There are many anecdotal reports on the importance of visual thinking by scientists. Perhaps the best known visual discoveries are the benzene ring and the helical structure of DNA. Friedrich Kekule envisioned the benzene ring as a snake biting its own tail. Computing can facilitate this experience.
Max Planck reinforces the idea how human reasoning coincides with, but exists independent of, the physical world we observe.
Even more vivid was Albert Einstein's explanation how human reasoning includes visual thinking.
A more contemporary example of visual thinking is given by James Gleick from "The Life and Science of Richard Feynman", Vintage Books, New York, 1992.
Feynman continues: "What I am really trying to do is bring birth to clarity, which is really a half-assedly thought-out-pictorial semi-vision thing. I would see the jiggle-jiggle-jiggle or the wiggle of the path. Even now when I talk about the influence functional, I see the coupling and I take this turn - like as if there was a big bag of stuff - and try to collect it in away and to push it. It's all visual. It's hard to explain."
Schweber: "In some ways you see the answer - ?"
Feynman: "The character of the answer, absolutely. An inspired method of picturing, I guess. Ordinarily I try to get the pictures clearer, but in the end the mathematics can take over and be more efficient in communicating the idea of the picture. In certain particular problems, that I have done, it was necessary to continue the development of the picture as the method before the mathematics could be really done."
The reader can witness how Feynman develops the theory of quantum electro-dynamics (Q.E.D.) using visual thinking, referred to above as "a half-assedly thought-out-pictorial semi-vision thing", in his Douglas Robb Memorial Lectures at the University of Auckland, New Zealand, 1979.
Feynman would often refer to how he used a graphical approach in his thinking but was at a loss to explain, when studying turbulent flow for example, how a variety of behaviors (three-dimensional structures) were observed to be "hidden" within but yet predicted by the same equations. Consequently it was difficult to explain how others could reproduce his visual thinking. This conflict was demonstrated in a summary on modeling viscous fluid flow (Chpt 41, The Flow of Wet Water, The Feynman Lectures on Physics, 1964), Feynman provides an example of Couette flow (41-6) that was used to explain the formation of different wavey "barber pole" formations of vortices that start out as bands of vortices caused by a differential rotation between two concentric cylinders. Feynman was fascinated how different three-dimensional vorticy structures result from different rates of rotation, e.g. Reynolds numbers, while the equations and boundary conditions are unchanged: "And no one knows why!". Feynman concludes,
The difficulty that we do not have the complete mathematical power to analyze and envision these different behaviors (three-dimensional structures) led Feynman to speculate on how the future may produce a method to envision these structures within the context of "understanding the qualitative content of the equations". To understand the "qualitative content of equations" it is necessary to at least read and study the ideas used in the creation of the equations developed in chapters 40 and 41. You don't need a Ph.D. in Physics to appreciate the anecdotal evidence of Feynman's visual thinking. However Feynman's lectures in chapters 40 and 41 did target undergraduate physics students who had already taken their freshman and sophomore physics and calculus courses. So at least an undergraduate level of education, perhaps more, is needed as an analytic foundation to develop such methods of understanding. When this analytic foundation is combined with creating new methods of understanding three-dimensional structures, Feynman hopes to envision his science. In short something's missing.
Until such methods of understanding are developed, the emphasis has been to, at best, create graphical models of these various structures by experimenting with graphical tools. Because these graphical tools are widely available and easy to use, these tools are typically used to graphically model structures predicted by complex equations without requiring an understanding of ideas used in the development of these equations. In this case results lack reproducibility in the scientific sense, and so we are all entitled to our own opinions when we analyze and interpret information embedded in equations or massive data sets by using graphical tools serendipitously without a method. If however graphical methods, not tool based models, are developed by scientists within the context of their scientific knowledge in new ways, understanding is possible so that they may "see" the content of their equations and results will be reproducible in a scientific sense. And so we seek reproducible graphical methods, not tools, that allow scientists to "see" and understand the content hidden within their equations or massive simulation/experimental data sets. It is important that scientists create these insightful graphical methods within the context as they understand their science. Creating insightful graphical methods is often dismissed by some as imaginative, subjective, and not reproducible. Feynman discusses this issue in, The Feynam Lectures on Physics, Vol. 2, 1977, Addison-Wesley, Chapter 20, pg.10.
Scientific reproducible results
Although informative and perhaps inspirational, anecdotal reports on visual thinking are not instructive in the sense that they do not allow the reader to reproduce these insightful experiences. There are however some examples where visual thinking was developed by scientists as a graphical method and published which enabled other scientists to reproduce geometric physical property relationships within the scientific context that they were created. In these cases creation of graphical property relationships transcends the use of graphics for presentation. Creation of these graphical methods becomes a cognitive experience that was used to understand (discover) and then develop the underlying theory. This idea was demonstrated by J. Willard Gibbs in his first two historic publications, "Graphical Methods in the Thermodynamics of Fluids" and "A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces", Transactions of the Connecticut Academy, Vol. II, Part 1, pp. 309-342 and Part 2, pp. 382-404, 1873, respectively. In the first publication, Part 1, Gibbs' objective, as the title implies, develops graphical methods that provide insight into the thermodynamic properties governed by the first and second laws of thermodynamics:
After carefully developing various reversible thermodynamic "graphical methods", pp. 310-341, Gibbs concludes:
Evidently, according to Gibbs, the equation of state derived at the beginning of his first paper was not as insightful as the graphical method. Interesting. To understand how the thermodynamic relationship of properties can be better understood graphically without the analytic expressions it is necessary to actually read and study Gibbs 1873 publications. After reading these two publications James Clerk Maxwell created a "sculptured" surface in 1874 showing the thermodynamic graphical relationship of energy, entropy, and volume, described in detail by Gibbs, but never drawn. Maxwell also graphically reproduced and extended Gibbs' original graphical method by constructing lines of temperature and pressure mapped onto his sculptured surface in figure 26d, pg 207, Theory of Heat, 1904, without using any mathematical relations as recommended by Gibbs. This is quite an endorsement, coming from Maxwell the mathematician. This also demonstrates scientific reproducibility. Maxwell used clay and plaster to make a "sculpture", a graphical model of Gibbs' graphical method. It was 1874, there were no graphical tools. Maxwell sent one of three sculptures to Gibbs at Yale in 1874, which is enclosed in a dusty display case next to an oil drum. Another of Maxwell's sculptures can be viewed in a display case at Cavendish Laboratory at Cambridge.
It is interesting to note that the graphical method, originally developed by Gibbs, was done so independent of any graphical tools or models built with these tools. With today's computer technology we have focused on using graphical tools, not developing graphical methods within the scientific/mathematical context understood by the scientist. A brief web summary describes how Gibbs' graphical method is related to a generalized graphical method used to envision total derivatives without reference to graphical tools. Another web site, created by Professor Kenneth Jolls and Dr. Daniel Coy, summarizes Gibbs' graphical method and highlights Dr Coy's Ph.D. dissertation, "Visualizing thermodynamic stability and phase-equilibrium through computer graphics", Iowa State University, 1993. These web sites and Dr. Coy's dissertation exemplifies how others can learn about Gibbs' graphical method and create energy-entropy-volume surfaces as originally described by Gibbs and graphically reproduced by Maxwell.
Although it was Gibbs intention to develop "graphical methods coextensive in their applications", selecting a specific example such as the thermodynamic theory of state should encourage us to seek other examples and perhaps other general graphical methods coextensive in their applications. Too often scientists stop once they find their specific example. Perhaps there are other interesting graphical methods that can be used to envision scientific information. This idea is presented as the theme in the class notes for ESM4714, "Scientific Visual Data Analysis and Multimedia: Create the graphical method -- discover the science".