Conveying a 4D object on a 3D medium

[metalhead1991]

Hi everyone, I'm Conor and I'm new to this forum and seeing as I'm only 17 I may say some rather stupid things, so please bear with me and feel free to correct me whenever appropriate.

I haven't been able to study this topic for very long, but one thing does stand out to me. A 3-hypercube (a regular cube to us) can be conveyed easily on paper (a 2 dimensional medium) with the use of what I suppose has to be called an optical illusion, as it gives the concept of 3D in 2D space. Therefore, could it be possible to convey a 4-hypercube by use of a 3 dimensioal medium, some sort of hologram for example, by the same use of optical illusion? Any and all opinions welcome, thanks for reading.

Conor

[papernuke]

Therefore, could it be possible to convey a 4-hypercube by use of a 3 dimensioal medium, some sort of hologram for example, by the same use of optical illusion?

Yes, it would be possible to project a 3D representation of a 4D object with a hologram.
it hasn't been done yet simply becuase it is hard to imagine and draw that up, even with computer aid.

[Eric B]

Wouldn't that just be the cube within a cube, or the isometric projection (Like your "tetronian" symbol)? That should be possible now, like with lasers in fog.

[pat]

Yes, it would be possible to project a 3D representation of a 4D object with a hologram.

The way holograms are done right now involves the interference pattern of two planar sheets of light... one reflected off of an object,
the other not. You could make a hologram of a 3D representation of a 4D object, but it would only be viewable from fairly close to
the angle at which the hologram was made. Additionally, there would be some artificial shadows... ie. you wouldn't be able to see
some parts of the 3D representation because other parts of the 3D representation blocked the light from getting to those spots or
blocked the light reflected from those spots.

Better would be to use computer-holography to simulate reflecting two 3-D sheets of light onto 3-D film. Of course, you would
then have a problem re-injecting 3-D sheets of light through the exposed film.... so I'm skeptical that you can do any better with
a hologram than you can with some tinted plexiglass.

[quickfur]

[...]Better would be to use computer-holography to simulate reflecting two 3-D sheets of light onto 3-D film. Of course, you would then have a problem re-injecting 3-D sheets of light through the exposed film.... so I'm skeptical that you can do any better with a hologram than you can with some tinted plexiglass.

Regardless of how successful we might be at reproducing the 3D image of a projected 4D object, whether it be holograms or otherwise, our eyes still only see in 2D. Unless there was a way to directly feed a 3D image into your brain(*), we'll still have to deal with 2D projections eventually.

(*) Feeding a 3D image into the brain assumes that the brain would be able to understand it as a 3D image, which is unlikely since we have no sensory faculty to see in actual 3D, so the brain is unlikely to have any faculty for dealing with actual 3D images.(**)

(**) Doesn't mean we don't try, though! For example, I've been visualizing the 120-cell as a folded 3-complex of solid white dodecahedra... you couldn't render this on any 2D screen and still have it comprehensible, but the brain is quite able to conceive of such a thing---or so it seems; when it comes down to it, we still think in terms of surfaces, whereas to really grasp 4D, one must think in terms of volumes.