When ambient light is reflected and reradiated in an image, finding the final distribution of light in the image requires the solution of a very large linear algebra problem. The total light R_i radiating from a face i is given by R_i = E_i + Σ F_ij R_j, where i is the light emitted by face i and the Fij are “view factors” describing the relative geometry of faces i and j. In the scene on the left, there are 64 polyhedra, each with 32 faces. The resulting system of 2048 linear equations in 2048 variables is solved in several tenths of a second using an iterative method implemented in Mathematica, but would take much longer with a standard solver. In a real application, like a scene from an animated movie, there would be a few million polygons in the scene and the resulting solution would require solving a system Ax = b where the matrix A was (say) 2,000,000 by 2,000,000. Even storing such a matrix would take on the order of 4 terabytes of memory! Luckily, such matrices are very sparse, so they are (barely) tractable with good computing hardware.
The Flocktree is a sculptural installation of a flock of pigeons cast in an expanding urethane foam supported and grouped by a collection of nesting aluminum frames. The interplay between the rigorous order imposed by the frames and the fluid organic shape of the flock raises questions about the way the viewer makes sense of collections of objects. The groupings of birds created by the frames form an octree structure.
The Flocktree was made possible by the financial support of the ICE program at UGA and installed in the courtyard at the Floor Group at 159 Oneta Street in Athens, GA. The Flocktree was created in collaboration with my brother Luke Cantarella a scenic designer and artist living in Brooklyn, NY.
The Flocktree structure divides a group of 17 bird sculptures into 6 groups by grouping them into front/back, left/right, and top/bottom groups, then enclosing each subgroup (such as the top/right/front) group in an aluminum frame. The entire flock is contained in a larger frame as shown below.
This process for dividing objects into groups is used in computational geometry and computer graphics to quickly process large collections of geometric objects.
Constructing the frames for the boxes poses an interesting challenge: how to build the best box possible from angle stock without mitering and welding it? Each of the corners are constructed in a triple-overlapping pattern shown in the gallery below. This means that around a face of each box, the pieces overlap in a spiral pattern, as shown in the middle frame. These spirals can be right-handed or left-handed. Can we pick a direction for the spiral on each frame so that they are all compatible?
As the pictures show, all of the spirals are compatible as long as each one points outward from the cube. In a frame built like this, all the corners are the same. In practice, this makes the cube design particularly strong and accurate, since each piece of angle stock is bent just a bit out of shape and the pieces all press together symmetrically at the corners.
This design is a simple example of a mathematical concept called orientability. A surface, like the cube is said to be orientable if on each face you can choose an outward direction so that all of these directions are compatible. Any surface like this can be built with the spiraling frames above. On the other hand, a nonorientable surface, like a mobius strip, couldn’t have been built this way!
In 2004, I created a new poster for the SouthEast Geometry Conference. I had just written some code to find shortest paths or geodesics on a polyhedral surface, and was anxious to try it out as an art project. So I started with a model of the Stanford Bunny and added a spray of geodesics from a point on the back of the bunny’s head. The resulting paths crisscross and make an interesting pattern on the surface. Sadly, I had to do the graphic design myself, so I don’t think it’s as good as a poster as it was as a rendering. You decide:
In the summer of 2004, I ran a summer program on Mathematics and Visualization which enrolled four art students and four science students for a series of joint projects in science and art. We started with the project of building our own image filters in MATLAB. Here’s an example of a filter created by Seth Dowling based on reshaping the pixel data in an image using a space-filling curve:
Most of the student projects focused on the idea of tensegrity, in which different elements in tension and compression are balanced against one another to create intricate and sturdy shapes. The summer ended with a group art show at the main gallery in the Lamar Dodd School of Art at UGA.
I have installed various versions of a piece based on the knot tightening movies available elsewhere on this webpage called Everything in its Right Place in the Contemporary Mathematical Photography and New Media Exhibition at the New Image Gallery of James Madison University, in the final show of the Math and Art summer program in 2004, and at the Joint Meetings of the AMS, MAA, and SIAM in 2005. I don’t have a good photograph of any of these installations, but the photo at left is ok. I’m not showing this piece anymore, but you can see the color version and the very high-resolution color version.
A main focus of my research work has been the study of the shapes of knotted tubes when they are pulled tight. With my coauthors Joseph Fu, Rob Kusner, John Sullivan, and Nancy Wrinkle, I found a tight configuration of a link known as the Borromean rings. These Electric Image pictures show the configuration:
These Borromean rings were built from a simpler configuration– a simple clasp formed by two ropes pulled across each other. That configuration turns out to have the surprising feature that there is a small space between the tubes. These pictures show the tubes and the space between them.
All of these pictures ended up in the paper Criticality for the Gehring Link Problem.
Every eight years, the University of Georgia math department holds a massive international conference on the subject of topology. In 2001, I got the chance to design the poster for the conference. I created the image below, and my brother Luke Cantarella did the layout. I originally had the idea of drawing a knot for the conference, but settled on the idea of drawing a knot inside out. The picture below is the space left when a knotted tube is removed from a 3-dimensional sphere. I created the surface as the tube around a knotted curve using my software tube, and then relaxed it using Brakke’s Evolver. The final render used the free raytracer POV-Ray.