Visualizing Mathematics

Post by Alan Singer

Implicit Surface

generated by

3D Xplormath Program

The book written by Sylvia Nasar and published in 1998 called: "A Beautiful Mind " is a biography of Nobel Prize winner John Nash, who was a mathematician of great distinction.   John Nash also had difficulty in life, and during the course of this book you can read about the ups and downs of his career, and through this narrative we meet fellow teachers like Richard Palais who helped Nash negotiate the academic world.  At that time, Dick Palais was a professor of mathematics at Brandeis University who later moved to the University of California at Irvine.  Palais came to my attention when he appeared at a presentation given at Rochester Institute of Technology ( where I teach ) so I went over to meet him.

I was interested in speaking with him because for my artwork I was using his program called 3D -Xplormath that I found on the web.  I had come across his site while exploring a different collective site called "Geometry Junkyard" which was filled with interesting paths to take.  The site that Dick Palais developed with his team was something I could introduce later to my art students.  The interface is so easy to use - you can sit down and get to work and produce something right away, and then you are hooked!

From my perspective the reason I got involved with his program is that it allowed me to not only learn more about 3D geometry, but it also gave me the tools to describe the forms that I wanted to see.  I took to working with user defined dimensional forms called implicit surfaces, a kind of three dimensional algebra.  The program that was posted on the web has been updated to include a geometric gallery of forms that have great visual impact - not seen in visual art before.  It didn't hurt that anyone could download his program for free, and I spent many happy hours at work, finding ways to create forms and light them the way I wanted.

Formula:  4*x^2+4*y^2+4*z^2+16*x*y*z-1

I include some samples of my work here, and I can recommend that you look at this program before getting involved in anything more costly that may have a steeper learning curve.  Here is a disclaimer:  I am an artist - my math skills are basic, so the way I write out my formula for these shapes may not be elegant, but they work for me.  Also I can state that the conditions for these images go beyond algebra that I can understand, into the realm of an interaction with form and light sources - so my images become a way to visualize a physical presence.

pi^.5*min(cos(x)+cos(.2*y)cos(z)^2,+x*cos(.3)y-sin(.3)-x*cos(.3)+sin(.3)-z)*pi^3* pi^.5*min(cos(x)+cos(.2*y)cos(z)^2,+x*cos(.3)y-sin(.3)-x*cos(.3)+sin(.3)-z)*pi^3*min(cos(x)+cos(2*y)cos(z)^2,+x*cos(2)y-sin(2)-x*cos(2)+sin(2)-z)-1.5

Century Man

The program that helped me develop this image

is called 3D-Xplormath

image by

Alan Singer

So, this program will let  the user define the surface here as long as you work within the parameters set for building an image.  This implicit surface can be rendered within a given sphere, which you can also determine as you work.  Then there are the many factors including lighting, which I can explore in another post.