Visualizing Rn and Some New Dualities
Speaker: Alfred Inselberg, Tel Aviv University
Location: Warren Weaver Hall 1302
Date: October 9, 2009, 11:30 a.m.
Host: Richard Pollack
With parallel coordinates the perceptual barrier imposed by our 3-dimensional habitation is breached enabling the visualization of multidimensional objects. The highlights, interlaced with interactive demonstrations, are intuitively developed. M-dimensional objects are recognized recursively from their (Mâˆ’1)- dimensional subsets. It emerges that a hyperplane in N-dimensions is represented by (N âˆ’1) indexed points. Points representing lines have two indices, those representing planes three indices and so on. In turn, this yields powerful geometrical algorithms (e.g. for intersections, containment, proximities) and applications including classification. A smooth surface in 3-D is the envelope of its tangent planes each represented by 2 planar points. As a result it is represented by two planar regions, and a hypersurface in N-dimensions by (N âˆ’1) regions. This is equivalent to representing a surface by its normal vectors. Developable surfaces are represented by curves revealing the surface characteristics. Convex surfaces in any dimension are recognized by hyperbola-like regions. Non-orientable surfaces yield stunning patterns unlocking new geometrical insights. Non-convexities like folds, bumps, concavities are visible. The patterns persist in the presence of errors. Intuition gained from the R3 representations leads to generalization for RN with beautiful new dualities like cusp in RN $(N âˆ’1) â€œswirlsâ€ in R2, â€œtwistâ€ in RN $(N âˆ’1) cusps in R2.
Alfred Inselberg received a Ph. D. in Mathematics and Physics from the University of Illinois (Champaign-Urbana) in 1965 and was Research Assist. Professor until 1966. He held research positions at IBM, where he developed a Mathematical Model of Ear (TIME Nov. 74), concurrently having joint appointments at UCLA, USC, Technion and Ben Gurion University. He is at the School of Mathematical Sciences at Tel Aviv University since 1995. He was elected Senior Fellow at the San Diego Supercomputing Center in 1996 and Distinguished Visiting Professor at Korea University in Seoul in 2008. AI invented and developed the multidimensional system of Parallel Coordinates for which he received numerous awards and patents (on Air Traffic Control, Collision-Avoidance, Computer Vision, Data Mining). His textbook on "VISUAL Multidimensional Geometry" was recently released by Springer in 2009.
Refreshments will be offered starting 15 minutes prior to the scheduled start of the talk.