Mass in 2d and 3d

[AmitX123]

Geometrically 1d, 2d, 3d, etc only imply geometric quantities (length, area, volume, etc)
A circle has no mass but a disc (circle with some thickness) has.
Where does the idea of mass come from? Does it imply some thickness in a higher dimension?

[Keiji]

Well, mass is essentially a count of particles such as neutrons and protons.

Those particles are three-dimensional in our universe, which means that for anything to have mass, it must be three-dimensional in order to contain them.

In a two-dimensional universe, the fundamental particles would probably be two-dimensional thus giving them mass too.

[quickfur]

There's a difference between rest mass and effective mass.

Rest mass is your "ordinary" mass, basically a count of particles that comprise the object (or more precisely, the sum of the rest masses of said particles).

Effective mass is rest mass plus part of the kinetic energy that an object acquires when it's moving close to the speed of light. Well, actually, this energy is always there, but in normal situations it's so small that it's negligible. But when you get to speeds close to the speed of light, this "extra mass" starts becoming noticeable. The faster you move, the "heavier" you become.

Also, at the subatomic level, mass is one of those things that is not so easy to analyze. Subatomic particles have an inherent rest mass that is completely independent of its "size" and "shape". It's sorta like energy that's been "bound up" inside the particle. A subatomic particle actually has no real "size" or "shape"; an electron, for example, is smeared across a very large area around the nucleus, like a cloud, whereas the protons in the nucleus actually overlap with each other and are compressed into a tiny dot. Nevertheless, the rest mass of the electron is far, far smaller than the rest mass of a single proton, even though in terms of its effective volume, it's thousands of times "larger". Furthermore, electrons cannot be broken down into smaller particles so you can't really define its mass by counting its constituent parts... it just has rest mass, an inherent mass that doesn't come from anything else. Protons can be reduced into quarks, although it's arguable whether quarks can exist freely or are merely mathematical entities... but regardless, at the quark level again it just has a rest mass, which is an inherent property that cannot be ascribed to any constituent parts or "volume" or anything like that.

Furthermore, when a particle collides with an antiparticle, they annihilate each other and their rest masses are converted into energy in the form of photons. This is Einstein's famous equation E=mc²: the mass gets multiplied by the square of the speed of light, and that's the amount of energy that's released. So in some mysterious way, the rest mass of a particle is just an inherent amount of energy that's "locked up" inside the particle, which exerts its effect by affecting how the particle responds to outside forces. So if you really want to boil it down, this is what mass is really all about. You can say that it's just the energy that makes up the particle itself.

[AmitX123]

Well, mass is essentially a count of particles such as neutrons and protons.

Those particles are three-dimensional in our universe, which means that for anything to have mass, it must be three-dimensional in order to contain them.

In a two-dimensional universe, the fundamental particles would probably be two-dimensional thus giving them mass too.

The fact that mass needs 3 dimensions may not be entirely true. A blackhole point singularity has 0 dimensions and a ring singularity 1 dimension, but both have mass.

[quickfur]

Well, mass is essentially a count of particles such as neutrons and protons.

Those particles are three-dimensional in our universe, which means that for anything to have mass, it must be three-dimensional in order to contain them.

In a two-dimensional universe, the fundamental particles would probably be two-dimensional thus giving them mass too.

The fact that mass needs 3 dimensions may not be entirely true. A blackhole point singularity has 0 dimensions and a ring singularity 1 dimension, but both have mass.

Not to mention that we don't know if truly fundamental particles like the electron are three-dimensional. For all we know, they could be actual 0-dimensional points, since we have not been able to discern any substructure to them.

[gunnar98]

The fact that mass needs 3 dimensions may not be entirely true. A blackhole point singularity has 0 dimensions and a ring singularity 1 dimension, but both have mass.

A ring with 1 dimension? Isnt that impossible seeing that everything has to have 3 dimensions, when you write on a paper, graphite takes up a fraction of a cm making a 3rd dimension even a tv has 3 dimensions light being the width. Also a ring as in a circle? I dont think that is possible seeing that a ring has to have a height and length to create a round side meaning a 2nd dimension. But also it could just be a 2d ring for in space it depends on your reference point that all objects are 3d. Concluding that mass might need to be 3d in order to measure, also again, it depends on your reference point. P.S. Im 13.

[quickfur]

The fact that mass needs 3 dimensions may not be entirely true. A blackhole point singularity has 0 dimensions and a ring singularity 1 dimension, but both have mass.

A ring with 1 dimension? Isnt that impossible seeing that everything has to have 3 dimensions, when you write on a paper, graphite takes up a fraction of a cm making a 3rd dimension even a tv has 3 dimensions light being the width. Also a ring as in a circle? I dont think that is possible seeing that a ring has to have a height and length to create a round side meaning a 2nd dimension. But also it could just be a 2d ring for in space it depends on your reference point that all objects are 3d. Concluding that mass might need to be 3d in order to measure, also again, it depends on your reference point. P.S. Im 13.

It is useful to note the difference between the inherent dimension of an object, versus the dimension of the space that contains it. For example, a point is 0-dimensional: it has no length, width, or height. Even though a point singularity may exist in 3D space, the 3D space does not make the point have any more dimensions than 0. It's still a 0-dimensional object; it just happens to be embedded in 3D space. The same point can be embedded in 2D space, or 4D space for that matter, and it's still just a 0-dimensional point.

In the same way, a ring is a 1-dimensional construct. To specify a point on it, only a single parameter is needed: angle. Or, if you like to think in terms of measurement, the ring is fully specified by its circumference. A ring is interesting because even though it is 1-dimensional in itself, you cannot embed it into 1D space (at least, not into Euclidean 1D space), because the topology of a line is different from a ring; the ring simply doesn't fit in it. But this doesn't mean that the ring itself is any more than 1-dimensional; it is still a 1-dimensional object. It just happens to need more than 1 dimension to be embedded in. A ring that exists in 3D space is still a 1-dimensional object. Just because its ambient space (the space it's sitting in) is 3D, doesn't make the ring magically acquire 2 more dimensions. You can stick the same ring into 4D space, and it's still just a 1-dimensional object.

Now, drawing a ring on paper obviously has more than 1 dimension, because your pencil makes lines with a non-zero width, and the graphite on the paper has non-zero thickness as well. But that is merely a representation of the ring. You shouldn't confuse the picture of an object with the object itself. For example, a photo of you is not the same thing as the real you. An actual mathematical ring has zero width and zero thickness; the only thing it has is circumference. So it's a 1-dimensional object.

[Keiji]

I shall just take the opportunity to point out that what you call the "inherent dimension" of an object I call its net space and have done so for years. This is as opposed to the bounding space, which is the number of dimensions of space it can be embedded in.

[quickfur]

I shall just take the opportunity to point out that what you call the "inherent dimension" of an object I call its net space and have done so for years. This is as opposed to the bounding space, which is the number of dimensions of space it can be embedded in.

Wendy's Polygloss uses the term "all-space" for this. Just sayin'. The point is that there is a difference between the number of dimensions in the object itself and the dimension of the ambient space.

[Keiji]

Yep, that's fine. I was just throwing that out there in case gunnar98 wants to research on the wiki, which frequently mentions it.

[gonegahgah]

Not to mention that we don't know if truly fundamental particles like the electron are three-dimensional. For all we know, they could be actual 0-dimensional points, since we have not been able to discern any substructure to them.

Sadly, I think science does say that. It's probably like the whole 0.9... = 1 silliness. That is that the particles are so tiny and close to being points that they can be considered to be points therefore. But I don't think that that is fair to the particles.