Surfaces

Biography

Richard Palais's is a Professor Emeritus at Brandeis now.

After 37 years in the in the Brandeis Department of Mathematics, in 1997 I retired to have more time to work in the area of Mathematical Visualization, and more specifically to develop my Macintosh program 3D-Filmstrip (now called 3D-XplorMath). In the Fall of 2004, my wife, Chuu-lian Terng, resigned from Northeastern Univ. to accept a position in the mathematics department at the University of California at Irvine (where she holds the Advance Chair) and we have now moved permanently to Irvine. I am continuing to work on mathematical visualization and in particular I am cooperating with David Eck of Hobart and William Smith College, helping with the design of his Java port of 3D-XplorMath, which will be called VMM---for The Virtual (or Visual) Mathematical Museum. However I have also partially "unretired" and accepted a position as Adjunct Professor of Mathematics at UCI, which means I will be teaching one or two courses per year.

Artist Statement

In recent years I have become interested in mathematical visualization, and one of my major ongoing projects is the development and continued improvement of a program called 3D-XplorMath for MacOS X. This is a tool for aiding in the visualization of a wide variety of mathematical objects and processes. Based on what I have learned from my experience in writing this program, I wrote an essay called "The Visualization of Mathematics: Towards a Mathematical Exploratorium" that appeared in the June/July 1999 issue of the Notices of the American Mathematical Society. With the help of Xah Lee, I have created a Gallery of visualizations produced using 3D-XplorMath.

Artwork Highlight:

LiveGraphics3D, JavaView

These families are parametrized by 4-punctured rectangular tori; they and their conjugates are embedded. We therefore suggest the associate family morphing, and also morphing of the modulus (aa) of the rectangular Torus, which changes the size

of the visible holes.

Computing Process:

For the visual appearance of these surfaces it is particularly important that the punctures are centers of polar coordinate lines. Formulas are taken from [K1] or [K2]. The Gauss maps for these surfaces are degree 2 elliptic functions. The cases shown are particularly symmetric, the zeros and poles of the Gauss map are half-period points and the punctures are there. In the Jd-case the diagonal of the rectangular fundamental domain joins the two zeros, and in the Je-case it joins a zero and a pole of the Gauss map.

Under suitable choices of the modulus of the Torus these surfaces look like a fence of Scherk SaddleTowers – with a vertical straight line (Je), respectively a planar symmetry line (Jd), separating these towers. The conjugate surfaces

look qualitatively the same in the Jd-cases and like a checkerboard array of horizontal handles between vertical planes in the Je-cases.

Above: “Borg cube”, a plot of the equation: Sin[x*y]+Sin[y*z]+Sin[z*x]==0.

Analysis:

Mathematical Surface gives people feeling more concrete. The grids help the object represent it as a mathematical object. For example, the last picture is also a mathematical surface however it is without any grids so it has not give people feel this is a mathematical surface, it just only a computer created object only. So the grids are very important. And the colors of the Richard Palais’s arts are also important. He uses the pure hue and light saturation, such as light blue, light brown. It gives people feeling more prototype and abstraction.