

4D objects Hyperspace
Name: Name
Status: N/A
Age: N/A
Location: N/A
Country: N/A
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:
4D objects are usually 'viewed' by considering lower dimensional
slices. The 'graph' of w = f(x,y,z) represents an object in 4D in the same
sense that y = f(x) reps a curve in 2D. Some references may be found under
'hyperspace' in a card catalog. In Linear Algebra, 4D Euclidean space is
just the set of all 4tuples (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 3D 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 higherdimensional 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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Update: June 2012

