Works by The illiMath CollectiveCubocta | Curved Minus 3 | Dirac Belt Trick | Etruscan Venus | Evert.Full | Eye.Full | illiSnailLorenz Strange Attractor | Minimax | Möbius | Rikitake | Quasi Crystals The illiMath Collective is a group of students, scholars, professors, artists, scientists, and others based at the University of Illinois at Urbana-Champaign and the University of Illinois at Chicago who have created mathematical artworks since the late 1980s. 
CuboctaThe illiMath Collective, 1992 immersive interactive environment A cube with corners cut off becomes a cuboctahedron. This polyhedral sphere turns inside-out in a motion called an eversion. Animation: Chris Hartman Code: François Apéry Mathematics: Bernard Morin Modeler: Richard Denner From "The Optiverse," Francis, Levy, Sullivan, 1998 top 
Curved Minus 3The illiMath Collective, 1992-1998 immsersive interactive environment Curved Minus 3 offers the chance to experience a world where the rules of geometry differ from the rules of Euclidean geometry which we experience everyday. This piece is an example of a hyperbolic space tiled by twelve-sided dodecahedra. Here the pentagonal surfaces of the dodecahedra have five 90° angles, an impossibility in the Euclidean world. Animation: Chris Hartman, Paul Chappell Geometry: Charles Gunn, Mark Phillips Syzygy: Ben Bernard, Matt Woodruff, Ben Schaeffer From "The Post-Euclidean Walkabout" SIGGRAPH 94 Download the Curved Minus 3 interactive computer application and Quicktime movie. top 
Dirac Belt TrickThe illiMath Collective, 1993 immersive interactive environment Dirac Belt Trick explores the rotation of objects connected by a belt or by strings to a fixed background. Such an object plus its connections to the background will be entangled by a 360° turn, yet returned fully to its original configuration after a 720° rotation. This combination of topology and geometry has applications to the physics of an electron. Animation: Chris Hartman Book: George Francis Mathematics: Lou Kauffman Production: Dan Sandin From "Air on the Dirac Strings," SIGGRAPH 93 Download the Dirac Belt Trick interactive computer application and Quicktime movie. top 
Etruscan VenusThe illiMath Collective, 1989 PHSCologram and immersive interactive application In the 1980's, when 3D computer-generated mathematical visualizations were created, they were generally created on graphical supercomputers manufactured by Silicon Graphics, computers often residing at National Science Foundation-funded supercomputer centers across the country. Two such locations shared a common, but geographically-disparate university, the University of Illinois at Urbana-Champaign and Chicago. Professor Donna Cox, George Francis and Ray Idaszak at UIUC and Ellen Sandor, Tom DeFanti and Dan Sandin at UIC collaborated on a PHSCologram titled "Etruscan Venus." Described by the artists as "a video portrait of a Romboy Homotopy, a four dimensional object," the viewer is able to see three of those four dimensions without the aid of stereo glasses due to the partial masking of the multiple images by a lenticular film in front of each image. An additional "Venus" is available in the CANVAS, where the viewer can maneuver the image through various 3D slices of the 4D object. Download the Etruscan Venus interactive computer application. top 
Evert.FullThe illiMath Collective, original 1998, reproduced 2006 UltraChrome K3 on Piezo Pro Matte Canvas media 59.5 x 39.25 in. The minimax sphere eversion from The Optiverse, a video by John M Sullivan, George Francis and Stuart Levy, with original music by Camille Goudeseune, produced at the Univeristy of Illinois (Mathematics Department and NCSA). top 
Eye.FullThe illiMath Collective, original 1998, reproduced 2006 UltraChrome K3 on Piezo Pro Matte Canvas media 59.5 x 39.25 in. An inside view of an everting sphere from The Optiverse, a video by John M Sullivan, George Francis and Stuart Levy, with original music by Camille Goudeseune, produced at the Univeristy of Illinois (Mathematics Department and NCSA). top 
illiSnailThe illiMath Collective, 1992-2008 immersive interactive environment All the forms in illiSnail are projections of ruled, minimal surfaces in spherically curved space, where the rules of geometry differ from those of our flat, Euclidean space. illiSnail morphs a Möbius band into such familiar shapes as Steiner crosscap, Roman surface, Clifford torus, Lawson bottle aka Brehm knotbox. Download the illiSnail interactive computer application and Quicktime movie. top 
Lorenz Strange AttractorThe illiMath Collective, 1998 immsersive interactive application The Lorenz dynamical system illustrates two features of chaos: the butterfly effect and a strange attractor. All particles start from approximately the same initial position and, although governed by identical laws of motion, the particles soon disperse wildly. Eventually their paths converge and they are destined to wander back and forth on the attractor in an unpredictable way. CANVAS Implementors: Chris Rainey, REU Summer 2006 Kyle Wilkinson, Math 198, Spring 2006 Stan Blank's high school class, 2005 top 
MinimaxThe illiMath Collective, 1998 immersive interactive application Bernard Morin's central model is a strongly contorted sphere that penetrates itself with a bending energy of four spheres. The Minimax eversion turns a single sphere inside out by morphing it through the central model. To do this, a blue-green inside/red-orange outside sphere must contort into the central model by raising its energy before sliding down the other side of the energy saddle to change colors. Geometry: Rob Kusner Surface Evolver: Ken Brakke Animation: Alex Bourd, Chris Hartman Sound: Camille Goudeseune Post-Production: Jeff Carpenter, Dana Plepys From "The Optiverse," Francis, Levy, Sullivan, 1998 top 
MöbiusThe illiMath Collective, 1998 immersive interactive environment Bernard Morin's central model pirouettes in four space to show her best face for projection into three space where we can admire her shape. Despite appearances, the bending energy of the surface does not change during this inversion. From "The Optiverse," Francis, Levy, Sullivan, 1998 top 
RikitakeThe illiMath Collective, 2008 immersive interactive environment Rikitake, like Lorenz Strange Attractor, illustrates a dynamical system where particles starting from almost the same point in space eventually follow their own separate, but never too separate, paths. CANVAS Implementors: Nicholas Duchnowski & Chris Rainey Kyle Wilkinson, Math 198, Spring 2006 Stan Blank's high school class, 2005 top 
Quasi CrystalsThe illiMath Collective, 2004-2007 immersive interactive environment Quasi Crystals shows a small part of a space packing by non-rectangular bricks which lacks all symmetry. It is computed as a 3-dimensional shadow of a 6-dimensional cubical lattice. It is necessary to show the shadow rather than the 6D object because it is impossible to say what the hypercube looks like; as it would be impossible to describe a 3D cube to a being living in only two dimensions, it is impossible for humans (who perceive three dimensions) to visualize a 6D object. Mathematics: DeBruijn, Gregory, Robbin. More information on Quasi Crystals. top 
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| The Traveling CANVAS is developed by the ISL at the Beckman Institute at the University of Illinois at Urbana Champaign. | |