As scientists we often first encounter the physical world by observation. We develop mathematical models that mimic the physical world in some meaningful sense and which are subsequently used to objectively identify the nature of physical properties not yet observed or measured (``discovery"). In the past this discovery process was reserved exclusively to an empirical process that was based on detailed observation, logical formalism, repeatability, reductionism, and rigor. The aesthetic consequence of the mathematical models was often ignored or at most put on the front cover of text books. Recently scientists, mathematicians, and artists have witnessed a common experience through the use of computer graphics or ``visualization" where the physical and artistic worlds merge into fractal patterns of chaotic behavior in almost every discipline of science. Although numerous books have now been written on the subject, much confusion persists about the objective use of these visual tools.
Many of our scientific models originated as psychical images whose forms could not be communicated to others except perhaps notionally. Because objectivity and reproducibility are the ultimate goals of scientific models, scientists and engineers who create and use these models rarely allude to this psychical process and hence effectively deny the motivation and inspiration behind this creative visual experience. J. W. Gibbs, James Clerk Maxwell, Albert Einstein, and Richard Feynman were exceptions in that they would openly talk about their ``visual mental models".
Recently the use of visual tools has helped the scientist and engineer to analyze and interpret massive data sets that have been generated by supercomputer simulations and computer controlled laboratory experiments. The discovery process reported by scientists and engineers using these visual tools closely parallels comments made by researchers who used a visual cognitive process to create scientific models. In both cases the ``minds eye" is used to gain insight into complex abstract processes. Visualization is that initial step taken to establish a clearer pictorial representation of the problem and continue the development of the picture before the formal mathematics is done.
Perhaps some general guidelines can be established to help the
researcher use visual tools in a systematic and rational approach. But
the visual creative experience itself is unique to each researcher.
Feynman referred to this visual process as a ``half-assedly
thought-out-pictorial semi-vision thing". The objectivity that is
necessary to explain the final results and observations to others with
mathematical models does not extend backward to the visual process that
was used to discover the results. To illustrate this point I will
introduce the audience to a simple visual method of ``pattern 
function extraction: tensor equation invariance". Although this visual
method will allow the researcher to extract simple functions from raw
data, this visual method will only recover the dominant terms where
more accurate mathematical models can then be constructed to
communicate this new knowledge to others. On this topic several
researchers have investigated using visualization with tensors of second
order [2,3].
If we trace out what we behold and experience through the language of logic we are doing science; if we show it in forms whose interrelationships are not accessible to our conscious thought but are intuitively recognized as meaningful, we are doing art.
A. Einstein