When I was in Jr. High School math classes were living nightmares. Whether I became lost from start to finish, or became confused midway through a problem, attempting to understand math theories, or do math exercises, regularly defined failure in my life. I envied a friend of mine who could not only compose flowing pieces of music on his piano, but was a math "wiz" as well. Early on I came to see that there might be a connection between math and music, but as a fine artist, and a failure at math, I was unaware that there was any more connection between math and art than a "paint-by-numbers" craft kit.

Many years later, I have returned to school to finish college and have discovered a side to the math sciences I had been unaware of, and am just beginning to appreciate. Therefore, I am commencing this project to explore some of the many connections between math and art.

There is very little on heaven and earth that cannot be somehow represented by artistic expression and likewise, almost everything within our realm of physical awareness–and even in a few realms beyond, can be expressed in mathematical terms. When an artist turns to a mathematician for their help in better understanding art as well as the world as a whole through mathematics, what will they discover? What are the ways in which, for example, the science of math is reflective of the ways of Chinese brush painter might approach their painting? How does a mathematical formula define the very essence of what is considered good design in classical Western arts? Can a design be defined in mathematical terms? Can mathematical concepts help artists in designing and producing their work? What do artists and mathematicians have in common?

Initially, when we consider the connection between math and art, we can start with observing important components of mathematics that are also found in works of art, these being points, lines, curves, angles, planes and geometric shapes (circles, squares, triangles, etc.) Even though most works of art do not simply consist of blatant geometric shapes, we do find that these shapes are often implied in the composition. The apples, oranges, and grapes in a still life are of a circular nature; the door and stairs seen in another composition are based on rectangles. The fine arts student who has used an artists wooden model of the human form has seen how the human body can be translated into geometric shapes. And if we advance in geometry to begin to work with (for example) a cube, we are concerned with the same term that the "life drawing" student strives to achieve in their rendering, and that is volume.

Beyond the basic geometric shapes, the artist is constantly working with points, lines, curves, and various planes. Any shape not easily defined by geometric shapes can be further described with these. After all, how often have we heard people who compliment the design of an object refer with admiration to its (elegant, sleek, classical, modern, or fluid) "line?"

Not only does math play a part in the mechanics of creating a composition, it can be helpful in guiding the artist to find the most pleasing approach, or the most accurate rendering. One of the most frequent subjects of artists through the ages has been of those found in nature, how is it that mathematics and nature are related?

Leonardo Fibonacci (1170-1250) was a mathematician, who after studying the Arabic language and Islamic mathematics in Northern Africa in his youth, later mastered the Hindu-Arabic system of numeration. He popularized this system in Italy, and his work eventually influenced all of Europe, as well as modern mathematics. "Leonardo da Pisa" is best known today for his discovery of a mathematical sequence, the Fibonocci numbers. (W. Ball 167, Pappas 134, Dunham 191). Based on a study of the results of pairs produced by the reproduction cycles of rabbits, Fibonacci discovered a numerical sequence: 1, 1, 2, 3, 5, 8, 13, 21, … that is reflected throughout nature. This numerical system could be described as: a sequence "in which each number after the first two is the sum of the preceding two numbers, for example, 8+13=21. (A sequence a₁, a₂, a₃,… is any ordered set of numbers where a_n is the nth number.)" (Parks 589). The Fibonacci sequence can be found in the numbers of petals a flower has and the double spiral growth patterns of pinecones and sunflowers. The relationship of these numbers to the study of phyllotaxy is quite extensive. Phyllotaxy looks at the growth patterns of, for example, leaves on a stem, or branches on a tree in relation to one another as they spiral up the stem or trunk. In figure 2 we can see that "there are 8 branches in three complete spirals. This is called a phyllotatic ratio of 3/8, which is the number of spirals divided by the number of branches. Most phyllotactic ratios are ratios of Fibonacci numbers." (Parks 592). Philip Ball in discussing the phyllotaxis pattern of monkey puzzle branches finds that they "show a phyllotatic relationships of 5, 8 and 3,5" (also 8,13) he further notes: "Straight away we can see the adjacent pairs (3,5), (5,8) and (8,13). But it turns out that the phyllotaxis classifications of leaves, petals or floret patterns in any plant species correspond to pairs in this series. A corollary of this is that the number of petals on most flowers corresponds to a Fibonacci number: buttercups have five, marigolds have 13, asters 21." Philip Ball continues to say: "More mathematical spookiness follows. The ratio of successive terms in the Fibonacci series gets closer and closer to a constant value the further along the series one progresses: 8/13=0.615, for example, and 13/12=0.619. This ratio approaches a value of 0.618034 to the first six decimal places. This number was well known to the ancient Greeks, who knew it as the Golden Section. It can also be expressed as (Ö5-1)/2. To the Greeks, this was a harmonious, almost mystical constant of nature. These proportions were considered by the Greeks to be pleasing to the eye and they based the dimensions of many temples, vases and artifacts on this ratio." (P. Ball 106). This magical number is referred to as the Golden Ratio, or Golden Mean. The ratio (Ö5-1)/2 is also known as: f=1+1/f. Multiplying both sides of the equation by f, we obtain the equation f_=f+1 which is equivalent to the quadratic equation f_-f-1=0. We can solve for f by using the quadratic formula which states: for an equation in the form of ax_+bx+c=0 (where a, b, and c are real constants) that x= [-b±Ã(b_-4ac)]/2a, substituting f for x, we get f=(Ã5-1)/2 this shows us that we can write our formula as: af_+bf+c=0 or, in this case, f_-f-1=0. Any real values can be plugged into the equation for x, for example, 5x_-162x+17=0, with a=5, b=-162, and c=17, therefore:

x= {162±Ã[(-162)_-4(5)(17)]}/2(5).

The proportions of what is known as the "golden rectangle," which the Greeks derived using the Pythagorean Theorem, was thought to be the most pleasing and aesthetic proportions for a rectangle. We can see the use of the golden rectangle in the construction of the ancient Parthenon, the Temple of Athena built in 5th century BC in Athens Greece. The use of these special proportions can be seen in art and architecture from the ancient Greeks up to present day. There is another special aspect of proportions in relation to the Fibonacci numbers and the golden rectangle. Take a golden rectangle of say, 34 inches in length and 21 inches in width. Break the rectangle into a square that is 21’’ x 21’’. Break the remaining area into a square of 13’’ x 13’’, and the area that remains into 8’’ x 8’’, continue this process as long as possible; it is a process that theoretically could continue to infinity. "If a quarter circle is added inside each square, the arcs fit together into an elegant spiral. This spiral is a good approximation to the so called logarithmic spiral often found in nature, such as in the shell of a nautilus mollusk. Successive turns of the spiral grow at a rate of approximately equal to the golden number." (Stewart 102). This logarithmic spiral defines many shapes in nature including shells, and many animal horns. Philip Ball informs us that this spiral can be expressed as: r=aθ, and further explains: "The logarithmic spiral has the unique property that the curve is everywhere ‘similar’ differing in size but not in shape. In other words, as the curve rotates through a fixed angle, it grows uniformly in scale." (P. Ball 12).

Some 200 years after Fibonacci brought a new approach to mathematics to the Western world, Europe, particularly Italy, was flushed with new ideas and approaches to artistic expression in a time well known as the Renaissance. Many artists and mathematicians (including Euclid), sought to understand and portray perspective depth and foreshortening in attempts of bringing more realism to artistic portrayals. Alberti is known for introducing the approach of using drafting lines to create grids to guide the creation of life-like perspective and proportion. Leonardo Da Vinci worked on an instrument he called a "trellis work" that assisted artists in placing subjects realistically in conjunction with each other and relative perspective. In Germany, an artist named Albrecht Durer, worked with a method of transferring small drawings accurately to larger sizes using his grid system. (Steven, 2000 Art Skills, 1,2). This system was so successful that it has been in use continuously since that time, helping to build everything from wall murals to old fashion billboards. (A variety of geometric proofs and instructions for perspective grids both for two and three-dimensional models can be easily accessed on the web under the search words of: "perspective art grid.") These new tools, embraced by the art world then, and still used today, were considered so important that Leonard Da Vinci was quoted as saying: "Perspective is the reign and rudder of painting." (Steven).

As the Renaissance with its vibrant re-birth of art, and many studies in realism and perspective flourished in Europe, other world cultures continued to create art through sometimes similar, but often differing viewpoints of art, design, and even nature. The indigenous cultures of the Middle East, Africa, and of the North American Indians (especially the Southwestern tribes), give us some of the most striking and clear examples of some of the basic forms of symmetry found in examples of basketry, pottery, textile designs, and decorations, (see figures 19, 20).

Basic symmetry is categorized into four types. The first is called reflective symmetry, also known as: mirror, axial, or bilateral symmetry. With reflective symmetry "the figure looks the same on both sides of a line, except that the two sides are mirror images of each other." Imagine a line drawn through the middle of a design and fold the design along that line, if the sides match, then the design is considered to be of reflective symmetry. (Garfunkle 254). Examples of bilateral symmetry can be found in the human form and many higher forms of life, since we are usually born with two eyes, two ears, two hands, and so on. Reflective symmetry can become more complex; if we took a square piece of paper folded it in half twice and cut out different shapes from the two folded edges, we would create a design that has a reflective symmetry along two different axes. In this way we can identify more complex reflective designs with multiple axes. It is important, however, that the axis lines are described by the "fold centers" and all cross at the exact midpoint (Wilson 7). A second of the classic (simple) types of symmetry is rotational. "A figure has rotational symmetry if a point exists about which the figure can be rotated less than 360û (a complete rotation), and it maps exactly onto itself. If a figure maps onto itself three times in a complete rotation, it is said to have three fold symmetry." (Seymour, Geometric Design, 95). There are examples of figures (designs found both in art and nature) that can exhibit both rotational and reflective symmetry. (Pappas). The third type of symmetry is known as translation, "which is a slide in a certain direction by a certain distance. Only infinite figures can be translated without changing their appearance. The infinite Kuba pattern in Figure 7 shows translation symmetry but no other kind. Similarly, an infinite chessboard could be moved by sliding (translating) all square two squares to the left; we cannot detect movement in either the Kuba pattern or the squares. Unlike the Kuba pattern, however the infinite chessboard had other symmetries-it could be rotated 90û about the center of any square or could be reflected in a line containing the diagonal of any square." (Garfunkle 15).

The fourth type of symmetry (see Figure 8) is known as "glide reflection because it is composed of a glide (i.e. translation) by a certain distance along some line, followed by a reflection in that line. An infinite row of alternating p’s and b’s has glide reflection," (see Figure 9). (Garfunkle 255). Garfunkle points out: "It is a remarkable fact that every rigid motion of the plane, no matter how complicated, can be built from one of these four symmetries… [This] leads to a simple classification of all patterns according to which combinations of the four fundamental rigid motions leave a pattern invariant. This kind of classification was originally developed by crystallographers in the nineteenth century who wanted to be able to classify and recognize the three-dimensional patterns associated with crystal structure. They proved that there are exactly 230 such pattern types in three dimensions." Here Garfunkle concludes that the patterns of past (and present) cultures found on textiles and pottery, are not as complex as three-dimensional patterns. Two-dimensional plane patterns when defined as: Patterns extending in all directions in a single plane, (such as wallpaper), have only 17 patterns. When a two dimensional pattern is confined to a strip or border traveling in only one direction, (such as a hat band), there are only seven varieties of patterns, (see figure 13). (Garfunkle 256, 258).

From ancient times to our 21^st century, the popularity and importance of patterns in our world can be seen in everything from mosaics, to glass work, brick patterns, wall and floor tilings; we enhance textiles, rugs, baskets, blankets, and other items with a tremendous variety of patterns. (Garfunkle 259). As noted by Garfunkle: "Such patterns have one feature in common: they use repeated shapes to cover a plane surface without gaps or overlaps." (260). These types of patterns are referred to as: tilings or tessellations. Within the category of tilings we can find both periodic and non-periodic tilings. Periodic tilings are distinguished by having each tile meet edge to edge, while with non-periodic tilings, the edges are offset from each other. The non-periodic arrangement of tilings can show us one of the many examples of the concept of infinity that can be found in math related art. If we were to offset one row of square tiles under another repeatedly by an amount increasingly smaller by one half of the previous amount, no matter how many rows of tiles were added, the edges would never meet! (Garfunkle 262). "Any elegant theorem in number theory assures us that _ + ⅓ + _ + … + 1/n never adds up to a whole number, so this type of tiling never repeats." (Garfunkle 262). The remaining small measure that restrains the rows of tiling from ever meeting edge to edge can be divided in half אּ number of times, and yet will never reach zero.

Math plays an invaluable part in organizing and creating symmetrical designs, especially when exploring tilings and tessellations. Tilings and tessellations might be seen as their own science using a variety of theorems, classifications, and properties in organizing defining and creating what may be an infinite number of possible designs. M. C. Escher is renowned for his optical illusions as well as his precise, imaginative, highly original, and organized approach to creating art. Escher might be considered the mathematician’s artist, and could be credited for bring an awareness of the art and science of tilings and tessellations to the public consciousness. "His works far surpassed the traditional tessellations of a plane. He gives motion and life to the objects he tessellates, as illustrated in such famous works as Metamorphosis, Sky and Water, Day and Night, Fish and Scales, and Encounter. Besides transforming the plane, the tessellated objects undergo a transformation themselves. In addition, one sees his mastery of the concepts of translations, rotations, and reflections in periodic tiling." (Pappas 79). A great many of Escher’s pieces illustrate examples of infinity. His piece entitled Smaller and Smaller, (see figure 14), shows the tiling of his lizard characters becoming infinitely smaller as they move towards the center of the piece. Although we cannot see, we can imagine the figures continuing to grow smaller and smaller moving into the (proverbial) "sunset of infinity." Another example of an Escher piece that portrays infinity is the Circle Limit, (see figure 15). This illustration not only displays symmetry, tiling, and infinity, but also illustrates hyperbolic art where a three-dimensional image is transferred to a two-dimensional surface.

The symmetry of designs whether naturalistic or man made have certain predictability to them, there is isometry, balance, regularity, and form; these meet expected definitions and fall into regular patterns that we can anticipate. Tilings and tessellations travel across planes with such regularity that we assume that they could continue in all directions infinitely, and yet we could be able to predict at what point we would find reoccurrence of their structure by observing their original form. There is a relatively new area of mathematics known as chaos that contends with aspects of our world that, up until recently, were thought to be entirely random occurrences and lacking predictability. The generally accepted definition sees chaos as completely random, however mathematicians define chaos as: "randomly appearing patterns that don’t appear to repeat themselves. You can’t predict a precise instant (or point) that the path will reach" in observing (for example) an oscillating path or pattern. However, there exists some underlying pattern; it’s just not always obvious. "Chaos was known of as early as the 17^th century, however, to be able to actually display chaos took our more recent access to computers." (Walton, interview). One area of the study of chaos is known as fractals. Chaos is considered more general since there are lots of "patterns" that fall under the category of chaos such as weather patterns, clouds, turbulent water flow, phenomenon in space even socio-economic "patterns." Infinity is inherent to chaos and fractals. Geometric sequences describing fractals will continue into אּ.

Benoit Mandlebrot introduced displays of fractals in the 1960s. Fractals which also exhibit chaotic behavior are usually discussed in a category of their own. The structures of plants, river systems and mountains are examples of fractals. Recently, scientists have been able to program computers to construct simulations of trees, mountains etc. using mathematical fractal models. (see figure 16). A fractal pattern will repeat itself on a smaller and smaller scale (Walton). Examples of this can be seen in examples of the Mandlebrot sets. (see figure 17). Color (Plate 38) (Figure 18) shows us a computer generated fractal pattern derived from: "The four basins of attractions of the four solutions of z⁴ - 1 = 0." (Garfunkle, Color Pates). Although it would take a highly skilled mathematician to explain this, we can see that this intriguing and beautiful, nearly symmetrical design was created directly from a math formula! Here is a striking example of art derived from math.

Therefore we can conclude that mathematics does indeed play a part in the mechanics of creating an artistic composition. We know that this can be shown in the creation of fractal designs, as well as in the creation of more classical design. Western artists have for some time used the ratios of the golden mean or the tools of grids and perspective lines to create esthetic and realistic proportions. The Fibonacci numbers and the related phyllotaxis pattern ratios can give artists valuable information in translating concepts found in nature to art. Chinese brush painting teachers when first introducing their students to painting trees are very specific in explaining how the forward branch is painted as opposed to the side branch as opposed to the invisible rear branch and so on. Without intuitive or actual knowledge of the special role of Fibonacci numbers play, artists would find it difficult to portray subjects found in nature. Symmetry is of course its own guideline. In understanding rudimentary symmetry, the artist is more knowledgeable and better able to approach design work. Artisans of almost all cultures have embraced and created patterns for centuries. Mathematical formulas can be used in describing any variety of tilings and tessellations. The artist M.C. Escher created intriguing new examples of art with his imaginative use of tilings and tessellations. The study of chaos is not only challenging and intriguing to mathematicians, many artists have experienced the challenge of portraying elements as illusive and fleeting, as clouds or water. Chinese brush painters have actually categorized a number of styles of brush strokes used solely for building the images of rocks and mountains. Is this so very different from a mathematician or scientist using a mathematical formula to "build" an image of a fractal? When interviewing mathematics professor Ian Walton, the following question was posed to him: "Given that very intricate and complex patterns, tessellations and rotational symmetrical designs are found in very early cultures when the explanatory math either did not exist, or was not accessible to all artisans, do you think that the human mind is capable of intuitive/artistic higher math functions where the math steps are not taken, but the art is still produced, i.e. the brain takes a short cut?" His response was: "…maybe. I say that because artists and mathematicians will both describe things as beautiful. The mathematician when looking at a formula will see beauty…" (Walton). Likewise, artists will see beauty in art, design and patterns, as well as in the world around them — almost all of which can be described by mathematics. It would appear that indeed artists and mathematicians do have a great deal in common.

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Walton, Ian, Ph.D. Instructor of Mathematics at Mission College, Santa Clara, CA