Visualization with Mathematica^®

"Who could dispense with the figure of the triangle, the circle with its center, or with the cross of three perpendicular axes? Or who would give up the representation of the vector field, or the picture of a family of curves or surfaces with its envelope which plays so important a part in differential geometry, in the theory of differential equations, in the foundation of the calculus of variations and in other purely mathematical sciences? The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas; and no mathematician could spare these graphic formulas, any more than in calculation the insertion and removal of parentheses or the use of other analytical signs."
          - David Hilbert
            Address to the International Congress of Mathematicians, 1900

"The reason for me to rush to graphics was because I am among those who reason best on what can actually be seen. . . Our graphics did more than inform. They made people dream. Colleagues flocked to tell us that we had made them see their own work in a different light, and had helped them by unveiling previously unnoticed analogies. For the first time, they felt that what they saw directly affected what they did next. . . A resort to illustration was viewed as strictly prohibited only a few years ago, but now it has become routine . . . it has proven a rich source of new mathematical facts, and therefore of new conjectures."
          - Benoit Mandelbrot
            Forward to Michael McGuire's An Eye for Fractals

As Hilbert pointed out, and as Jacques Hadamard argued at length in his famous book, The Psychology of Invention in the Mathematical Field, visualization is an essential component in the mathematician's work. The extent to which a careful, detailed rendering of images in an external medium facilitates mathematical discovery is open to debate; but certainly, such renderings are useful to the student, and aesthetically pleasing. (Geometry and the Imagination, by Hilbert and Cohn-Vossen, is a beautiful example. Mandelbrot's Fractals: Form, Chance and Dimension is another. Other examples abound, from fields such as topology, differential geometry, dynamical systems, and chaos theory.)

It is possible that the availability of tools like Mathematica may facilitate an "experimental mathematics" of the sort that Mandelbrot alludes to, in which computer graphics is used in the discovery process as it is in the empirical sciences. In any case, we hope the images below are a convincing demonstration of Mathematica's capabilities.

All these images, except the ones labeled "VisualDSolve", were produced using standard facilities of Mathematica. VisualDSolve, a book and Mathematica software by Dan Schwalbe and Stan Wagon, is a landmark in mathematical visualization. Once you've used it to visualize differential equations, it's hard to imagine how you lived without it. (Available as an e-book and software from Wolfram Research. The original hardcopy book is out of print, but used copies may be available through Amazon.com.)

Click on any of the small images below (or on the text links in the description column) to view the full-sized image. Click on the text links in the description column to view the Mathematica code used to generate the images. These were generated using Mathematica 4.1, saved as bitmaps or encapsulated PostScript, and converted to JPEGs in Adobe PhotoShop.

Polyhedra: Union of tetrahedron and stellated dodecahedron. Rendered with ambient light and three light sources. Full-size image (31 Kb); Mathematica code.

Polyhedra: Union of dodecahedron and stellated icosahedron. Rendered with ambient light and three light sources. Full-size image (38 Kb); Mathematica code.

Shadow plots: Plot of cos x cos y, shadowed above parallel to the x-y plane. Coloring codes the height of the graph. Full-size image (47 Kb); Mathematica code.

Shadow plots: Logarithmic spiral evolving in time, (.001e^{1.4 t}cos t, .001e^{1.4 t}sin t, .15t), and projections on the x-y, x-z, and y-z planes. Full-size image (30 Kb); Mathematica code.

Contour plots: The hyperbolic paraboloid x² - y² modulated by cos(x⁴ y⁴); contours are color coded by hue:



Full-size image (142 Kb); Mathematica code.

Contour plots: plot of
cos(x³ y³) e^{-Abs(Tan(x y))} in the rectangle [2, 5], [-5, -2]; contours are coded by brightness (lighter is higher). Full-size image (97 Kb); Mathematica code.

VisualDSolve: Solution curves of x' (t) = x(t)² + t² - 1 for given initial values. The background is shaded to show the slope of the curves; the black circle corresponds to zero slope (nullcline). Full-size image (60 Kb); Mathematica code.

VisualDSolve: Solution curves of x' (t) = sin(t x) in color. The background is shaded to show the slope of the curves; the black lines correspond to zero slope (nullclines). Full-size image (80 Kb); Mathematica code.

VisualDSolve: Orbits of x' (t) = y(t), y' (t) = - x(t) - y(t)/10 for a range of initial values. The orbits are color-coded from blue to red as t increases. Full-size image (83 Kb); Mathematica code.

VisualDSolve: Phase plot of x' (t) = x (t) - y (t)²cos(y (t)), y' (t) = - y (t) + x (t)sin(x (t)). Shaded regions show where the vector field has a common direction; "Fish" shapes code the direction and magnitude of the vector field. See also comments in the Mathematica code. Full-size image (106 Kb).