The desire to understand the underlying geometry of multidimensional problems motivated several visualization methodologies to augment our limited 3-dimensional perception. After a short overview, Parallel Coordinates are rigorously developed obtaining a 1-1 mapping between subsets of Euclidean N-space and subsets of 2-space. It leads to representations of lines, flats, curves, intersections, hypersurfaces, proximities and geometrical construction algorithms. Convexity can be visualized in ANY dimension as well as non-orientability (Moebius strip) and other properties of hypersurfaces. This is a VISUAL Multidimensional Coordinate System with applications to Air Traffic Control, Visual and Automatic Data Mining, Interactive Models of Complex Systems.

Sphere is represented by two hyperbolic regions. There is a translation -- rotation duality (center). An N-dimensional sphere is represented by N-1 regions analogously to the 5-D hypercube (right).

Moebius strip – non-orientability in 3 and higher dimensions.

A family of "close" planes is represented by two convex hexagons and in N-dimensions by N-1 convex 2N-agons (right).