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Date: Around 1993


Question:
Hi, can you give me any kind of example or something about 4 dimensional objects i.e., a function that would make one, or any information about it.



Replies:
4-D objects are usually 'viewed' by considering lower dimensional slices. The 'graph' of w = f(x,y,z) represents an object in 4-D in the same sense that y = f(x) reps a curve in 2-D. Some references may be found under 'hyperspace' in a card catalog. In Linear Algebra, 4-D Euclidean space is just the set of all 4-tuples (w,x,y,z) and objects therein may be described as any subcollection of points. An equation relating the entries is one way to produce a 'hypersurface'. For example, w^2+x^2+y^2+z^2 = r^2 may reasonably be called a hypersphere of radius r with cross sections (via a constant coordinate) that all spheres in 3-D have.

Thomas Elsner


I heartily recommend two books: Flatland by A. Abbott (Dover Press), and The Fourth Dimension by Rudy Rucker. These are both nice, popular books about higher-dimensional spaces. However, my understanding is that these days many mathematicians believe that it is very difficult to generalize certain theorems in N dimensions, i.e., just because something is valid for N=2,3,4,5,6 does not mean that the same theorem will hold for N=7,8,9...just a minor caveat. I really like these books, and I have done a lot of work in many dimensional systems as a theoretical chemist.

Topper



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