We will just need a new axis Shouldn't be too hard... All the points are, in reality, on the x/y axis, just shifted according to the Z-value and the angle of view...

Something I thought about recently... In many 3d-modelers, you got 4 views of your object: 3d, and three 2d screens...

Just thinking now... Can't you simplify a 4d object into three 3d screens, and those into three * three = twelve (I can count! Jippie!) screens?

[EDIT]

Hmmm... The irony... I state i can count, yet I screw up... Just noticed it... Hmmm, okay, make it nine then

Edited by - ronin_54 on November 18, 2001 12:16:33 PM

I am not familiar with the theories behind processing 3 dimensional data into a 2 dimensional picture, but I wonder: could that theory be adapted to display a 4 dimensional picture? Or has this already been done?


rk

Might take a few more images I think...

Splitting a 3d image into 3 2d images, requires you to just remove one coordinate from it, so you get a x/y, z/x and y/z image.

For 4d to 3d, that would become: x/y/z, x/y/''u'', x/z/''u'', y/z/''u''. Giving you 4 3d-images...

Hey, my 12 might be right after all!

This discussion is getting very interesting. I've always found 4D objects to be really confusing, but doing searches for 'hypercube' and 'hypersphere' brought up these pages which really cleared things up for me:
http://www.geom.umn.edu/docs/outreach/4-cube/
http://www.cyburban.com/~mrf/hierarchy(1).html
Google is such a wonderful tool.

That last site has some very relevant information for this thread. Basically what it says (under the heading: "Myth #2: The hypersphere can't be visualized easily by those of us confined to three-dimensional thinking.") is that the surface of a 3D sphere can be represented by two 2D circles, one for each hemi-sphere. In 4D, the 'surface' volume of a hypersphere can be represented by two 3D spheres, one for each 'hemi-hypersphere'.

Still, this stuff makes my 3D brains hurt.

EDIT: typos

Edited by - Scarab0 on November 19, 2001 12:07:42 PM

Very interesting discussion!

Anyone ever hear of a tesseract? I first learned about these reading the old Madeleine L'Engle novel "A Wrinkle in Time" when I was young. A tesseract is a 4-dimensional cube, and there are graphical representations of it. I found a few sites of interest:

http://www.geom.umn.edu/docs/outreach/4-cube/

The next one has a very interesting interactive tutorial graph. The plot is always projected into the 2D screen, of course, but the shape you see does change as it rotates. And it allows you do view "cross sections" of the tesseract.

http://www.cut-the-knot.com/ctk/Tesseract.html

There's also a page on 4D geometry at Mathworld.wolfram.com:

http://mathworld.wolfram.com/4-DimensionalGeometry.html

Also, anyone here read "Flatland" (or the more recent "Flatterland")?

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

Edited by - grhodes_at_work on November 19, 2001 1:38:12 PM

Not 12 to go from 4d to 2d, only 6 (4 choose 2): xy, xz, xu, yz, yu, zu

Since x^2+y^2+z^2=r^2 is a spherical surface then wouldn''t w^2+x^2+y^2+z^2=r^2 be a spherical solid, i.e. a set of spherical surfaces where the radius is r^2-w^s? So couldn''t you just animate the sphere as a shrinks or expands, i.e. time as the fourth dimension? It would be like passing a plane through a sphere. It seems somewhat metaphysical to say what a 4D object would look like if we could see 4 dimensions, i.e. 3d on a 2d screen is intended to look like a 3d object, but what does a 4d object "look" like? It seems you could say it "looks" like anything you please since you can''t be proven wrong.

Was boared, so went from square to cube to 4d-object... Looks great! And mathematicly correct

Am going to show it to my math teacher tomorrow, and will post it on-line as soon as I can get my webhosting back on-line... grmbl...

Here''s sort of a visual representation of a hypersphere...



And the POV-Ray code I used:

global_settings {  ambient_light color 3}camera {  location <0,0,-3>  look_at <0,0,0>}// main lightlight_source {  <-10,10,-10>  color rgb 1}// under lightlight_source {  <0,-10,0>  color rgb 0.5}sphere {  <0,0,0>  sin(clock*pi)    pigment {    color rgb <1,0,0>  }} 

With the .ini settings:

Initial_Frame = 1
Final_Frame = 9
Initial_Clock = 0.0
Final_Clock = 1.0
Subset_Start_Frame = 2
Subset_End_Frame = 8

That''s just an animation. I figured out a way to draw an n-dimensional object on a 2 dimensional plane

But am at school right now, so can''t upload it...