There seems little doubt that Art and Mathematics can complement one another. Mathematics exists profusely in works of many artists. Also, Art resides in many mathematical applications that are related both to the formulas and programs. One can see many reasons for cooperative work. Aesthetics of animation, comics, and manga production influenced the web page design, and the omnipresence of the web projects makes this impact noteworthy. When talents of mathematicians and artists act as a unit, a mathematician's great program formula and elegant solution of a problem can be supported with an artist's input.
MATHEMATICAL ORDER AS INSPIRATION FOR ART
In contradiction to seeming dichotomy of the mathematics- and nature-derived art, mathematicians and electronic artists can find beauty in mathematical description of natural order, whatever they choose to examine. We can find mathematical order in Greek meanders made by square shapes, sculptural forms that can be compared to programmed sculptures, Iranian pottery, or patterns woven into African textiles. Artists, especially folk artists, often act instinctively in accordance with mathematical rules and create forms that follow the formulas derived from geometry, fractal geometry, or various approaches to programming, even if being unaware of it. Natural beauty, for example, the beauty of plants, provides inspiration both for mathematicians and artists using computers. As it is widely known, regular patterns are perceived as beauty in nature: it may be a natural meander of a river; it may also be a fractal design of a mollusk shell, a snowflake, a pinecone, a sunflower, or a foliage arrangement on a tree.
PROGRAMS AND THE ESSENCE OF ART
Artwork resulting from writing a program is often as pleasing and impressive as nature itself. It is so, regardless of any possible remarks and disagreements about the essence of art and about conditions that identify what art is. For example, nature is not considered art by most of us. Controversies about the definitions of art arose in the second half of the 20th century. Morris Weitz (1956, 1989) presented his opinion in a paper "The Role of Theory in Aesthetics" that artists and aestheticians failed to define the nature of art because there is no essence that would be common to all artworks and restricted only to artworks. However, anyone can conclude what is both sufficient and necessary to call something art.
NEW INTEREST ABOUT ART AND MATHEMATICS CONNECTION
The art and mathematics connection gains more attention due to a variety of web sites that are designed to be comprehensible to a non-trained person. Web presentations about a golden section, fractal geometry, or mathematics-based architectural details enhance curiosity about this connection. Testimony about the order of the world would be incomplete without showing the beauty of mathematical solutions that are invisible in natural objects. Forms that present solutions of mathematical equations are often enjoyable and pleasing to the eye, maybe due to intuitive understanding of their origin. They include art works derived from random selection of numbers, permutations resulting in more or less regular patterns, equations visualized as sculptures (for example, a Moebius knot), or stereo geometry transformations. They provide the viewer with aesthetic satisfaction coming from a feeling of pleasure that the artistic goal has been fully achieved.
Abstract thinking in creating art
Higher level of thinking involves capacities to successfully perform problem solving and problem finding. Understanding of the structured type of thought that goes into computer science could possibly improve everyday interactions and work environments for artists and scientists. It may be valuable to those who are perusing a career in computer graphics, animation and web. This requires the ability of abstract thinking. Images, symbols, and words lie along a continuum from the concrete to the highly abstract, from representational to abstract image. Abstraction refers to any simplification or idealization of form, for example, achieved by converting an image into regular, geometric form. On a long way from generic to meaningful, we can organize our knowledge, look for relationships, select irreducible properties, and use only what we need. To achieve this, we take in those features of a category that best identify it, and suppress those features that are not basic to comprehending it, thus showing the scissorness of the scissors. In order to give a meaning to our image, we get rid of similarities and features (visible or semantic) that are not crucial. For example, knife and fork belong to 'silverware' - does not matter if it is green or wooden. Thus, we select features that are good for creating our categories and suppress other features.
AN EXERCISE BASED ON AN ABSTRACTION LADDER EXAMPLE
An abstraction ladder of categories described by Hayakawa (1990) consists of several steps.
1. A general physical reality, everything we see: all cow's characteristics are infinite and ever changing.
2. Reality perceived (a particular cow - the object selected by our senses: many of the characteristics are left out.
3. The word "Bessie" is the name of the cow, an object in (2) which stands for the object as a name. Bessie does not mean the features that were omitted.
4. The word 'cow' stands for the characteristics abstracted for a common cow, but not exactly for Bessie.
5. When Bessie is referred to as 'livestock', only those characteristics she has in common with pigs, chicken, goats, etc., are referred to.
6. When Bessie is included among 'farm assets', reference is made only to what she has in common with other salable farm items (corn, wheat, and grain).
7. When Bessie is referred to as an 'asset', still more characteristics are left out.
8. The word 'wealth' is an extremely high level of abstraction, omitting almost all reference to the characteristics of Bessie.
GRAPHICAL REPRESENTATION OF A C++ PROGRAM
This exercise in using abstract thinking was applied in a "Visual Thinking" course. We perceived relationships and selected irreducible properties of a cow Bessie, starting from a characteristics of a particular cow selected from the totality, using the word "Bessie" as the name of the cow, referring it to as 'livestock,' 'farm assets,' general 'asset,' and coming to 'wealth' as a high level of abstraction, omitting almost all reference to the characteristics of Bessie. Next, the exercise included visualizing abstract data with a simple C++ program. A C++ program was designed by an interdisciplinary student (computer science and art). He connected the user-defined elements of a program with visual symbols. Graphical representation of a C++ program was intended to teach a visual learner with little or non-programming experience how the basic structure of C++ works. Graphics described some simple aspects of the C++ class as an abstract data type. The class was used in a program that asked for the pounds of food that one wished to feed a cow named Betsy, and told how many steaks can be made from her.
The Cow program
Example of a C++ header file or class declaration. This is a declaration of the class called cow or more formally known as a class declaration. Defined as an ordinary every day cow. Nothing fancy, just some of the basic stuff about a cow.
Fig. 1. Example of a C++ header file or class declaration
Example of a C++ class definition. This is a definition of the class called cow that was declared above in the cow.h file. More formally known as a class definition this file describes exactly how the abstract data type, cow, works.
cow.cpp (The definition of the cow class)
Fig. 2 Example of a C++ class definition
Example of a C++ file that uses the cow data type. The abstract data type cow that we declared in cow.h, and defined in cow.cpp is used as a data type to collect information about an instance of the cow class called Betsy (black and white cow with a little more character)
betsy.cpp (Implementing the cow class in a program)
Fig. 3. Example of a C++ file that uses the cow data type
FACILITATING COGNITIVE PROCESSES WITH A SYMBOLIC CONTENT
Highly abstracted drawings that show no realistic graphic representation become symbols. Cognition makes possible a distinction between a signal and its referent. Symbolic shortcuts in human perception make a meaning of what we see or hear. Symbolic drawings may convey messages about universal forces of which we are not always conscious, such as a ying-yang symbol. And signals of that kind are grounded deeply in neural system. Symbolic shortcuts exist also in animal perception. Ethologists are aware of actual symbolic shortcuts and making a meaning of what animals see or hear. For example, for small animals the rustle of leafs may mean danger. An elongated object on the sky that is moving along its axis (like a goose) seems safe for small animals on the ground. When this object is flying sideways (like a hawk), it can be perceived as a predator by frogs, snails, or bunnies.
THE USE OF SYMBOLS AND METAPHORS IN DESIGNING A MESSAGE
Symbols may support developing the systems of ideas, recognizing particulars, discerning relations and concepts. By introducing symbols and metaphors, an artist may help to incorporate some concepts and complex data into a larger entity and hold the user's attention on cognitive processes. A metaphor indicates one thing as representing another, thus making a concept visible. Metaphors allow for understanding an abstract or unfamiliar domain in terms of another, more familiar concrete domain (Lakoff, 1990). We may find a pervasive use of metaphors in literature, visual arts, and music. For example, one can envision a continuum encompassing sound qualities: silence, sound, and noise as a grayscale.
According to Desain and Honing (1996), metaphors in music theory inform and shape the ways we theorize about music. Many believe that musical analyses are not scientific explanations, but metaphorical ones. In music theory, when one talks about rhythm, timing and tempo, often an analogy with physical motion (like walking or moving) is made.
"Two Directions for Two Trombones" created by Mathew Tolzmann is an example of a music data visualization. Students created interactive visualization of trombone playing. A scanned, transformed, and mirrored silhouette of the Denver downtown served as inspiration for two trombone players. The outline of the image served as a guide for music improvisation while changing pitch and volume.
Fig. 4. Mathew Tolzmann "Two Directions for Two Trombones"
Visual guide for an orchestra conductor created by Paul Grimes is another example of music data visualization. Graphic representation of four compositions shows:
a. Orchestration of forms composed for flutes, oboes, clarinets, sax, cornets, horns, trombones, euphonium and tuba presented along the time axis (measures),
b. Visualization of composition dynamics and architecture
c. Formal analysis of compositions with graphic presentation of their parts.
Fig. 5. Paul Grimes, "Visual guide for an orchestra conductor"
The use of suitable metaphor is crucial for successful program visualization - the presentation of pictures showing easy to recognize objects that are connected through well-defined relations. When we deal with several kinds of data of the same importance, we may describe the structure and the relations among these data with metaphors. A 3D visual city metaphor is often put in service of data visualization. A city metaphor presents static and dynamic information about the program. Selecting functional issues can reduce an effort and time, but also can provide fun when navigating through a city in 3D. For example, the size of a building shows the amount of lines of a code, the density of buildings tell about the amount of coupling between components, while the building's structure shows the quality of system implementation. There are also roads as a metaphor of connections. Cars moving through the city indicate how the program runs, leave traces in different colors, and show the origin, destination, and density of communication between components. Images of streets, waterways, boats, other vehicles, and clouds are also used for static and dynamic visualization.
Yasuhiko Saito (2002) produced artwork using an information visualization technique for financial analysis. He defined portfolio textures to visualize financial data. Saito presented graphically the stock price data in 1989 and 1990 from 23 Japanese automobile companies. In 1989, most of textures contain more red, orange and yellow dots than green and blue dots. In 1990 textures contain green and blue dots. The difference between 1989 and 1990 implies a turn of the tide in the Japanese stock market. Other visualization, which looks like a tapestry or an abstract painting, shows portfolio textures generated from stock price data of about 2,500 Japanese companies.
THE ESSENCE OF THE MESSAGE AN ARTWORK
While cognitive scientists are usually focused on looking for the sources of human communication that are grounded in precise thinking, artists often use purposefully transformed simple signs to direct thoughts of the viewers. To distill essential elements to be revealed in an artwork, artists transform an exact realistic representation into an abstract work that has no model in nature at all, for example, into the art of pure geometry. The essence of the message can be reduced to a simple gesture of a conductor, a selected fragment of a computer program, or a few wooden sticks symbolizing a warrior. In each case there is a signal to the receiver.
WHERE IS A PLACE FOR ARTISTIC INPUT?
Art application in mathematics may relate to working on mathematical formulas or programming. We still do not know what 'talent' means, especially 'talent' in a specific area. We do not know what cognitive processes make a mathematician or an artist a talented individual, for example, how some painters possess a sense of color. Maybe talent is needed to convey messages in essential, synthetic way.
WAYS TO CONVEY THE ESSENCE OF THE MESSAGE
It seems even more difficult to tell where is a place for talent in working on a formula or a code, or what makes one torus or knot looking dramatic and beautiful. How does a visualization of a formula become a way to convey an intense message? Why do some hand drawings with hardly any lines, make a gripping statement while other look arrogant or cynical? Mental shortcuts, synthetic signs, humor, caricature, or grotesque make a message even sharper. A realistically drawn dog has nothing to say to the viewer unless it becomes a character in a story where imaginary plot shows what is invisible in nature. Metaphors address cognitive abilities to abstract the essence of the message.
On the other hand, deformations in electronic art result in transforming the initial image aimed to depart from that what the eye can see. One can see many reasons for such transformations, either through programs or software. With purposeful deformations, we make mental shortcuts, react to synthetic signs, and imply connotations to symbols or icons.
APPROACHES TO CREATING AND DESIGNING
Everyone uses coding in one's own way. There is often some tension between the two types of thinking: precise or expressive.
Technology-oriented people respond to the market and make use of art in their technological applications. They use the Internet, available tools, and applications. Several personalities in electronic art have found a way of developing their individual artistic style that way.
Artists are determined to create art with any tool: a computer or a broom, whatever is at hand. Their powerful use of the elements and principles of design evokes an outburst of strong emotions. They often break the rules of composition. Many viewers believe they could do similar works consisting of a few simple strokes.
MAKING MENTAL SHORTCUTS AD CONNOTATIONS
Completely different outcomes follow these two approaches. We may contrast works of electronic design completed with first-rate tools with expressive art. For technology-oriented people teamwork with artists may bring a conflict between precision and accuracy of ready solutions and the artist's individual style.
In advertising, messages are often successful due to impressive imaging that gives a shorthand summary of ideas. Maybe there is a place for art to de-formalize the websites that are loaded with knowledge, and evoke some cognitive processes that would facilitate absorption of their content.
Desain, Peter and Henkjan Honing (1996) 'Physical motion as a metaphor for timing in music: the final ritard', San Francisco: ICMA, Proceedings of the 1996 International Computer Music Conference: 458-460.
Hayakawa, S. I. (1990) 'Language in Thought and Action' (4th Ed) Harcourt Brace.
Lakoff, George (1990) 'The Invariance Hypothesis: Is Abstract Reason Based on Image-Schemas?' Cognitive Linguistics 1(1): 39-74.
Saito, Yasuhiko (2002) 'Using a Visualization Technique for Financial Analysis to Produce Artworks', YLEM Journal, Artists Using Science & Technology, Issue Art and Programming 22 (10): 2.
Weitz, Morris (1956) 'The Role of Theory in Aesthetics'. The Journal of Aesthetics and Art Criticism 15: 27-35.
Weitz, Morris (1989) 'But is it Art?' Sonus 10(1): 1.
Anna Ursyn has created over 20 single art shows, participation in over 80 juried and invitational fine art exhibitions, such as: Immaginando '97-Digital Visions of Etruria (Honorable Mention), Grosetto, Italy, juried. Association for Computer Machinery/Special Interest Group for Computer Graphics (ACM/SIGGRAPH) Art Galleries: 1989, '90, '94, '95 (two works), '98, '99, '01 (two images), '02 (7 images), '03, '04 (two images and animation)'05 (three works) and ACM/SIGGRAPH traveling shows. Der Prix Ars Electronica 1988-2004. Eurographics 1991, 1996 (Artistic Merit award), 1998. (ISEA 1990, '92, '94, '96. SCAN 1990, '92, '93; ArCADE (Brighton, England) 1995, '97, 2001, 2003; New York Digital Salon 1995, '96, '98. 20th Century Matrix, NTT Museum, Tokyo, Japan (2000 images and 5000 texts describing 20 Century). Her artwork has been published in 'Digital Printmaking' by George Whale & Naren Barfield A&C Black (Publishers) London 2001, Watson-Guptil 2003, ISBN # 0-82301398-7. CD-Rom for "The Best of 3-D Graphics" by Vic Cherubini, Rockport Publishers, 2003. To be published in the spring of 2005 "Art in the Digital Age" by Bruce Wands, Thames & Hudson, Ltd. of London. Assignments and student work published in Deborah Greh "New Technologies in the Artroom", Davis'99 and 2003 Updated Edition ISBN # 87192611-3, and Stephen Pite "The Digital Designer" Thomson, 2003 ISBN # 0-7668-7347-1 She has received several grant awards for developing computer graphics technology. Research and pedagogy interests include integrated instruction in art, science, and computer art graphics.
Anna Ursyn, Ph.D. Professor
Head of Computer Graphics
School of Visual Arts
University of Northern Colorado
Greeley, CO 80639
firstname.lastname@example.org / email@example.com
Phone (970) 351 2476
Fax (970) 351-2299