If there was a 3 dimensional plane existing within a 4d space and something forth dimensional casts its shadow into this 3d plane would it be physical to 3d beings on that plane?

Now theoretically a 4d object's shadow is three dimensional. But is it just a 3d image of one version of that object or is it a physical three dimensional object? Logically it wouldn't be able to be acted upon(meaning you couldn't effect the 4d object by pushing its shadow) but could it act on something else? Could a 4d object's shadow push a 3d being? Using the same logic could a 3d object's shadow push a two dimensional being? Would it be like a wall to it?

Side note: If anyone knows a better way to theoretically give mass to a shadow feel free to explain or point me in that direction.

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    $\begingroup$ A shadow is simply the volume of space (or the area of a plane) where there is less light than elsewhere. It is not an object. And shadows of three-dimensional objects are also three-dimensional; we commonly see them as two-dimensional because it's easier to make out the difference in the amount of light on a flat surface, but occasionally we do see them in three-dimensions, when the object is immersed in a dispersive medium such as mist or a dusty environment. $\endgroup$ – AlexP Apr 26 '18 at 21:32
  • $\begingroup$ If you have particles traversing 4th dimension, then you will see some peculiarities in their behavior seen in the usual 3 dimensions. But these traversing particles won't turn into a "solid mass", if this is what you have in mind. $\endgroup$ – Alexander Apr 26 '18 at 21:35

The standard tesseract shape you're probably familiar with is the 3D shadow of a 4D object.

enter image description here

We can't represent 4D shapes in the world any other way. It wouldn't have any more physical presence than your own shadow does. It would simply be an area with less light.

You can't give mass to shadows. They represent an absence of something.

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    $\begingroup$ "The standard tesseract shape you're probably familiar with is the 3D shadow of a 4D object" for certain unsual meanings of the word "shadow". I think that you mean "projection". $\endgroup$ – AlexP Apr 26 '18 at 21:40
  • $\begingroup$ @AlexP Not when it's qualified with '3D'. Check the Wikipedia link: "Similarly, if the wireframe of a tesseract were lit from “above” (in the fourth dimension), its shadow would be that of a three-dimensional cube within another three-dimensional cube." In either the 2D or 3D case the shadow is the result of a projection of a higher dimensional object onto a lower dimensional medium. $\endgroup$ – Samuel Apr 26 '18 at 21:50
  • $\begingroup$ so overall the answer is no, a shadow of a 4th dimensional object wouldn't have any sort of force on lower dimensions, but it may still create a three dimensional image of sorts. $\endgroup$ – Snaebrm Apr 26 '18 at 22:02
  • $\begingroup$ @Snaebrm Right, a shadow is a shadow. There isn't anything special about it coming from a 4th dimensional object. The shadow is still n-1 dimensions. $\endgroup$ – Samuel Apr 26 '18 at 22:10

I'd like to address some controversy both in the question and in answer by Samuel. I'd try to supply some further information and actually answer the question, but bear with me first.

Shadows are volumes

Let's stay in our 3D world first. The shadows are not flat, as one would actually assume from looking on them and from the typical all-day usage of the word. A shadow is a 3D form, a volume.

Basically, everything behind an occluddee in absence of other light sources is in shadow. Look at the typical explanation of lunar eclipse, for example.


The thing we colloquially call a shadow in an intersection of the actual shadow (a volume) with some other object. So we basically see where the shadow ends. When comparing the occluddee with its "shadow on the wall" we can draw some conclusions. This "shadow on the wall" is sort of a projection of the object (the word "silhouette" is related), or an intersection of the (real, 3D) shadow with the background object (wall, ground, Moon, etc).

Now to the question

4D objects in 4D world would have 4D shadows. No win here.

An intersection of a 4D shadow with a 4D wall would probably produce a 3D intersection. We might learn something about the shapes of the 4D object and its 4D shadow when studying various such intersections. So-to-say 3D silhouettes of 4D objects.

In a sense (back to our 3D world and 2D displays), something similar happens when a 3D scene is displayed on a usual computer screen or when a 3D projection of a cube is sketched on a very 2D sheet of paper.


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