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What troubles me is how "nothingness" can have any attributes. The way we perceive space has a lot to do with light moving in straight lines and that we see this.

But is there some non-trivial way in which the number of dimensions is more or less than 3? I don't mean rolled up, tiny dimensions that I have read about but that some creature or device with different sensory apparatus would naturally understand our space to be 4 dimensional (not including time)?

The Pythagorean distance between two points in space involves 3 coordinates but could it, for a particle other than photons, require a fourth?

It is very easy for me to imagine a sort of example of 4D coordinates -- if you live in an apartment building than to completely describe where the apartment is requires the normal 3D coordinates but also apartment number. However, this is not really the natural 4D space I am looking for.

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The question of dimensionality of the world we live in is still an area of active research.

From the point of view of the human ability to see and perceive - the world is 3-dimensional, and this was the accepted picture of world before the Einstein's special relativity. But the way to see and perceive is a subjective property of a person. Imagine some world on a 2d plane, like in the book "Flatland" https://en.wikipedia.org/wiki/Flatland, where inhabitants are different polygons. They have ability to see and feel only the actions and events, occuring in the 2d-plane under consideration, despite the fact, that this plane is embedded in the some higher dimensional space.

However, physicts can check the validity of conservation laws and measure the strength of the forces. If one notices flux of energy, and some other quantity, which is believed to be conserved, it may be the case, that it flows into the extra-dimensions unseen.

At the classical level at least, one can check the Coulomb law for the electromagnetic and gravitational forces. The potential energy, decaying with the law of inverse power of distance: $$ V(r) \sim \frac{1}{r} \qquad \vec E = \frac{1}{r} $$ indicates, that existence of 3 spatial coordinates. The easy way to get the power law decay is from the Gauss's law.

Consider a pointlike charge $q$ and cover it with a some surface. The surface area will grow as $\sim r^{d-1}$ with the increase of radius $r$, where $d$ - is the dimensionality of space. And the relation between the total flux through the surface and the charge: $$ q \sim |\vec E| r^{d-1} $$ implies the desired power law fall-off.

However, from the point of view of superstrings, the critical dimension of the space is $D = 10$ (this is requierement of consistence of string theory - conformal invariance of the worldsheet invariance).

Nevertheless, the 10-spacetime dimensions of spacetime can be consistent with the Coulomb law, provided 6 of them are compact, such that their size is below the precision of apparatus available for us in present to figure the deviation from the Coulomb law in 3 spatial dimensions.

So in principle, the world can be $10$-dimensional, some stuff like $\mathbb{R}^4 \times Y$, where $Y$ - is some 6-dimensional compact manifold (actually it is a Calabi-Yau manifold, in order for the supersymmetry to take place https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold)

In order to check out whether it is the case or not - one needs to investigate the gravity force on a small distances. It as an are of active research since long ago. Gravity, being a wery weak force, in comparison to other interactions, present in nature is extremely weak, so the experiments have to very precise and accurate. A recent experiment in this field - https://arxiv.org/abs/2002.11761v1.

So the final answer is - that we don't actually know by far.

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  • $\begingroup$ the radiation intensity falling off in a way consistent with 3 dimensions maybe has to do with the way radiation works. my point about nothingness implies to me that the number of dimensions that "space" has is based on how you measure and perceive things and maybe there is a real way in which our space is every bit as 4D as it is 3D. $\endgroup$ – releseabe Nov 26 '20 at 19:20
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The question is not the number of dimensions, but what we do with them. Indeed, it is not uncommon for physicists to deal with 20, 30, or even 5000 dimensions. Fluid mechanics and quantum field theory are typically handled using infinite dimensional constructs.

So what we have to do is narrow in on what makes the 3 dimensions you speak of so special. Obviously we can both agree on what those dimensions are, and that they are intuitively different than other dimensions.

The most useful discriminator between them is that there are translational and rotational symmetries in these three dimensions. This means that if you take any problem you can set up, and either translate every object by the same amount, or rotate every object the same amount around a point, or both, you end up with the same result, only rotated. An ice cube remains an ice cube, even if you rotate it, or throw it across the room.

These symmetries are very interesting because of Noether's Theorem. That theorem states that for every continuous symmetry, there is a conserved value. You can show that the translational symmetry that lets you move the entire system and get the same result directly corresponds to the concept of momentum -- there is conservation of momentum if and only if there is translational symmetry. Likewise, the ability to rotate everything and get the same results corresponds to the concept of angular momentum -- there is conservation of angular momentum if and only if there is rotational symmetry.

Now if there was a 4th dimension which operated in exactly this same way, one of two things would come of it. The first possibility is that we would see it in the conservation of momentum. You would see what Dvij D.C mentions. Masses would grow faster than they do as the radius of an object increases.

The second is that there is a 4th dimension but something is fundamentally constraining us to this 3 dimensional space. There is actually no way to tell the difference, until whatever constraint in the 4th dimension releases its hold on us and everything around us. But even if this happens, we will still consider the 3 dimensions we know to be "special," at least until we discover new physics which brings this 4th dimension into this translational/rotational symmetric group.

To date, we have found no need for a 4th spatial dimension.

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There are many ways to verify that the space we live in is $3$ dimensional, at least macroscopically. One very simple way would be to do this:

Consider a region of space which lies at a distance $r$ or smaller than $r$ from a given point. Fill it up with a uniform material. Now do the same exercise for a distance $kr$ where $k$ is some positive real number. Compare the masses of the materials required in each case. You would find that the ratio of the masses of the materials in the two cases is $k^3$, thus showing that volume grows as $r^3$ in our space, i.e., our space is three dimensional. If we were to be living in a $5$ dimensional space, it would grow as $r^5$, etc.

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    $\begingroup$ You don't even need material for this argument. Consider light: if it radiates uniformly from a source (say the sun) then the intensity falls as r^2 (because the emitted energy is spread across the surface of a sphere). But if there were other dimensions that we somehow couldn't percieve, it would fall off faster because energy would be spreading in that extra dimension as well. $\endgroup$ – jamesqf Nov 26 '20 at 17:07
  • $\begingroup$ @jamesqf Sure, I just thought it would be easier to measure the mass of water than the intensity of light ;) $\endgroup$ – Dvij D.C. Nov 26 '20 at 17:12
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What troubles me is how "nothingness" can have any attributes

This problem gets away once you start seeing space as set of possibilities for different configurations of objects, not some real material entity. The "space" is mathematical abstraction. At least in physics.

In your everyday life, space is psychological abstractions. I read some observations made on very small children and it seems, they do not yet perceive space the way we do. Only by observation of many situations do we abstract away that there are only 3 main directions and space is continuum and so on. It gets so ingrained in our brain, that we usually think this is the way it obviously is, but the reality is this is not what we see directly. This is only the image our brain puts together from our sensory perceptions as the best model for putting the objects in their relation.

I think one should get rid of this notion of trying to understand objective reality in some absolute sense. Our everyday life suggests there is some objective reality, but what we really have are only raw data without any interpretation. The interpretation - be it the picture you see in your consciousness, or physics model - is only that - a model.

The questions is not "how nothingness can have any attributes" but "what are the raw data we have that lead us to this particular model of reality". In case of space, the raw data are measurements of distances between objects. Just bunch of numbers. The curious thing is, that if you have 3 objects, the 3 distances you measure are not without relation between each other. This relations between all distances is easily understood if we model universe as being 3 dimensional euclidean space (forget gravity for now). The moment we measure some distance that is not in the correct relation with other distances, we know the space cannot be 3D euclidean space and need to look for another model, if such even exists.

The Pythagorean distance between two points in space involves 3 coordinates but could it, for a particle other than photons, require a fourth?

It could, but as far as we know, it doesn't.

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This is a philosophical question.

We do not even know for sure if we are really real or if we are a computer game being beamed in by holograms from some other universe.

The universe as we seem to see it is a ten dimensional brane on an 11 dimensional object. Six of the dimensions are not visible to us. We always see three of the dimensions and some of us know/feel the 4th dimension of time which we can see in some physics experiments.

A few of us mathematicians can envision things in 4D. Most of us, but not all of them, have problems seeing things in their mind that have higher dimensions.

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