As scientists and engineers we have traditionally used graphs and tables to communicate our ideas and mathematical functions to others in publications and presentations. However graphics is also creatively used to envision complex mathematical ideas and scientific information. Before we construct an algebraic function we imagine its shape graphically, i.e. "it goes like?" -- the idea of linearity is imagined as a straight line, exponential as an increasing sloped line, etc. We are taught in calculus to imagine multi-dimensional functions as shapes where geometric definitions of derivatives are envisioned as iconic patterns of slopes or planes tangent to these functional curved lines or surfaces and integrals of these same functions are envisioned as iconic patterns of areas or volumes. Iconic patterns representing derivatives and integrals enhance our understanding of physical properties that are of no consequence to the pure mathematician, e.g. the derivative of velocity with respect to time is acceleration and the integral of stress as a function of strain is energy. Using graphics to organize simple and complex ideas is an innate cognitive ability that can be used to envision scientific information and extract complex functions embedded in massive experimental or numerical data sets.
Often scientists (i.e. Gibbs, Maxwell, Einstein, Feynman) reported that visual thinking occurred before formalizing their ideas into words or symbolic script (equations). Their "productive thought" was first to imagine their functional relationship of physical properties as a "combinatory play" of images*. This psychical imaging thought process is an innate visual cognitive ability that we all experience, which is self contained, rarely discussed, and typically not shared with others**. Although this experience is subjective and not as accurate or reproducible as the final mathematical form, it is however a valuable creative component of how the human mind discovers and presents complex ideas.
These web pages attempt to elaborate on how the cognitive creation of psychical images by scientists can be facilitated using computer graphic technology that did not exist fifteen years ago. If such images can be created, this cognitive-psychical experience can be shared with others. Code fragments are provided with examples on how to create meaningful (reproducible) images. It is important that applied scientists and engineers learn how to create these computer images as part of their creative process. Enhanced understanding is realized when graphical methods are created together with the development of the mathematical methods within a scientific context (Gibbs 1873). Create the graphical method -- discover the science. Sometimes graphical and mathematical methods coincide and enhance our understanding of complex ideas that can not always be explained with either just images or math. Here an attempt is made to combine the use of graphical and mathematical methods by developing three general visual methods that work well with massive data sets encountered when working with supercomputer simulations and computer controlled laboratory equipment. Because of recent advances in laboratory and high performance computers, "we are data rich but information poor"***. It is the intent here to create an information rich experience by creating insightful graphical methods within a meaningful (reproducible) scientific and mathematical context.
The visual methods created here trade off the more accurate and traditional quantitative techniques for more qualitative but comparative visual techniques. Both techniques are important. These visual methods indeed parallel how we first imagine our functions that mimic our first impressions of possible functional relationships. This "combinatory play" of images does indeed seem to be the "essential feature in productive thought"*. Such an approach is shown here to work well when visualizing the three-dimensional gradient of a scalar function by cognitive visual data compression, extraction of simple algebraic expressions embedded in distributed numbers, and envisioning n-th order tensors and invariance of tensor equations that exist at points within the compressed data format. With regard to tensor equation invariance it was demonstrated that the quantitative mathematical invariance associated with tensor equations can be used to qualitatively envision the same invariance associated with physical laws. Consequently envisioning invariance enables scientists to see and understand the qualitative content of equations associated with physical laws, e.g. equilibrium in Cauchy's equation was envisioned and understood graphically as a glyph at a point or 3D gradient of glyphs surrounding that point.
Visual methods developed here required exploitation of new graphical features such as rotating voxel volume translucency and planes with color gradients moving through three dimensional space. These methods were developed in an attempt to create images and animations of "psychical entities" that represent complex physical property relationships in insightful new ways for development of new theories (no data), discovery of new knowledge embedded in massive data sets, and for presentation of these new theories or knowledge to be shared with other scientists. We are still data rich but perhaps a little less information poor.
** Here the word psychical is used in the cognitive sense (not the paranormal sense) that describes imaging as an individual self contained experience.
*** Professor Michael Vorster in a private conversation with the author Ron Kriz: "We are data rich but information poor". Professor Vorster is the Director of the Constuction Management Program in Civil Engineering at Virginia Tech.