

4D Objects
Name: Name
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1993
Question:
Hi, I could not find the 2 books that were recommended to me,
about the 4D objects. But, maybe someone can help me. I made a computer
program to draw 3D Objects and project them in the 2D 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 3D or 2D coordinates, so I can draw a 4D 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 3D objects, but what about different kinds of
projections?
Replies:
Pedro, I really hate to seem discouraging, but this does not seem
possible to me. It is theoretically possible to "project" a 4D object into a
3D space (just like we can "project" a 3D object, like a cube, onto a 2D
surface, which is a plane). However, to project a 3D object onto a 2D
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 4D object,
but it does not seem possible to do it in 2D (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
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 2D color graph could give 5 dimensions of
data. Actually, there is a 6dimensional 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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Update: June 2012

