Shadows and Structures

Timothy Faulkner

Chapter four concerns itself with shadows and introduces the concept of the simplex: smallest figure that contains n+1 given points in n-dim. space and that does not lie in a space of lower dimension. N-simplexes are an interesting concept, and it is strange that the hypercube is so lauded in four-space. The hypercube gets such star status for its interesting rotations and also because people are more likely to understand a hypercube. . . or rather a cube in four-space than the most simple figure of a higher space--so the hypercube remains the figure for understanding four-dimensional geometry: is there any true merit to this distinction or is it essentially political (in the sense of selling 4-d to a non-mathematical world)? An exrecise would be to create an animation of a 4-simplex and its rotations, projections, and slicings; however, I think this is beyond what we are capable of doing in class right now. Anyway this chapter also provides the best method for constructing the hypercube. A good exercise would be to follow the construction for other figures: the 5-cube, etc. . .

The end of the chapter is devoted to data visualization, one of the most practical and understandable usages of higher dimensions in the real world. If you have more than three variables, how do you represent your data in a graphical manner which is easily understood? Use a computer!