Art
From JasonCantarellaWebpage
Violet and the Gordian Unknot
We are making a short animated film about knots and knot tightening, starring an young girl who accidentally ties a mysterious knot in her shoelace while trying to invent the ultimate shoelace knot and a mathematically inclined reptile who knows a bit about how knot theory might help her. Here are some test renderings showing versions of the characters.
You can [download a short clip from the movie]. We are working on this project with a talented group of students from the Animation program at UGA: graduate student Josh Marsh (lead animator), undergraduates Jamie Sichel and Heather Byrd (animators), and Alliene Bouchard (storyboard artist). The movie features the voice acting of Katie Robert (as Violet) and Scott Higgs (as Colin).
Atelier Blink Knot Chairs (and Cushions)
A friend of mine pointed me to these very cool chairs made from tight(ish) knotted cushions. The designer's webpage is [Atelier Blink]. They also seem to have some smaller standalone cushions.
2009 Georgia International Topology Conference Poster
My brother [Luke Cantarella] did the layout below for the 2009 GITC poster, using one of my tight knot images as a starting point. I think it's a very cool poster!
Number Butterflies Desktop Pattern
The Flocktree
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.
Subdividing a flock of birds
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.
Building a frame
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?
 A corner of a frame |
 Each side spirals |
 All spirals point outward |
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!
The Sculpture Installed
The gallery below shows some images of the installation at the Floor Group.
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Talk on Mathematics and Quilting
In August of 2007 I was lucky enough to give a talk on mathematics and quilting to the Cotton Patch quilters here in Athens. The talk was a lot of fun, and taught me quite a bit about quilts! You can read the slides at the link above.
Images for Cabinet Magazine
In 2006 I did a series of images of tight knots for the art magazine [Cabinet]. They appeared in issue 20 Ruins, accompanying an article by Kenneth Millett about knots. These are all rendered with the flaky but amazing Electric Image Animation System. I think they look pretty good:
 The 86 knot |
 The 818 knot |
 The 72 knot |
These also appear on the web page of [Excelscient], which is a recruiting firm for physicists and mathematicians in the United Kingdom.
2004 SEGC Conference Poster
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:
Mathematics and Visualization Summer Program 2004
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:
 An original image by Jack Kirby... |
 ...filtered by Dowling's algorithm. |
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. These images show the progression of the group's work through the summer.
The summer ended with a group art show at the main gallery in the Lamar Dodd School of Art at UGA.
From left to right, the students and faculty were Kit Hughes, Alessa Ellefson, Cody Van der Kaay, Catherine Covington, Julie Orlemanski, Rusty Wallace, Jason Cantarella, Seth Dowling, (lower) Ryan Thompson, and (top) Chad Mullikin.
Practical Introduction to Mathematical Visualization
In 2003, I visited Berlin, Germany for a mini-sabbatical. While I was there, I had the chance to give a lecture on visualization in mathematics to an audience of mathematics students. The slides for the talk, which I've given several times since, are posted on the web.
Knot Tightening Animations
I have installed various versions of a piece based on the knot tightening movies available elsewhere on this webpage; 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 below left is as good as any I've got.
We also made a short video called "Animating self-contact in tightening knots", for IEEE Vis 05, and a longer one for a conference in Tokyo later that year. You can see a screenshot above right. The movie is available [here] in mp4 form. If you are some sort of video fiend, you can download a very large (240mb) and very high-quality short version of the film [here].
Images of Borromean Rings and Tight Clasp
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.
 An exploded view shows the two tubes and the chamber between them. |
 In this view, the tubes are together, but transparent, showing the chamber. |
 This closeup shows only the empty chamber between the tubes. |
All of these pictures ended up in the paper Criticality for the Gehring Link Problem.
2003 SouthEast Geometry Conference Poster
In 2003, my wife Tammy made a conference poster for the SouthEast Geometry Conference. I had the idea of doing something inspired by this painting by Walter Obholzer which I saw at the art museum MassMOCA.
The knot on the poster is not a computer rendering, but an actual wire sculpture which Tammy made and photographed with a digital camera.
2001 Georgia Topology Conference Poster
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.
(This also appears on the web page of [Excelscient], which is a recruiting firm for physicists and mathematicians in the United Kingdom.) Luke cropped and edited the image to come up with the poster:
I think that I still have a few hundred printed copies of this around my office. If you want one, feel free to email me and I'll send you one.