size of dimensions

[papernuke]

Are 1D dimensions a just an infinetely long line of atoms? or is it just infinitely small?

Same for the 2D. ?

[houserichichi]

Depends who you talk to. To a mathematician, for the most part, a 1D piece of a 3D world is an infinitely small, infinitely long, continuous "thread" so to speak. There is a "point" at every possible place on that line and then there are infinitely many "points" between that place and the next. Thus we say that the 1D real line is dense, that is there are infinitely many points between every two points on that line.

If you now take this 1D line and extend its definition to 2D so that there are infinitely many points in both the "x" and "y" directions then you have an infinitely dense 2D plane. There is always an infinitude of numbers between any two points in that plane no matter how close those two numbers you choose are. Same goes for 3D, 4D, ..., etc.

There are other 1D lines that don't have this property to a mathematician. If you can "count" (or at least enumerate) each point on the 1D line then we say that the 1D line is countably infinite but not dense. The line of integers is like this...you can make a line, put a digit (1, 2, 3, 4, ..., 1 million, ..., etc) at every point but there will always be holes between, ie: between 1 and 2 there is 1/2 but that's not an integer so it's not on the 1D line.

If you extend this reasoning to 2D you get what's called a 2D lattice. There are a countably infinite number of points in both the "x" and "y" directions but there will be holes between any two points you choose in any direction. The exact size of these holes is determined by some rather abstract mathematical constructions.

If you talk to a physicist there are two schools of thought. To a relativist you talk of great expanses of spacetime like a continuous mathematical manifold; that is, a continuous "shape". Thus, relativity simplified treats spacetime as our first mathematical space above: It's infinitely dense in all directions. Thus spacetime, in (1+1)D, ie: 1 space and 1 time dimension, is our first example above.

To a quantum physicist the universe is an ugly "foam" at the small scales. It can even be argued that in some circumstances the quantum scale is just that, quantum...space comes in little packets when you get real small. It's still infinitely long but it's not dense; there are holes in space. Those holes are regulated by something called the Planck length in which there is no theoretical model (that I'm aware of) that can explain space on such small scales, thus the universe is not dense that small, thus space is like the second mathematical model above; it has holes between points. What's in these holes? Science doesn't pretend to have the answer as far as I know.

I'm studying quantum field theory as we speak which is an amalgamation of special relativity and quantum mechanics. As to what it says I'll get back to you.

[wendy]

A thing of one dimension is as thick as one of three dimensions, or eight dimensions, when the view is taken around (not in the space of) the fabric.

In terms of solids in these spaces, (ie living in the relevant space), the thickness orthogonal to the space has no bearing on what happens in the space, so it is a mute point to talk about whether 3d space is 1 inch or 1 mile thick. These lines are realised as a single point of the fabric, and behave as such.

The polygloss makes a distinction between, eg "hedrid" = 2d-like + exact vs hedrous (2d-like + approximate). In terms of our real world, a peice of paper is essentially "hedrid", because it fairly accurately represents a hedrix. A mattress is "hedrous", because while it has a large 2d extent, it has a sizable depth around the hedron (2d patch) that forms the sleeping surface.

[zero]

Two comments on the above posts:

First, for wendy, thanks for providing such a helpfully explanatory review of the relevant terminology.

On a more trivial note, I don't think physicists universally perceive quantum "foam" as being ugly. It's a subjective point of view that people may easily see differently. One may even distinguish discrete and continuous conceptualizations without attributing negative feelings to either of them.

[wendy]

I don't necessarily follow the physics, but for what it is worth:

Distances at the order of smaller than the planck lengths, are generally unresolvable, although these must exist.

I heard even wilder theories about the nature of space/time, in as far as that it is not extensive but intensive. That is, space-time is largely a kind of property held by individual co-existant quanta. There are some rather frightening ideas one must untangle to get this right.