INTRODUCTION

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.

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"