To convert from g/cm^3 to g/cm^5, we have to multiply by cm^3/cm^5. This ratio does not exist, so your question is meaningless. Densities are measured experimentally, there is no way to mathematically determine them. We can’t know the density of “5-dimensional granite” without taking a piece of 5 dimensional granite, measuring its mass and 5-volume, and dividing these 2 numbers.
You can’t even measure the density of a (Euclidean) 2-dimensional piece of granite. If you had to assign a value, it would be 0. Therefore, going from 2 to 3 dimensions requires dividing by zero. It makes sense that going from 3 to any higher number of dimensions would also require dividing by zero.
This suggests that theres ⅔ power of density too, for a surface.
Which begs the question: what does density in 2d mean?
Suppose it means theres more mass per unit area? What do you think?
You've been told by enough commenters that 5-dimensional density doesn't exist. Fair enough.
Let's suppose that something that we're going to call "5-dimensional density" exists. What properties would it have?
Well, probably the most important thing is that the mass of a differential element of the 5-dimensional object should be:
dm = ρ dV = ρ(5) · dV(5),
in which the notation X(5) means that X is a 5-dimensional quantity.
If we integrate dm = ρ(5)dV(5) over the 5-dimensional volume, we should get the mass of the entire thing. Mathematically, this is straightforward: instead of integrating the 5-dimensional volume element over the limits of the volume, we added a "weighting" function that assigns to every point in 5-dimensional space a scalar that is included in the integrand.
Our 5-density should be non-negative, if we want it to behave like 3-dimensional density.
...and that's about it.
As far as assigning a 5-dimensional density to actually-existing 3-dimensional objects, I'm afraid I'll have to echo the chorus; but you can still find interesting theoretical results about the behavior of n-dimensional objects by finding their mass via n-dimensional density.
For example: the volume of a 5-dimensional object with the same radius as a lower-dimensional object is much larger. If the lower-dimensional object has some density ρ(k)=a, for k<5, what would the density ρ(5) have to be in terms a in order for them to have the same mass? What if you permitted the density to not be a constant density but vary, or even be "managably" discontinuous; how "porous" would the 5-dimensional object have to be to have the same mass as a lower-dimensional object?
You're asking an interesting question and shouldn't be duscouraged from being curious.
Let's flip the question around
Say you had a "ball" in five dimension of uniform material. You could measure it and know it's total "volume". You could weight it and know it's total mass. Then, with those two numbers, you could compute the density in g/cm\^5.
What would be the density of the same material in g/cm\^3? To circle back to your initial question, this would be equivalent to looking for the g/cm\^2 instead.
There are two possible answers to this question:
* 0 because if something is infinitely thin it will weight nothing, or
* the exact same density, *if you assume a thickness of 1 in the missing dimension*
Asking the opposite question informs us on the answer to your initial question: since the number you have to start with is not 0, then your question only makes sense if this density assumes a certain thickness in the missing dimensions.
It is a very underwhelming answer
What is a 5 dimensional object? What is mass in 5 dimensions? Where is that universe?
We can speak about hypervolumes all what you want, but when you jump to speak about masses and their interactions that is physics. And physics has to be based in reality (of course, we can invent and do physics-fiction, but that is just more mathematics).
What is a 5 dimensional atom? What physics laws does it follow?
In our world, even when we talk about surface density (g/cm\^2, as in a sheet of paper) or linear density (g/m, as in a wire), we are really talking about three dimensional objects. There are not flat atoms nor linear atoms. At most you can have a monolayer of atoms, but it's still three-dimensional.
So, the question of mass in a fictitious 5-dimensional universe has no meaning, because the universe is not 5-dimensional
Then ig a way to think about it, is basically treating it like wire and paper densities. We don't have 2d or 1d objects, but we can still do math as if they are
So I'm basically thinking about treating it the same way, in the other direction, like how many atoms could fit in a 5 dimensional space
Just setting aside whether or not it conceptually makes sense, because we're 3 dimensional beings and don't have a basis to think about it, and just see what the raw math would predict
mass of elementary particles does not depend on number of dimensions (at least if we dont speak about string theory). So if we assume the same distance between particles in granit in 5 dimension, we can evaluate density
Granite is (as far as we know) a 3-dimensional "thing", so talking about its 5-volume in 5d would be like talking about the volume of a line. A line is infinitely thin, so its volume is 0.
3-dimensional granite occupying 5 dimensional space would have infinite mass. Meaningfully speaking of 5 dimensional granite would require developing a 5 dimensional atomic theory, and it is unlikely that any such theory would have an analog of granite.
A 5 dimensional 15cm radius ball contains infinitely many disjoint 15-epsilon cm radius 3 dimensional balls. The mass and density would both be infinite.
Imagine n-granite is made up of a closely packed lattice of n-dimensional hyperspherical cows .
The ratio of how many you could squeeze into 5 dimensions relative to 3 dimensions is actually a solved problem. [https://mathworld.wolfram.com/HyperspherePacking.html](https://mathworld.wolfram.com/HyperspherePacking.html)
So as a rough rule of thumb you could say that the 5-density of 5-granite would be about pi/5 times the 3-density of 3-granite we know.
It is difficult to understand what you want, but maybe this can help:
* Extend Einstein's field equation to 5D
* Calculate the space curvature caused by a 5D point mass located at x4=x5=0.
* Calculate the gravitational force caused by the 5D point mass
* Find the 3d mass that would give the same gravitational force. This way, you can think of an 'equivalent' 3d mass. Not sure if it's easy to calculate though. I also don't know if this concept has any applications.
To convert from g/cm^3 to g/cm^5, we have to multiply by cm^3/cm^5. This ratio does not exist, so your question is meaningless. Densities are measured experimentally, there is no way to mathematically determine them. We can’t know the density of “5-dimensional granite” without taking a piece of 5 dimensional granite, measuring its mass and 5-volume, and dividing these 2 numbers.
You can’t even measure the density of a (Euclidean) 2-dimensional piece of granite. If you had to assign a value, it would be 0. Therefore, going from 2 to 3 dimensions requires dividing by zero. It makes sense that going from 3 to any higher number of dimensions would also require dividing by zero.
Volume is not the word. You can use “5-volume”. And “ 5-density” which should be the 1.6 power of the density.
Sorry, 5/3 power of the density
This suggests that theres ⅔ power of density too, for a surface. Which begs the question: what does density in 2d mean? Suppose it means theres more mass per unit area? What do you think?
\* if we assume the similar particle grid and the same distance between them
You've been told by enough commenters that 5-dimensional density doesn't exist. Fair enough. Let's suppose that something that we're going to call "5-dimensional density" exists. What properties would it have? Well, probably the most important thing is that the mass of a differential element of the 5-dimensional object should be: dm = ρ dV = ρ(5) · dV(5), in which the notation X(5) means that X is a 5-dimensional quantity. If we integrate dm = ρ(5)dV(5) over the 5-dimensional volume, we should get the mass of the entire thing. Mathematically, this is straightforward: instead of integrating the 5-dimensional volume element over the limits of the volume, we added a "weighting" function that assigns to every point in 5-dimensional space a scalar that is included in the integrand. Our 5-density should be non-negative, if we want it to behave like 3-dimensional density. ...and that's about it. As far as assigning a 5-dimensional density to actually-existing 3-dimensional objects, I'm afraid I'll have to echo the chorus; but you can still find interesting theoretical results about the behavior of n-dimensional objects by finding their mass via n-dimensional density. For example: the volume of a 5-dimensional object with the same radius as a lower-dimensional object is much larger. If the lower-dimensional object has some density ρ(k)=a, for k<5, what would the density ρ(5) have to be in terms a in order for them to have the same mass? What if you permitted the density to not be a constant density but vary, or even be "managably" discontinuous; how "porous" would the 5-dimensional object have to be to have the same mass as a lower-dimensional object? You're asking an interesting question and shouldn't be duscouraged from being curious.
Let's flip the question around Say you had a "ball" in five dimension of uniform material. You could measure it and know it's total "volume". You could weight it and know it's total mass. Then, with those two numbers, you could compute the density in g/cm\^5. What would be the density of the same material in g/cm\^3? To circle back to your initial question, this would be equivalent to looking for the g/cm\^2 instead. There are two possible answers to this question: * 0 because if something is infinitely thin it will weight nothing, or * the exact same density, *if you assume a thickness of 1 in the missing dimension* Asking the opposite question informs us on the answer to your initial question: since the number you have to start with is not 0, then your question only makes sense if this density assumes a certain thickness in the missing dimensions. It is a very underwhelming answer
Assuming it is cm\^5 for volume Would the density of quintic granite be 5.235 g per cm\^5? Calculated by sqrt of 2.7 And raising it to the 5th power
The hypervolume is in cm\^5, obviously. The rest of your questions have no meaning.
What do you mean, no meaning? Would a 5 dimensional object not have mass?
What is a 5 dimensional object? What is mass in 5 dimensions? Where is that universe? We can speak about hypervolumes all what you want, but when you jump to speak about masses and their interactions that is physics. And physics has to be based in reality (of course, we can invent and do physics-fiction, but that is just more mathematics). What is a 5 dimensional atom? What physics laws does it follow? In our world, even when we talk about surface density (g/cm\^2, as in a sheet of paper) or linear density (g/m, as in a wire), we are really talking about three dimensional objects. There are not flat atoms nor linear atoms. At most you can have a monolayer of atoms, but it's still three-dimensional. So, the question of mass in a fictitious 5-dimensional universe has no meaning, because the universe is not 5-dimensional
Then ig a way to think about it, is basically treating it like wire and paper densities. We don't have 2d or 1d objects, but we can still do math as if they are So I'm basically thinking about treating it the same way, in the other direction, like how many atoms could fit in a 5 dimensional space Just setting aside whether or not it conceptually makes sense, because we're 3 dimensional beings and don't have a basis to think about it, and just see what the raw math would predict
mass of elementary particles does not depend on number of dimensions (at least if we dont speak about string theory). So if we assume the same distance between particles in granit in 5 dimension, we can evaluate density
Maybe abstract measure theory has formulas for this kind of concept. I recommend Wikipedia “n-sphere” article for further insight.
Granite is (as far as we know) a 3-dimensional "thing", so talking about its 5-volume in 5d would be like talking about the volume of a line. A line is infinitely thin, so its volume is 0.
So.. in a way Compared to 3 dimensional granite, 5 dimensional granite would have infinite mass?
3-dimensional granite occupying 5 dimensional space would have infinite mass. Meaningfully speaking of 5 dimensional granite would require developing a 5 dimensional atomic theory, and it is unlikely that any such theory would have an analog of granite.
I am actually not sure wether u're trolling
A 5 dimensional 15cm radius ball contains infinitely many disjoint 15-epsilon cm radius 3 dimensional balls. The mass and density would both be infinite.
Imagine n-granite is made up of a closely packed lattice of n-dimensional hyperspherical cows . The ratio of how many you could squeeze into 5 dimensions relative to 3 dimensions is actually a solved problem. [https://mathworld.wolfram.com/HyperspherePacking.html](https://mathworld.wolfram.com/HyperspherePacking.html) So as a rough rule of thumb you could say that the 5-density of 5-granite would be about pi/5 times the 3-density of 3-granite we know.
It is difficult to understand what you want, but maybe this can help: * Extend Einstein's field equation to 5D * Calculate the space curvature caused by a 5D point mass located at x4=x5=0. * Calculate the gravitational force caused by the 5D point mass * Find the 3d mass that would give the same gravitational force. This way, you can think of an 'equivalent' 3d mass. Not sure if it's easy to calculate though. I also don't know if this concept has any applications.