Davide Cervone, Department of Mathematics, Union College
"Decomposing the Four-Dimensional Hypercube and Hypersphere"
The familiar breakdown of a sphere in 3-space into two hemispheres can also be applied to the hypersphere in 4-space: while in 3-space we use two disks joined along their boundaries, the analog in 4-space is to join two solid spheres by gluing them along their bounding spheres. There is, however, another interesting decomposition of the hypersphere into two congruent parts. It turns out that the hypersphere can be broken down into two linked solid tori glued together along their surfaces. In this talk, we analyze this decomposition by first looking at the hypercube as a polyhedral model of the hypersphere, and develop the corresponding breakdown within it. To return to the hypersphere, we "smooth out" the polyhedral version to obtain the desired decomposition of the 3-sphere in 4-space. This leads to both beautiful mathematics and beautiful images, with connections to a variety of mathematical fields, from dynamical systems to complex analysis.
Ron Stern, Department of Mathematics, UC Irvine
"How mathematicians view dimension four"
For over 20 years, the study of 4-dimensional smooth manifolds has been a hot area that has drawn together most areas of mathematics and theoretical physics. Despite spectacular advances in defining invariants for 4-dimensional manifolds and the discovery of important qualitative features about these manifolds, we seem to be retreating from any hope to classify smooth 4-dimensional manifolds. The subject is rich in examples that demonstrate a wide variety of disparate phenomena. Yet it is precisely this richness which gives us little hope to even conjecture a classification scheme. In lecture I will indicate how mathematicians work and play in dimension 4, indicate the recent major advances, and plot future directions.
Richard Palais, Department of Mathematics, UC IRvine
"Time as an Extra Dimension in Mathematical Visualization"
Presentations at this conference for the most part focus on representations of 4D objects via static projections into two or three dimensions. I will review the more prosaic idea of using time as a tool for showing an extra degree of freedom in data. One major advantage of this temporal approach is that the interpretation is a lot more natural and immediate. As I will illustrate, the related concept of morphing allows one one to wander in time through moduli spaces of very high dimension. Moreover, the remarkable improvements in graphics cards and the enormous gains in CPU speeds over the past few years have made it feasible to do real-time experimental research with this technique, using average desktop or even laptop machines. The mathematical visualization tool, 3D-XplorMath, that I have been working on for many years uses this approach heavily and I will use it to illustrate my remarks.
Tom Banchoff, Department of Mathematics, Brown University
"Four-Dimensional Worlds: From 'Flatland' to Interactive Hypergraphics
New developments in computer graphics on the internet open up powerful ways to visualize, interpret, and manipulate phenomena not just in two- and three-dimensional space. How will these insights change the ways we view geometric structures, multi-dimensional data sets, imaginative literature, and surrealist art?
Linda Henderson, Department of Art History, University of Texas at Austin
The Spatial Fourth Dimension Comes Back in Art and Culture, 1950s-2000
�There is hope for humanity as long as great minds ponder the fourth dimension,� declared the NewYork Times in April 2003. Readers in the early 20th century would have understood immediately this reference to the highly popular spatial �Fourth Dimension.� Yet by 1920 Relativity Theory's temporal fourth dimension had begun to displace the purely spatial fourth dimension, and during the 1930s through the 1950s it was almost completely eclipsed by the popular understanding of the fourth dimension as time in the �space-time� world of Einstein. As a result, my book The Fourth Dimension and Non-Euclidean Geometry in Modern Art (Princeton University Press, 1983; new ed., MIT Press, 2006) focused on art of the period 1900-1940. I concluded, however, by noting the pioneering contemporary work in visualizing four-dimensional space of mathematician Thomas Banchoff and painter Tony Robbin and suggested that perhaps the fourth dimension was �on the verge of a new phase of influence.� Little could any of us have imagined the resurgence of interest in higher dimensional space that was about to occur with the emergence of string theory as well as cyberspace and accessible computer graphics programs during the course of the 1980s.
In tracking the fate of the spatial fourth dimension from the 1950s to the present for the new introductory essay for MIT's reprint of my 1983 book, I have recovered some of the lost history of the spatial fourth dimension during its mid-century eclipse. This talk recounts aspects of that history, including the active interest in the fourth dimension and complex space in the 1960s by Robert Smithson and the artists of the Park Place Gallery, an episode completely excluded from the standard narratives of avant-garde American art in this period
Tony Robbin, Artist and Mathematician
"Artistic Strategies for Depicting Four-Dimensional Worlds"
After discussing how Picasso used the technical drawing of hypercubes available in his time to invent Cubism, Robbin discusses how he uses the technical drawing of our time, the computer modeling of four-dimensional figures, to create his art. Over the last 30 year, he has developed three distinct formalisms that allow four dimensions to be experienced in less.
William Lindgren, Department of Mathematics, Slippery Rock University
"Some newly discovered dimensions of Flatland"
We discuss recent discoveries made in our study of the life andwork of Edwin Abbott Abbott. In particular, we consider several primary documents, which suggest new possibilities for the origin and meaning of Abbott's classic Flatland. The most notable document is a satirical essay, �A new philosophy,� which appeared without attribution in the City of London School Magazine (December 1877). In this remarkable essay the author describes a model of religion couched in the language of dimensional geometry. We also discuss the significant changes and additions to the 1st edition of Flatland that were occasioned by a review (now known to have been written by Arthur John Butler) in The Athenaeum (15 Nov 1884) and the ensuing correspondence in the Athenaeum between A Square and the reviewer.
George Francis, Department of Mathematics, Univ. of Illinois, Urbana-Champaign
"Why some things remain difficult to do in 4D"
Since 1992 I have been experimenting in the CAVE virtual environment tomake navigation in 4D both intuitive and learnable. The reasoning for doing this follows the customary dimensional dialectic: Just as we can experience persuasive 3D scenes in accomplished 2D pictures (from Greek vase drawings to the movies), so a persuasive real-time interactive 3D environment should make the fourth dimension intuitively accessible to us.
My presentation, using real-time interactive but 2D animation (of necessity) will analyze experiments created by my students over the past decade, including Mike Pelsmajer's 4D maze, a 4D Tetris by Greg Kaiser and Ben Bernard, and Mike Daniels and Pelsmajer's realization of Scott Kim's 4D Penrose illusion.
Mike D'Zmura, Department of Cognitive Sciences, UC Irvine
"Navigation in 4D Virtual Environments "
Cognitive scientists have emphasized, in recent years, the importance of our embodiment in the world for skill acquisition and cognitive processing. In mathematical education, this emphasis has led to exploration of the way people use their bodies to learn and to exploit mathematical concepts. Following this line of thinking, we reasoned that immersion in an interactive 4D virtual environment (VE) might be a far better way for most people to learn about 4D space and objects than gazing at diagrams, pictures or rotating hypercubes. We are happy to report positive results with such methods. Greg Seyranian, Philippe Colantoni, Barb Krug and I developed software for presenting VEs with four spatial dimensions in a manner similar to that in which 3D VEs are presented in action games. The software renders a 3D cross-section of a 4D VE and, in real-time, changes the position and orientation of the 3D cross-section in response to user input. In a first experiment, we tested whether people can learn to move efficiently from one location to some remote location in a rich, immersive 4D VE. All participants in this experiment improved their search and navigation skills dramatically. Yet it was clear that they used landmarks to select efficient routes, a navigation technique common in the real world (�turn left at the Shell station�), rather than some high-dimensional map of the environment. In a second experiment, we used non-immersive, maze-like VEs to test whether people can learn to point towards an unseen, remote location in 4D space and to estimate its straight-line distance. Ability to perform such tasks is often taken as evidence, when working in 2D and 3D environments, for use of a more global, map-like representation of space. Results depended on the individual participants. While all improved their performance in what are initially extremely difficult tasks, at the end of training only some were able to point immediately and accurately towards a remote location. Interestingly, certain participants in these experiments, particularly the first, immersive one, reported a strong feeling , when walking about afterwards, that the real world was but a 3D cross-section of a 4D one and that they should have been able to move their bodies to explore the missing dimension.
Andrew J. Hanson, Department of Computer Science, Indiana University
"Touching the Fourth Dimension"
By analyzing the methods we use to gain intuition about the properties of curves and surfaces embedded in 3D, we can draw deep conclusions about the modes of perception that are possible for 4D worlds. Our first observation is that 3D worlds of many kinds are actually sketched, annotated, and explored in 2D projections. For example, the most familiar means of representing knotted curves and links in 3D is to use crossing diagrams on a piece of paper or a blackboard. We first examine how this 2D space can be explored using visual feedback from the graphics display, tactile feedback from a haptic interface, and auditory feedback from a sound-generating interface. Following the fundamental visualization principle of utilizing redundant information cues, we can create a compelling 2D interface; even while restricting ourselves to 2D motion, a computer-controlled haptic interface can provide entirely new ways of perceiving a knotted curve by allowing us to perform continuous motions that �contradict� the self-occluding visual representation by adhering to the local continuity of the actual topological structure. In addition, even in 2D, sonic cues can flag over/under knot crossings, permitting a complete examination of the knot even in the absence of visual cues and the absence of full 3D haptics. These multimodal computer methods empower us to accomplish interactions that are impossible in the physical world, since our proposed interface moves the haptic stylus in 2D in such a way that we can �feel the continuity� of the 3D structure even though the projected visual representation contains apparent obstructions due to occluded segments. What can we do with 4D worlds? In fact nearly every feature of the 2D interface for 3D knotted curves can be carried over to a 3D multimodal interface for 4D surfaces, including knotted spheres, projected to a 3D haptic environment. Our first step is typically to create a graphics image that is a projection from 4D to a 3D image. A standard surface rendering hides all the internal detail, while transparency and screen-door rendering expose the internal geometry, but still obscure the fact that in 4D nothing touches anything else; the 3D projection typically contains self-intersecting surfaces that cannot be removed from the representation we �see�. We can, however, create the 4D equivalent of a knot diagram; by identifying the surface portions that are �nearer� the projection point in 4D, and allowing those to cut away the portions that are �farther� from the projection point in 4D, we get results precisely analogous to a 2D knot crossing diagram. Now we implement a 3D haptic interface programmed so that the hand-held probe is attracted to the projected 3D self-intersecting surface of a 2-manifold embedded in 4D. First, suppose that the probe adheres to the 3D surface but cannot cross a visual 3D intersection; the surface is broken up in a way that completely conceals the continuous topological structure. Next, we suppose that the haptic device navigates a crossing-marked surface: now we have agreement with the visual obstructions if the haptic device is allowed to pass freely only along the continuous surface, being forbidden to jump across the crossing-marked gaps. In either case, our exploration of the surface is interrupted. We can touch it, but to explore the entire space, we must remove the probe and reattach it to each separate portion of the 3D surface. The final step is to completely contradict the visual discontinuities of the surface, and allow the haptic device to take advantage of the hidden, unseen, but �feelable� structure of this surface; we now choose to �override� the visual obstructions in the projection and adhere to the �continuous� underlying manifold. This allows us to slide through the visual intersections as though they were ghost images, and keeps us stuck to the smooth surface that exists in four dimensions but cannot be seen without interruption in three. Both visual and auditory cues can redundantly indicate a passage across a crossing-marking that is closer in 4D (probe going under) or farther in 4D (probe going over). We thus argue that it is possible to explore 4D worlds using a multimodal interface to �feel� (and perhaps hear) the four-dimensional continuity and its crossing structure, even though the topological continuity cannot be directly �seen� in any depiction visible to the human eye.