Imagine that you need to fill-up a 4D unit hypercube with 3D unit cubes of water.

At first, I thought you'd need an infinite amount, arguing that it would take an infinite number of stacked 2D squares to fill up a 3D cube.

But then, I realized that given that each square had zero height, an infinite number stacked on top of each other should still have zero volume. Essentially, one can't talk about the volume of a 2D object. Similar reasoning can be applied to argue that 3D matter cannot fill up 4D space, regardless of whether it is infinite or not. Is this reasoning sound?

Consider these hypothetical scenarios: suppose a unit cube of space, relative to Earth so that it 'moves along' with the planet... suppose, a unit cube of such a space submerged in the oceans opens up into a 4th space dimension. Assume that the surround universe is not immediately destroyed.

  1. Can the water be pushed into the 4th dimension via say natural movement of the oceans? Why I'm asking this is because an object in 2D space cannot be pushed into 3D space by forces in the 2D space itself.

  2. Suppose it can be pushed by 3D forces: will it start draining immediately? How fast would it drain?

Basically, I'm writing a story about a 4th spatial dimension interfering with everyday life, and trying to make it as logically coherent as I can.

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    $\begingroup$ This depends a lot on your definition of your fourth dimension. Usually, the fourth dimension is "time", which would be simple to fulfil, but yours seems to be something else entirely. $\endgroup$
    – user10945
    Jan 13, 2017 at 8:52
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    $\begingroup$ IMHO Interesting questions for physics.se or math.se $\endgroup$ Jan 13, 2017 at 8:56
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    $\begingroup$ 4th is another spatial dimension here. Time, in this case, is then the 5th (4+1). $\endgroup$
    – Abs
    Jan 13, 2017 at 8:56
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    $\begingroup$ Relevant pictorial explanation of 4D objects: math.stackexchange.com/a/1254375/360255 Also remember that $0 \times \infty$ is an undefined number, not necessarily $0$ $\endgroup$ Jan 13, 2017 at 9:20
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    $\begingroup$ See also: interacting with higher dimensions. Answerers, please don’t repeat general principles here that duplicate the older post (ref and excerpt ok), but focus on the specific scenareo only for this post. Feel free to add more general answers or edit answers to the other question too! $\endgroup$
    – JDługosz
    Jan 13, 2017 at 9:43

8 Answers 8



Look at space filling curves. You can apply the same idea to filling a 4th dimension with a 3d ribbon.

You still need an infinite amount, though.

You might also consider how 3d matter exists in the 4d world. Just like sheets of paper (or even ink on a sheet of paper) in our 3d world is not really 2 dimensional but meerly extremely thin in the 3rd dimension, you can postulate that the 3d objects, in order to exist at all in the 4d realm, are actually paper thin in the 4th dimension rather than having zero extent.


See my answer to interactions with higher dimensions for details. Of note:

you don't have to stand beside something at w=5 inches for example to push in the −w direction. The physics is not "closed" over the domain of the dimensions of the current arrangement of particles. Effects can operate at right angles to the participants. That is the general thing you see with cross products. Gyroscopes would produce torque in that direction, electromagnetic effects would have more right angles to reach out to.

  • $\begingroup$ I use n-dimensional space filling curves in various search optimisation problems, and the statement 'in order to exist at all in the 4d realm, are actually paper thin in the 4th dimension' is very important, otherwise there isn't actually any extra dimensionality to consider with your 'object'. $\endgroup$
    – Joe Bloggs
    Jan 13, 2017 at 9:33
  • $\begingroup$ Paper thin strips of 2D objects are only represented as 2D objects mathematically because it simplifies the object down to less dimensions. The only reason you'd fill up 3D space with them is that they ARE actually 3D all along. You can not fill up 3D spaces with 2D objects no matter how much you try even with infinite amount of 2D objects. The same thing is true for 3D objects in 4D. $\endgroup$ Nov 3, 2017 at 16:38
  • $\begingroup$ @A.C.A.C. see the link on space-filling curves. $\endgroup$
    – JDługosz
    Nov 3, 2017 at 17:51
  • $\begingroup$ @JDługosz Even if your entire space is filled with an infinite number of space filling curves, the curves have no physical interaction with 3D matter itself, which means you can still put a person in your 3D universe filled with an infinite number of 2D space filling curve. If you substitute your curve with physical matter, you are using 3D objects and only representing them as 2D. $\endgroup$ Nov 3, 2017 at 18:05
  • $\begingroup$ @A.C.A.C. that’s up to the author to define. Maybe they interact with gravity but not other forces. Hey…dark matter could be patches of 2D matter, and Flatland could be discovered! $\endgroup$
    – JDługosz
    Nov 3, 2017 at 21:47

Disclaimer, none of this should be regarded as proof since it's all theory.

The 4th Dimension

The 4th dimension is by humans incomprehensible, the same way that the 3rd dimension is incomprehensible for a 2D entity and the 2nd dimension is incomprehensible for a 1D entity.

A 2D entity simply can't view its existence from a 3D perspective. Just like we are incapable of imagining viewing something from the 4th dimension, quite similarly how we cannot imagine a color outside our spectrum simply because it's outside of our perceived reality.


A 1D object simply has one dimension, "forward" and "backward" along a line. (so to speak)

A 2D object has two dimensions, "forward", "backward" and "side-to-side".

A 3D object has the dimensions of a 2D object with an added "up" and "down",

In short, 1D has one axis, 2D has two and 3D has three.

enter image description here

Now, some argue that the 4th dimension is "time" but that does not quite represent the entire picture.

We, as "3-dimensional" beings, comprehend the world and our universe as infinite 2D "projections" so to speak. In essence, we are viewing an infinite number of 2D planes. The same way a 2D being would observe their universe as an infinite number of "lines" of the first dimension. (this is not a 100% correct statement, but I hope you understand what I mean.)

So, A 4-dimensional being would be viewing their universe as an infinite number of 3-dimensional "projections". An "angle" or perspective we are simply unable to comprehend, the same way a 2D being is incapable of comprehending the angle a 3D being would observe it.

Thus, the 4th dimension would be an infinite number of 3D "instances" or projections. Now, some argue that this is what "time" means; "An infinite number of 3D projections over time". In other words, moving through a 3D space and through time.

However, for us to comprehend this fourth dimension, the fourth dimension needs to be presented in a "3D" format. The interstellar movie had a pretty good take on this, as can be seen here: enter image description here

Essentially, the fourth dimension would be presented as infinite instances in 3 dimensions; "up, down, forward, backward, left, right", that we would be able to traverse.

Note that these 3D "instances" should actually exist in the exact same place at the exact same time.

But that would mean that we would perceive it as only one instance, the one we are currently in.

In regards to OP's question:

If this theory is correct, your 4D cube of space would not "fill up" or suck water into it. It would probably merely be an infinite number of instances of the same water. (through time? maybe? nobody knows)

What this would mean that if the 4D "cube" was 1x1x1 meters in 3D size, the water inside it would be an infinite number of 1x1x1 meter instances of the same 3D water. In essence, an infinite number of the same 1 cubic meter of water.

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    $\begingroup$ "The 4th dimension is by humans incomprehensible" - in a way true, but mostly NOT true... we live in a universe with at least four dimensions, and we deal with them constantly and have a perception of them. We perceive time and can operate in it and perceive an object, or even ourselves, at a different time, through memory. We quantify time, measure it, study it, see how it interacts with gravity, space, in relativistic and non-relativistic frames. Time is no more incomprehensible to us than distance. $\endgroup$
    – user11864
    Nov 6, 2017 at 2:55
  • $\begingroup$ A key point you miss is that memory is a perceived interpretation of reality, not an instance of reality itself. A 4dimensional space would be a literal near identical copy of an existential instance, at every point of "time" at every possible state. $\endgroup$
    – DannyBoy
    Nov 6, 2017 at 9:10
  • $\begingroup$ Any observation of space or time is "a perceived interpretation of reality". When you measure the initial height of a candle, you have no more comprehended the candle or even its height, than you have comprehended the candle or its time dimension when you measure the time it takes it to burn down to a nub. When you say "a 4 dimensional space would be a literal near identical copy of ...", you fail to see that the time dimension would be different and the effect of non-static contents of that space. $\endgroup$
    – user11864
    Nov 6, 2017 at 15:38
  • $\begingroup$ You could say the same like this: "every 2d layer of a 3d cube is 'literal near identical copy' of the same square". but that ignores that they are not the same square and that given a more complex object, they would be nothing alike. Similarly, a 4d space at any point in time may be "a literal near identical copy" for a static object, but they are not the same, and that becomes all the more clear when you have a more complex 4d object, even one as simple as a burning candle. $\endgroup$
    – user11864
    Nov 6, 2017 at 15:41
  • $\begingroup$ Take a video of a burning candle. Pause it an any point in time, and you are looking at a 3d scene (2d representation, but you should get the point). If not for the technical limitations of cameras, you would have an infinite number of 3d spaces recorded. $\endgroup$
    – user11864
    Nov 6, 2017 at 15:53

The thing is that you have to realize that there is no such thing as a 2d object. Even sheets of paper, no matter how thin, are 3d. Looking at it from a 3d perspective, we live in a 3d world and all objects in it are 3d. 1d and 2d are theoretical constructs that we use to incrementally understand our universe.

So then, when you start thinking of our universe in 4 dimensions, you realize that nothing in it would have zero length in that 4th dimension. Say you think of time as that fourth dimension. If an object had zero length in the fourth dimension, it would exist for 0 seconds, i.e. it would never exist. Ergo, no such object can exist, and it should not be hard to see how this would extend to any dimension (1st or 2nd or 3rd or 4th or ...).

Here is another thought experiment to help you think in terms of four dimensions - Our perception allows us to look "backwards" through that time axis, so we can "see" an object projecting back in time. I can see my car out in the parking lot, and I have a vivid memory of it when I was getting into it in my garage. But our perception does not allow us to look forward, so we do not see the extent forward of an object in that time axis, but that is ok as a backward look is sufficient for this experiment. Now, look at your computer; it's there. Then remember it at a time before, and realize that you are looking at a 4d object.

As far as filling a 4d hypercube with 3d cubes of water, realize that there is no such thing as 3d cubes of water. They are 4d, because they exist in a universe that has more than 3 dimensions. If your 4th dimension in your 4d hypercube is a real dimension that exists in the same universe as those 3d cubes of water, for example "time", then those "3d" cubes of water are going to have some sort or extent along that dimension in order for them to exist in that universe. Therefore you could extend their length or stack them, end to end, along that dimension and fill that hypercube.

  • $\begingroup$ ...then if we assume that a water molecule's thickness in the 4th dimension is roughly equal to its size in 3space (0.29 nm), then we could fill a 1x1x1x1 meter hypercube of water with 3,448,275,862.07 cubic meters of water. $\endgroup$ Jan 19, 2018 at 3:12
  • $\begingroup$ @Draco18s - why would you assume such a thing? And what makes you think that the fourth dimension shares units with the first three? If time is the fourth dimension, then it does not. Any other fourth dimension is just as likely to use entirely different units. And more than likely that the span of a water molecule in that fourth dimension is entirely different. If you use time as that fourth dimension, the span of a water molecule could be in the billions of years. $\endgroup$
    – user11864
    Jan 22, 2018 at 19:05
  • $\begingroup$ Because 1) I don't have a way to know what a water molecule's size is in a non-existent 4th spacial dimension and 2) because friction-less spherical cows make for a decent approximation. And 3) Time is not the 4th spacial dimension. Time is a dimension of time, not space, and dimensions of time have no relevance to a 4 spacial dimensional structure. Why do people keep doing that? $\endgroup$ Jan 22, 2018 at 19:21

Your reasoning is sound. This is impossible, almost by definition of what a dimension (and multidimensionality) is.

  1. There are an infinite amount of points (0-dimensional) on a given line (1-dimensional), as a point has no width.
  2. There are infinite lines on a given plane (2-dimensional), as a line has no thickness and therefore doesn't quite "stack".
  3. There are infinite planes in a space (3-dimensional), as a plane has no thickness and therefore doesn't quite "stack".
  4. Logically, you can therefore fit an infinite amount of 3-dimensional objects in a 4-dimensional space (note that this logical consequence is consistent regardless of whether you consider the 4th dimension to be time, or something else).

In order to fill an N-dimensional space, your object must have a defined "thickness" in all N dimensions.
But an (N-1)-dimensional object inherently does not have a defined "thickness" in all N dimensions, since it only has N-1 defined dimensions.

If it did have N defined dimensions, then it would be an N-dimensional object.

As for your hypothetical scenario's, we simply cannot say. As it stands, we perceive the universe as having exactly as many dimensions that we ourselves consist of.

Every object with a different amount of dimensions (points, lines, planes, 8-dimensional space) are all just abstract theoretical constructs. We can reason about them, we can make representations of them, but we can never truly see one.

From a tangible point of view of an N-dimensional observer, non-N-dimensional objects are as abstract as concepts such as love or awkwardness or cynicism.
They are not tangible in any way, even though they can be represented by a related N-dimensional object e.g. we could construct a cylinder, and then point at its base and say that it's a circle. But we cannot create a circle by itself, without it being part of an N-dimensional representation.


Hmm. Do you really need to "fill up" 4d space, or do you just want 4d objects?

If you just want 4d objects, that's easy. First note that you can create 2d objects from 1d objects. Behold the triangle

a rigid 2D object made from 3 1D objects. In 3D, we get the tetrahedron

a rigid 3D object made from 4 1D objects. We can even get to 4D this way. Behold the 5-cell

a rigid 4D object made from 5 1D objects.

So, you only need 5 1D objects to make a 4D object. If you insist on using 3D objects, there are other ways you glue them into rigid 4D objects. It won't fill 4D space, but it allow you to make rigid objects.

  • $\begingroup$ Question is explicitly about filling up hyper-volume. This does not answer the question as asked at all. $\endgroup$
    – Mołot
    Jan 19, 2018 at 7:29
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    $\begingroup$ @Mołot oh, the titular question is different from the body question. In any case, it makes sense that 3D objects might form rigid 4D ones, even if drained randomly. $\endgroup$
    – PyRulez
    Jan 19, 2018 at 18:37

I think you are right with the 2d analogy. Draw the vectors. How can a 3d object produce a vector force into a 4th dimension? You are right about the stack of squares too. A bummer thing about the infinite number of third dimensions is that you (3d person) would not be able to see from one to the next unless you give light some special properties.

Other weird thing is that all infinite number of stack of squares are in the same place. There is no order to them. If you have infinite number of 3d dimensions and you leave yours, be sure to take your stuff with because you are never finding your original one again.


My thought is this: all of the fundamental particles (electrons, quarks, and photons (NOT protons or neutrons)) are zero dimensional if you don't believe in string theory, and one dimensional if you do. This means that protons, neutrons, and atoms are all just collections of zero/one dimensional particles, relative to eachother in space. The only reason atoms are three dimensional is because these fundamental particles are only situated in, and moving in, three dimensions. If a force were applied to a three dimensional atom, in a direction perpendicular to the normal three, the fundamental particles would settle into positions in four dimensions. You would then have a four dimensional atom which could, with enough of them, fill a tank.

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    $\begingroup$ Apart from the correctness of considering the photon a fundamental particle, and from the fact that one doesn't "believe" in physical theories, how is this answering the question? $\endgroup$
    – L.Dutch
    Feb 20, 2018 at 16:21
  • $\begingroup$ I was replying to the question, "Is this reasoning sound?" $\endgroup$
    – Ian
    Feb 22, 2018 at 1:49

A cube of arbitrary dimension that is higher than 3 can be 'filled' with 3-cubes (but not with 2-cubes or 1-cubes, no analogy with Peano and Hilbert). This is a so called 3-cube theorem proved in the forties by L.V. Keldysh and published in 1957. Find a place to read about it.

The article is hard to find and written in Russian, but see this summary.

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    $\begingroup$ Please find a reference to this paper (I can't find one on Google) and summarize the results here. Otherwise this is a link-only answer....except with no link. $\endgroup$
    – kingledion
    Feb 20, 2018 at 2:16
  • $\begingroup$ Apparently this article is really hard to find, and written in Russian. You may be able to get a hold of it if you have a subscription to some of the scholarly libraries. Here's a Math.SE summary: math.stackexchange.com/questions/1692266/… $\endgroup$
    – AndyD273
    Feb 20, 2018 at 14:45
  • $\begingroup$ I don't know how to help the poor mathematicians who don't know German, French and Russian. Seriously. As to the link - here's the link The downloadable pdf of the article in Russian $\endgroup$ Apr 8, 2018 at 17:00

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