Many people associate mathematics with the opposite of art - organized, straightforward and logical - whereas art is abstract and freeform. When one thinks of art, the word "aesthetics" or "beauty" often comes to mind, but what exactly does that mean? The terms are so broad, and so diverse that "beauty" can be applied to nearly any topic, whether it be in literature, film, music or even mathematics. At the basis, patterns have been used as a form of creative self-expression and to appease our (or most people's) sense of aesthetics. One of the most common type of patterns, often found in nature, involves fractals.

The word "fractal" was coined relatively recently. A Sterling Professor of Mathematical Sciences and IBM Fellow, Benoit Mendelbrot became deeply involved in the research of the significance of the equation: z = z² + c, in which the two variables measure values in the complex plane. In 1980, he used IBM's high powered computers to compute and graph thousands of outputs. As it would turn out, the graph would be an example of a fractal: a geometric figure with smaller sized recurring patterns. Mandelbrot realized the importance of his discovery, and published his findings in his book, The Fractal Geometry of Nature, in 1982. In addition to explaining the meaning of the word "fractal," he legitimizes his findings by providing examples of fractals in nature - his most basic example being the similarity in patterns in the splits of tree branches. Mandelbrot's work on fractals became a classic on chaos theory, the study of the capturing unpredictable into simple equations, since fractals display never ending patterns that recur on different scales.

However, artists have observed fractals long before Mendelbrot was known, implementing fractal patterns in their paintings, sculptures, and artwork. With the mathematical knowledge known today, one can gain a deeper appreciation for the talent, precision and patience required to create these intricate pieces.

Jackson Pollock

Jackson Pollock was a gifted Abstract Expressionist artist who had a talent for creating fractals. At first glance, his art seems like a random conglomeration of paint. Critics complained that Pollock's work was utterly insignificant and could be easily replicated by simply splattering paint on a canvas. However, through closer examination, each of Pollock's movements
Pollock's painting, 'Number 14'
Pollock's painting, 'Number 14'
were deliberate, each paint drip carefully planned, to create a harmonious artwork.

But how does one determine whether his work contains fractals? One way is to measure the degree of fractal dimension - the measure of self-similarity - and "fractal displacement" - the level of fractal dimension at different locations. Below is a graph a measurement of both of these properties of Pollock's painting, Number 14 through the use of the "box-counting" method in which sections of Pollock's work is covered with a number of computer-generated squares (represented as N(L)). This is repeated with differing square sizes (represented as L). Using an equation linking N(L) and L, values for fractional dimension (represented as D) can be calculated and graphed. As seen by the graph, the D values are linear, but its gradient is what determines whether a pattern is fractal. The typical accepted fractal dimension value for natural fractal patterns range from 1.25 to 1.3. Analysis of the graph reveals that Number 14 has a fractal dimension value of 1.45, significantly larger than the accepted fractal dimension value and thus, more complex patterns.
Measure of fractional dimension of Pollock's painting, 'Number 14'

Perhaps what is more interesting is the method in which Pollock created fractals. He introduced chaos through two radical differences in the way he applied paint to canvas. First, Pollock did not limit his movement to his hand and arms but included his body, providing an entire range of different lengths in which paint appeared on the canvas. Second, he painted by allowing paint to drip from the brush. In conjunction with varying trajectories due to his body movement, he generated an unpredictable, chaotic motion. Over the years, Pollock perfected this dripping technique as seen by increasingly larger fractional dimensional values in his work. On the right is his most fractal-heavy work, Blue Poles. With such a high fractal dimension (1.72), Pollock challenges the limits of what human eye can determine something as beautiful.
Pollock's painting, 'Blue Poles' (1952)
Pollock's painting, 'Blue Poles' (1952)

 Figure 1: Detail of non-chaotic (top) and chaotic (middle) drip trajectories generated by a pendulum and detail of Pollock's 'Number 14' painting from 1948 (bottom).
Figure 1: Detail of non-chaotic (top) and chaotic (middle) drip trajectories generated by a pendulum and detail of Pollock's 'Number 14' painting from 1948 (bottom).

Menger's Sponge

menger sponge.jpg
Created by Dr. Jeannine Mosely in 1995, this Menger Sponge is comprised of over 66,000 business cards.
The Menger Sponge was invented by Karl Menger in
an investigation in topological dimension, and is quite succinctly best described as a fractal cube. Although not an artwork in the typical sense, it has inspired others to create their own versions of Menger Sponges through different mediums due to its unique properties.
One of its unique properties is its infinite surface area but with zero volume. How is that possible? The answer lies in the construction of the sponge which involves taking a cube, slicing it into 27 individual cubes and then removing the cube in the center of each face and the cube in the center of the whole (in this scenario, 7 cubes). Repeat this process indefinitely. With every iteration, 7/27 of the volume of the sponge is removed and consequently, the volume is zero when done an infinite number of times. Conversely, the surface area will keep increasing as more cubes are added.

Easily the most interesting property is its cross section: cutting the sponge at an angle produces six sided stars in a symmetrical pattern - quite an unexpected result given the fact that all of the holes of the sponge are rectangular.
Cross section of a Menger Sponge
Cross section of a Menger Sponge

To create many levels of the Menger Sponge is an arduous task. Dr. Jeannine Mosely successfully created the behemoth level 3 (3 iterations) Menger Sponge out of 66,048 business cards. Realizing that creating such a large structure would be impossible from the ground up, she devised a method in which she would create modular components that would be synthesized at the very end. This basic unit was the "tripod" - a corner cube that is attached to three adjacent cubes. In order to create the level 3 sponge, over 448 tripods were linked.
The different types of tripods

Islamic Work

Central dome of the Great Mosque of Cordoba
Central dome of the Great Mosque of Cordoba

The Great Mosque of Cordoba, located in Spain, is universally known for its decoration, unity and combination of old and original architecture. Under Umayyad caliph Al-Hakam II, the Cordoba was transformed to reflect the new caliph and his beliefs, with
A closer look of the central dome. Also shows the circle inscribed inside a square
A closer look of the central dome. Also shows the circle inscribed inside a square

history being rewritten in the view of the current caliph. One such example is found in the central dome with its flowing Kufic script, blue-brown-yellow color scheme and 8-latticed shape. The Kufic script, while fractal-like in the uniform vertical stripe pattern, is also a universal call to fulfill religious obligations - an emphasis on predestination: "O you who believe, bow down and prostrate yourselves, and adore your Lord, and do good, that you may prosper. And strive in His cause as you ought to strive, He has chosen you and has imposed no difficulties on you in religion, it is the cult of your father Abraham; it is He who has named you Muslims, both before and in this (Revela-tion), that the Apostle may be a witness for you." The art of calligraphy was considered the highest art of all in Islamic culture, so it is not unusual to see excerpts from the Qu'ran or other written texts in Islamic art. Besides the Kufic script, the beauty of the arabesque patterns is a testament to Islamic mathematics and are notable for their seemingly infinitely extendable form (not much unlike a fractal). These patterns are based on basic geometric shapes which symbolize the principles that create order in the world. For instance, the four equal sides of the square is symbolic for the four elements: earth, water, air and fire. The physical world (represented by the circle) which is inscribed in the square, would cease to exist if one the square's sides collapsed.

The mihrab is also decorated with Kufic script - this time, a call to "guard strictly your prayers, especially the middle prayer, and stand before God in a devout frame of mind," and to "whosoever submits his whole self to God, and is a doer of good, has grasped indeed the most trustworthy handhold, and with God rests the end and decision of all affairs." Framing the niche is an inscription of belief of total submission to God and his omniscience: "Such is He, the Knower of all things, hidden and open, the Exalted, the Merciful. He is the Living, there is no God but He; call upon Him, giv- ing Him sincere devotion, praise be to God the Lord of the worlds." Like the dome, the mihrab is predominantly composed of a flowing arabesque patterns, but this time, based on plant forms, symbolizing the nature of life giving, or in this context, the nature of giving for the duties of religion.

Mihrab of the Great Mosque of Cordoba
Mihrab of the Great Mosque of Cordoba
Screen Shot 2014-06-09 at 2.19.01 AM.png
Uniformity of the Kufic script

Medieval Work

chi rho.jpg
Chi Rho monogram

The Book of Kells, considered as Ireland's greatest national treasure, is one of the earliest pieces of art that contains fractals. Created in 800 AD, the Book of Kells was used for sacred rather than educational purposes, as it would have been only used for the reading of the Gospel during Mass.

Like the Great Mosque of Cordoba, much of the illuminated manus
Screen Shot 2014-06-09 at 1.30.56 AM.png
cript is flowing arabesques and text. The term "moresque" is used to describe this Western adaptation of arabesque decoration. Defined as "a rude or anticke painting, or carving, wherin the feet and tayles of beasts, are intermingled with, or made to resemble, a kind of wild leaves," (Oxford English Dictionary) the moresque style is evident in the Book of Kells. For example, the Chi Rho monogram, is embedded with three angels at different section of the 'chi'. They are representative of Jesus' birth and also suggest the presence of the four Evangelists: Luke the calf, Matthew the man, Mark the lion and John the eagle. Also present in the Chi Rho monogram (in the bottom left corner) are snakes which symbolize Christ's resurrection because of the belief that youth was renewed when they shed their skin. Notice how the complicated knotwork the snakes form bear a similar crisscrossing shape which is repeated
on smaller scales. Thus, the monks
Chi Rho 2.jpeg
Snake interwined into complex patterns
who created this beautiful artwork managed to put in animals and people into a complicated arabesque-like design while maintaining fractal patterning.

Screen Shot 2014-06-09 at 2.11.38 AM.png
The crisscrossing shape of the snakes
Screen Shot 2014-06-09 at 2.12.02 AM.png
Screen Shot 2014-06-09 at 2.11.54 AM.png

Screen Shot 2014-06-09 at 2.11.54 AM.png

Fractals are a mathematical concept that one would not expect to find associated with art, but the inner beauty of the mathematics itself is what makes it a transcendental element in many art pieces.

Works Cited