One of the most computationally intensive parts of animation is rendering, which involves determining how the light leaving or reflecting off one point affects another, for all such pairs of points in a scene. DeRose stated that Pixar has 10,000 CPUs in its "Render Farm" to solve the integral equations connected to rendering. He concluded his talk by describing problems in animation which are still hard and which the motion picture industry would be happy for mathematicians to solve:
Over 1200 people, a record number, attended MathFest, the summer meeting of the Mathematical Association of America (MAA), in Providence, RI August 12-14. The Rhode Island Convention Center and the Westin Hotel were the locations for invited addresses, presentations, meetings, and exhibits.
Some of the MathFest events are summarized below.
Pebbling Results by Undergraduate Researchers, Aparna Higgins (University of Dayton), MAA Invited Address
Higgins' lively invited address was the first talk of the meeting. Pebbling is a fairly new area of research which began with a paper by Fan Chung in 1989. Pebbles are non-negative integer labels on the vertices of a graph. In a pebbling move, two pebbles are removed from a vertex, and one is placed at an adjacent vertex while the other is discarded. The pebbling number of a vertex is the least number of pebbles so that any graph with that many pebbles allows one pebble to reach that vertex through a finite sequence of pebbling moves. The pebbling number of a (connected) graph is the maximum of the pebbling numbers for its individual vertices. Higgins gave a good introduction to the topic, including showing extreme cases for graphs: The pebbling number for the complete graph of n vertices is n and the pebbling number for a path of length n is 2n-1. She talked about a conjecture by MAA President Ron Graham that says that the pebbling number of the Cartesian product of two graphs is bounded above by the products of each graphs' pebbling numbers, and gave examples of results in the field discovered by undergraduates.
Ramanujan Graphs, Peter Sarnak (Princeton University and the Courant Institute), The Hedrick Lecture Series
This three-part series was held each morning of the meeting. Sarnak first spoke about expander graphs which are highly-connected but very sparse. Most of the audience knew about sparse graphs but the concept of highly-connected was new and harder to define. Sarnak said that researchers first wondered if expander graphs existed, but their existence was eventually proved using the probabilistic method. Ramanujan graphs are expanders that are optimal in a certain sense. The Ramanujan graph to the left, which can be constructed by shrinking pentagons on a dodecahedron, has 80 vertices and is the second-largest planar Ramanujan graph. Sarnak pointed out applications of the graphs to communications networks, error-correcting codes, number theory, and computational group theory. (Interested readers can read about expander graphs in a short article from the August issue of the Notices, "WHAT IS ... an Expander?" by Peter Sarnak. The cover for that issue illustrates a calculation involving the Ramanujan graph pictured here.)
The Mystery of the Missing Tangents, Steven Sigur (The Paideia School [Atlanta]), MAA Invited Address
Sigur spoke about triangle geometry and opened his lecture with the quote: "Triangle geometry has more miracles per square meter than any other area of mathematics." One of the miracles Sigur showed, using Geometer's Sketchpad, was the behavior of a right triangle's incircles as a vertex was allowed to approach, and then pass through, the vertex at the right angle. Later in the lecture, he discussed Gergonne points and Nagel points, and showed how introducing "missing" tangents to a diagram with a triangle and circles, which may seem unusual, led to a familiar diagram associated with the Gergonne and Nagel points.
AMS Open House
On Thursday afternoon, the AMS hosted an open house for MathFest attendees. Participants got a chance to see AMS headquarters and enjoy a lunch featuring some favorite food and drink of Rhode Islanders, including clam cakes, clam chowder, saugies, iced coffee, and frozen lemonade. The afternoon provided a counterexample to the statement, "There's no such thing as a free lunch."
| How Computer Graphics is Changing Hollywood, Tony DeRose (Pixar Animation Studios), MAA Invited Address (Images in this section are courtesy of Tony DeRose, Pixar Animation Studios.) |
Pixar is responsible for some of the most popular animated films ever: A Bug's Life, Toy Story, Toy Story 2, Monsters, Inc. and Finding Nemo. DeRose, who is senior scientist in Pixar's Tools Group, gave an overview of how an animated film is made and provided specific examples from the previous films and from the Oscar-winning animated short film Geri's Game (DeRose was Minister of Geometry for Geri).
DeRose emphasized that animators must not interfere with the story: the most important part of a film. To prevent seams, which can occur where different features come together (for example, where the nose joins the face) and which could distract viewer attention, animators employ subdivisional surfaces. These surfaces approximate a three-dimensional object and are improved iteratively by subdividing the approximating surface at a given stage and averaging. Geometry is used in the surfaces, but so are linear algebra, to represent transformations at an iteration, and limits, which are applied to the sequence of iterations. DeRose noted a strong connection between subdivisional surfaces and wavelets.
- Fast lighting: currently, getting the lighting right in one frame can take two hours,
- Directable dynamics: directors would like to dictate how animated smoke or fire behaves, and
- Animated power tools: which would do many of the tasks that animators now do automatically.
MAA Prizes and Awards
MAA President Ron Graham (below, left) presented the following prizes and awards late Friday morning.
- Carl B. Allendoerfer Award to Charles I. Delman and Gregory Galperin for "A Tale of Three Circles," Mathematics Magazine, Feb. 2003
- Trevor Evans Awards to Douglas Dunham for "A Tale Both Shocking and Hyperbolic," Math Horizons, April 2003; and Hugh McCague for "A Mathematical Look at a Medieval Cathedral," Math Horizons, April 2003
- Lester R. Ford Awards to Noam Elkies for "On the Sums ∑∞k=-∞ (4k+1)-n," The American Mathematical Monthly, Aug.-Sept. 2003; Charles Livingston for "Enhanced Linking Numbers," The American Mathematical Monthly, May 2003; R. Michael Range for "Complex Analysis: A Brief Tour into Higher Dimensions," The American Mathematical Monthly, Feb. 2003; and Rüdiger Thiele (above, right) for "Hilbert's Twenty-Fourth Problem," The American Mathematical Monthly, Jan. 2003
- George Pólya Award to Greg N. Frederickson for "A New Wrinkle on an Old Folding Problem," College Mathematics Journal, Sept. 2003
- Chauvenet Prize to Edward B. Burger for "Diophantine Olympics and World Champions: Polynomials and Primes Down Under," American Mathematical Monthly, Nov. 2000
- Henry Alder Awards for Distinguished Teaching by a Beginning College or University Mathematics Faculty Member to Francis Edward Su of Harvey Mudd College and Zvezdelina Stankova of Mills College
Visit the MAA website for more information.
MathFest 2005 is in Albuquerque, New Mexico. Also available on this site are highlights from MathFest 2002.