based on symmetries.
12
13. Figure 9: p2
Figure 10: p6
7.3 M.C. Escher
Maurits Cornelis Escher, born 1898, is most famous for his “impossible structures”, such as Rela-
tivity, Ascending and Descending, and his Metamorphosis pictures. Eschers work was not limited
to lithographs and paintings, he also designed murals, postage stamps and tapestries. Be was born
in Leeuwarden, the Netherlands, and was the youngest son in his family. Escher went to the School
for Architecture and Decorative arts in Haarlem, after a week, he told his father he would rather
study graphic art. [3] Escher’s art was the art of his mind, not what he saw in the world around
him. Escher would create impossible ”realities” in his mind and then draw them. His work had a
very strong mathematical component.
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15. Figure 12: Waterfall, M.C. Escher
Figure 13: Relativity, M.C. Escher
References
[1] Ronald Staszkow, Robert Bradshaw, The Mathematical Palette, Brooks/Cole-Thomson
Learning, 2004.
[2] Morris Kline, Mathematics for the Nonmathematician, Dover Publications, New York, 1967.
[3] http://www.mcescher.com/
[4] Xubin Zeng, Roger A. Pielke, R. Ekykholt Chaos Theory and its Applications toe the Atmo-
sphere, Department of Atmospheric Science, Colorado State University, Ft. Collins, Colorado,
April, 1993.
[5] http://en.wikipedia.org/wiki/Fractalart
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