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illiMath2006: an REU of
Geometrical Visualization in Virtual Environments

11 June - 4 August
Program Director: George Francis

Visualization of mathematical phenomena and their transmutations is a maturing field of geometical computer graphics. Though opportunities to learn and practice this discipline are now plentiful, few universities have as long a tradition and as broad a base of application as Illinois.

Students in this program work with current researchers in diverse areas of mathematics, science and technology, who use a common programming platform, Syzygy. This distributed graphics system supports cluster based virtual environments such as the CAVE, CUBE and CANVAS at UIUC, the DiVE at Duke, and the PORTAL at the Technical University of Berlin.

In this program students with strong interest and some experience in mathematical graphics participate in research programs of their associated mentor (AM). The program provides the facilities (and any necessary training in their use) for the student to assist their AM in creating visual materials illustrating research in a variety of media, including the production of custom software, webpages, animations, print quality images, etc.

Common to all projects is Syzygy, Python, and OpenGL. Syzygy is the distributed graphics system, developed by the Integrated Systems Lab (ISL), for cluster based virtual environments such as the CAVE, CUBE and CANVAS at UIUC, the DiVE at Duke, and the PORTAL at the Technical University of Berlin. The program also provides the student with tutorials in the mathematical background and lectures by specialists in the subjects of their projects.

Similar REU programs, Audible Sketchpad for the CAVE, illiMath2001, illiMath2002, illiMath2004 and were conducted in recent years. Here is a report on a typical summer.

Projects:

Projects of illiMath2006 Research Experience for Undegraduates, Mathematics Department, University of Illinois at Urbana (UIUC), directed by Prof. George Francis, are the work of the student principal investigator (SPI) and their assiated mentor (AM) and/or corresponding mentor (CM). Other student investigators (SI) assist.

"Syzygy/Povray"; SPI: Paul Prue, UC Davis; AMs: Jim Crowell and Camille Goudeseune, ISL, Beckman, UIUC; CM: Dave Zielinski, DiVE, Duke University.

In this project real-time ray traced scenes are displayed in Syzygy based virtual environments (VE) such as the DiVE at Duke, and the CUBE, CAVE and CANVAS at Illinois. A number of raytracing techniques are tried out for possible animation speeds. These experiements illustrate concepts in geometrical computer graphics.

"Sakubo"; SPI: Mimi Tsuruga, Hunter College, NY; AMs: Ulises Cervantes-Pimentel, Wolfram Research, Champaign, IL, and Stuart Levy, NCSA, UIUC.

The next release of Mathematica has radically improved 3D graphics in the kernel, including adaptive mesh subdivision and animation tools. In this project Mathematica calculates deformations of surfaces, such as the Romboy Homotopy of the Etruscan Venus, F. Apery's Gastrulate Sphere Eversions, and J. Dalbec's contraction of Zeeman's Duncehat. Mathematica computes the 3D stages of the homotopy (topes) and exports them as either OOGL or VRML files. OpenGL applications Geomview and AVN, and Syzygy application ZAVN or VRMLView animate these topes on a console or in virtual environments.
Sakubo will be presented as a PME talk at Mathfest, Knoxville, TN, August, 2006.

"Quasicrystals"; SPI: Matt Gregory, UIUC; AM: George Francis; CM: Tony Robbin, New York.

DeBruijn proposed two methods for realizing quasicrystals as projections of sub-lattices of the 6D unit lattice into 3-space. In this collaboration with mathematical artists, Tony Robbin, deBruijn dual method is used to create real-time interactive CUBE animations (RTICA). As a distributed scenegraph (DSG) and Python application of Syzygy, the program can be modified at run-time in collaborative sessions among remote peers (Myriad).
A poster of Quasicrystals will be presented at Young Mathematicians Conference at Ohio State University. Columbus, OH.

"Parallel Worlds"; SPI: Emily Echevarria, UIUC; AM: George Francis; CM: Ben Schaeffer, New York.

Ben Schaeffer's Syzygy application Myriad is a peer-to-peer sharing of scenegraphs and their subgraphs for real-time interactive visualizations of virtual worlds. With avatar.py many users each acquire, modify and navigate their avatars in a common world. With tieSpline.py users may simultaneously create a spline in the same 3-space, interlacing the splines and sharing a ride on each spline as desired.
A poster of Parallel Worlds will be presented at Young Mathematicians Conference at Ohio State University. Columbus, OH.

"ATP Synthase" SPI: Dave Bergman ; SI: Chris Rainey; AM: Prof. Paul McCreary, Evergreen College, WA ; CM: Jim Crowell, ISL.

This is a Syzygy port of McCreary's RTICA with which the user can navigate elaborate molecular models and a series of animated examples of Kleinian groups. The several formats of standard molecular data can be imported by the user. The navigation paths are Bishop framed and many features of the molecule are highlighted on demand.

"illiDyne" SPI: Chris Rainey; SI: Emily Echevarria; AM: George Francis;

This project greatly expands upon several previous Syzygy RTICA where users view and interact with a series of chaotic and non-chaotic 3D dynamical systems in the CUBE. These include the Lorenz, Roessler, Rainey, Rabinovich-Frabrikant, and 2D non-linear systems such as the Lienard-VanderPol. As a DSG-Python application, the user can modify the code by inserting new dynamical systems, and investigate these collaboratively with Myriad. This project builds upon previous applications written by Stan Blank, Ben Schaeffer, and Kyle Wilkinson.
ATPsynthase and illiDyne will be presented as an MAA talk at Session 5, Room 200A, 3:40pm Friday 11aug06, at Mathfest, Knoxville, TN, August, 2006.

"The Kiosk of Oz"; SPI: Abby Watt; AMs: Rose Marshack, CANVAS, Krannert Art Museum, and Jim Crowell, ISL, UIUC.

The CANVAS, CAVE and CUBE are fully immersive virtual environments that currently require users to navigate to different applications with a non-immersive computer kiosk. The purpose of the Kiosk of Oz is to allow the user to navigate to different applications using an immersive, museum-like environment.

"illiNav: Gesture Mapped Navigator for Virtual Environments"; SPI: David Bergman; AM: George Francis.

A Syzygy RTICA is navigated in the VE using a head tracker and a wand tracker. The wand has several buttons. In some setups, the console keyboard can be used for additional interactive input. In addition, all Syzgygy applications run on a console in standalone mode using a simulator of the VE control inputs. The design of gesture-assisted VE navigation and its mapping to the mousepad and keyboard in standalone mode is a difficult problem with many trial solutions, none of them satisfactory. This project makes some progress on improving the standalone input simulator.