I am not entirely sure if this question is supposed to be posted on Physics SE, but I'm going to post it anyway.
We have been told that shining a light on a three-dimensional object will always produce a two dimensional shadow. This is unchanged. However, I noticed something that may counteract this idea. The hypothesis is as follows:
If a one dimensional object were to shine light on another one dimensional object, it would produce a zero-dimensional shadow, a point, as a one-dimensional line would only perceive everything in view as a point, as we three-dimensional beings perceive everything in 2D. Now, if a two dimensional object were to shine light on a two-dimensional object, the same thing would occur: a line, a one-dimensional shadow, as two-dimensional objects only perceive one dimension. The same thing happens with a third dimension. Now, this is my finding: If a two dimensional object shined a light on a one-dimensional object, the object's shadow would be a one-dimensional line. If a three-dimensional object shined light on a two-dimensional object, the object would be two dimensional. Here we see that the brane, when viewed from a higher dimension, would produce a shadow of its own dimension. Would it be possible for a cube, or even a human being, to have light shined on it from the fourth dimension and produce an equal model of itself in a shadow? If so, holograms could be produced simply by a fourth-dimensional light.

  • $\begingroup$ Pedantic point: in a one-dimensional universe, there aren't enough dimensions for there to be any sort of shadow, as light can't actually move past any object. A one-dimensional object embedded in a two- or three-dimensional universe will produce either a one-dimensional or a zero-dimensional shadow, depending on its orientation relative to the light source. Likewise, a two-dimensional object embedded in a three-dimensional universe will produce either a two-dimensional or a one-dimensional shadow, again depending on its orientation. $\endgroup$ – probably_someone Oct 23 '18 at 19:31

Since we live in a four-dimensional universe (counting time as a dimension, of course), any spatially-extended light source that lasts for more than an instant of time is a four-dimensional light source. For clarity, let's suppose that we have a light source shaped like a hollow sphere that illuminates all sides of what's inside; inside, we put, say, a wobbling blob of green Jell-O. The Jell-O is transparent, as Jell-O ought to be, so we can see its insides.

We then allow the Jell-O to move in four dimensions, which, in this case, means it wobbles around in space as time passes. There are a couple of different ways to construct a three-dimensional "shadow" of the Jell-O's four-dimensional motion. For instance, we could simply take an ordinary video camera and film the wobbling blob. The image is two-dimensional in space and changes as time passes, which means it's a three-dimensional projection, or "shadow," of the Jell-O. We could also take a bunch of photographs from different angles at the same instant in time; since we can see the entirety of the Jell-O's three-dimensional bulk, we can use these photographs to make a three-dimensional model of the shape of the Jell-O at that instant in time, which is, again, a three-dimensional "shadow" of the Jell-O's existence.

EDIT: (in which things are made much more complicated, and probably more confusing)

To extend the analogy to extra spatial dimensions, imagine that, in addition to existing in four-dimensional spacetime, you also had a set of sliders that changed the objects around you as you moved them (what they're doing is transporting you along the various axes of higher-dimensional space). Any ordinary four-dimensional object in our universe would look "pointlike" to these sliders; assuming that all of the objects in our current four-dimensional universe are only four-dimensional, our current universe would only be visible to you when all of the sliders were tuned to exactly the right point. But this is not the only way to embed a four-dimensional object in higher dimensions.

To illustrate this, let's jump back to our ordinary four-dimensional world (forget about the sliders for a moment), and consider a three-dimensional object embedded in it. This object can be embedded in at least two ways: it can have two spatial dimensions and exist for some length of time (and will look "pointlike" in the other spatial dimension, having no thickness), or it can have three spatial dimensions and will exist only for an instant (having no "thickness" in time). The version of the object with two spatial dimensions can, further, be oriented in such a way that it appears like a one-dimensional object to the eye ("side-on"); when looked at from different angles, it appears to have a different area (i.e. two-dimensional volume) to an observer. It should be clear, with a little thought, that the concept of "angle" here refers to how an object is distributed across the various dimensions. Objects that are oriented differently in space are obviously positioned at different "angles" from each other, but it should also be at least somewhat intuitive that this concept of "angle" extends to how the object is distributed between space and time; in other words, this generalized "angle" not only tracks an object's spatial orientation, but also how its spatial volume is related to its duration of existence in time.

So, getting our sliders back, we can see that there are even more possibilities for objects that are four-dimensional, but embedded in higher dimensions. There is the mundane possibility that the object is confined to the currently-known four dimensions, and is pointlike in all the others. Alternatively, the object could exist only for an instant (being "pointlike" in time), but could persist and even change apparent shape as you moved a slider. The object could also exist for some interval of time, and continue to be visible with motion of a slider, but be only two-dimensional in the currently-known spatial dimensions. The object could also be some combination of those, with the distribution of its existence across the various dimensions represented as a set of generalized "angles"; however, for a four-dimensional object, it is guaranteed that, when looked at from the right "angle," it will be pointlike in all but four of the dimensions.

When you change a slider's position, it takes you into a four-dimensional universe that is a "shadow" (a projection) of some higher-dimensional form, just as an individual photograph is a two-dimensional "shadow" of a four-dimensional object. Viewing a bunch of different "shadows" of the same object from multiple "angles" gives you the ability to reconstruct its shape in higher-dimensional space, just as filming a plate of Jell-O from multiple angles allows you to reconstruct the four-dimensional objects involved from their three-dimensional "shadows" on each video.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.