Ask A Scientist Mathematics Archive

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. 
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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
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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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