(loosely inspired by the discussion of mirrors in this thread
If I’m looking at a mirror, or a image on holographic film, I can tell that the image I’m looking at is in some sense 3-d, even though it appears on a 2-d surface. For instance, I can observe parallax – if I move side to side, objects in the mirror or hologram appear to move relative to each other. This doesn’t happen if I’m looking at a photograph, which (I think) is a genuine projection of the 3-d object onto a two-dimensional surface.
So what I’m wondering is this: suppose I had an object that had four spatial dimensions, and I created a 3-dimensional holographic image or reflection of the object. Are there any properties of this image that I could observe that would indicate to me that it’s really an image of a four dimensional object?
At first I thought I’d have to be viewing it from the fourth dimension, just like I look at a two-dimensional mirror from outside of its two dimensional plane. But then it occured to me that there’s no reason the three dimensions of the image would have to be the same as my three spatial dimensions. For instance, a two dimensional being who lived in a horizontal plane could look at a vertical two-dimensional mirror, and even though he’d only see the slice of the mirror that intersected his plane, he could still observe parallax. Similarly, I’d think a 3-d being could share two spatial dimensions with the 3-d hologram, and have his third dimension be the viewing dimension.
I’m not sure I follow you completely, but I don’t think our brains are equipped to fully comprehend a 4D object. The reason a 2D surface like a mirror or hologram can produce an image that appears 3D to us is simply because our brains are wired up to comprehend a 3D illusion (those mediums allow each eye their own unique paralax), which the brain then puts together and says, “AH! a 3D object!”.
I think the key difference, would be that there is no likewise function of the brain that exists, that would “see” a 3D illusion depicting a 4D object, where the brain would then say, “AH! a 4D object!”.
We don’t have eyes (or brains, I expect) that work in four dimensions, and light waves don’t seem to work in four dimensions, either. We’re in a 3-D world. I can see a 3D analogue of a hologram in a 4D world, but I think we’d be as unable to see the 4D aspects as the inhabitants of Flatland or Dewdney’s planiverse (2D universes) would be unable to see and interpret a 2D hologram of a 3D scene.
What we can do is see 4D objects unfolded into 3D space or projected into 3D space. There are examples of both – Salvador Dali’s painting “The Crucifiction”, for example, is a 2D representation of a 3D unfolding of a 4D tesseract.
When you look at a hologram or mirror you are not looking at an image on a 2D surface, you are looking at wavefronts from a 3D object that have been either stored in a holographic material or bounced off a reflective surface.
Without getting too technical: for a reflection the light you see did not originate at the mirror plane; it orignated at the 3D object and contains all of the phase (think depth here) information of the object. With a hologram the phase information has been stored in the holographic material and recreated perfectly (well, almost) so that the wavefront you see is indistiguishable from the light that originated from the 3D object.
Objects with 4 spatial dimensions are impossible for me to imagine in this context.
Now if the 4th dimension is time you might be able to do something. In this case you could create a hologram by integrating the image in the material over a period of time. Assuming the hologram diffraction efficiency is constant over the exposure (this is impossible, you would have to shedule the energy while writing the hologram, i.e. you would have to turn up the lights over time to compensate for the holograms already written), I would guess this would just blur the holographic image, just like the pictures of cars at night where the headlights blur into long lines. What this would look like I have no idea, but I would guess it would look like crap. Now if you change the angle of the reference beam used to make the hologram while you integrate and map your 4th dimension (time) into angle (angular multiplexing) you might get something neat.
In fact, I think people already do this. For example with some credit card holograms you can get the bird to flap its wings by viewing the card from different angles. This can be viewed as mapping time into angle.
Even if you were observing the reflection of a 4-D object in a 3-D space, as a 3-D observer, the most you could hope to see is a 3-D slice of the 4-D object.
A 3d object like a sphere projected into 2d would be a circle.
If you as a 3d being take that sphere and move it through a 2d space populated by fuzzy, green 2d critters, those critters would see a circle appear out of nowhere, grow into a big circle, and shrink back into nothing.
If a 4d being took a 4d sphere (hypersphere?) and moved it through our 3d space, we would see a normal sphere appear from nothing, grow in size, and shrink back into nothing again.