Ask A Scientist Mathematics Archive

Question: Hi, I could not find the 2 books that were recommended to me, 
about the 4-D objects.  But, maybe someone can help me.  I made a computer 
program to draw 3-D Objects and project them in the 2-D monitor.  It also 
draws 3D functions.  I made a function to convert the 3D coordinate (x,y,z) to 
a screen coordinate (x,y), and it works really well.  What I want to do now, 
if it is possible, is to make a function to convert 4D coordinates (x,y,z,w) 
in 3-D or 2-D coordinates, so I can draw a 4-D object (or a function) on my 
screen, and see what it looks like.  Can I do that?  My program works fine in 
projecting and rotating 3-D objects, but what about different kinds of 
projections?
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Pedro, I really hate to seem discouraging, but this does not seem 
possible to me.  It is theoretically possible to "project" a 4-D object into a 
3-D space (just like we can "project" a 3-D object, like a cube, onto a 2-D 
surface, which is a plane).  However, to project a 3-D object onto a 2-D 
surface (like a computer screen) we also have to use lines, colors, and/or 
shading to present us with an illusion of what the third dimension looks like.  
In principle, one might be able to do something like this with a 4-D object, 
but it does not seem possible to do it in 2-D (on a computer screen) with a 4-
D object. If you can solve this problem, publish it in a computer graphics 
journal!  Good luck.
Topper
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Actually, it is not that hard to display 4 dimensions on a 2 
dimensional screen (or sheet of paper) as long as your data has the right 
form.  There is a book on ways people have displayed up to 6 dimensions of 
data - I think it was called The Visual Representation of Information.  A 
simple example of 4 dimensional display is placing arrows (which represent 2 
dimensions) on the points of a two dimensional grid.  This is often done to 
represent currents, or magnetic fields, for example.  Color can actually 
represent 3 dimensions (1 for redness, 1 for greenness, and 1 for overall 
brightness, for example), so a 2-D color graph could give 5 dimensions of 
data.  Actually, there is a 6-dimensional graph in the April Scientific 
American, displaying possible origins of new universes (see p. 24).  Actually, 
it looks of uniform brightness, so maybe there are only 5 dimensions on 
display.  The main problem with displaying higher dimensions is that it is 
impossible to treat all the dimensions in the same way.  So 2 (or at most 3) 
will look like spatial dimensions, and the others just have to look like 
something else (arrows or colors).  Hope this helps!
Arthur Smith
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