QUESTION: I was wondering if dimensions are ever equivalent to one another or if they never are? Assuming I have a square that is 2 units by 2 units. It is 4 square units of space. And if I have a cube that is 2 units by 2 units by 2 units, which would be 8 cubed units of space. Is the cube like any other 3D shape in that it is infinately more space than any 2D shape? Maybe each 2 x 2 square slice is 1/infinity thick and therefore you would need an infinite number of them to fill up a cube? Or would you have to assume that something in 2D has no Z axis, which just as easily would mean you would need an infinate number to reach 3D space? I ask this question because some people believe that 3D objects when entered into a blackholes event horizon, all the 3D information is now spread over a 2D space, and somehow preserved that way. Is there anything that makes higher and lower dimensions equal to one another?
ANSWER: There are a few ways in which to approach a question of dimension.
I will share two reasons why lower-dimensional objects are not
equivalent to higher-dimensional ones. There are ways -- sometimes -- in which the opposite could be argued, but it is more reasonable to believe that an object that has two "dimensions" can never be as "big" as one what has three.
First, and foremost, the notion of "dimension" in mathematics is complicated. There are different ways of defining dimension. The most reasonable way, however, is the one you mention. Two dimensional shapes have area
while three dimensional shapes have volume
Area is a measure of two linear dimensions, each of which contributes one dimension of length. For that reason, we would use something like cm² to measure area.
Volume is three-dimensional measure, and would be something like cm³
If we had a shape that was a square, say it has area 1 cm², then it has zero volume.
You see, the square can exist
in three dimensions, but it is flat. For that reason, it's volume is (1 cm) × (1 cm) × (0 cm) = 0 cm³. A two dimensional object has zero volume.
There are tricky mathematical understandings of "area" and "volume" (which are concepts in a field called "measure theory") that make this matter subject to some debate. Sometimes a weird object (one that would not exist as a real life object!) seems to be three dimensional, but has no volume. Such a set demonstrates that, although the idea of distinguishing "area" and "volume" as two- and three-dimensional properties (respectively) is good, it is not perfect.
Another way is to consider the dimensionality of an object in whatever space. In real life, we have three dimensional space. Every object has length, depth, and width, meaning that every real life object is three dimensional.
If we had a perfectly flat object in real life, it would have only length and width. This makes it two dimensional. It would exist in what is called a subspace
of our three dimensional world. If you had an object that was a perfectly thin line, it would be a sort of perfect one-dimensional object.
This is a concept from linear algebra -- the idea that directions like "length" and "width" have to do with the dimension of an object (or, more precisely, the space that object occupies). Even for objects that are bent, like this perfectly flat object if it were curved, we can say that it is (effectively) still two dimensional, even though it has been bent.
There is a book titled Flatland
which has a very interesting narrative about the life of lower-dimensional beings. It is a clever work of fiction that explores interesting mathematical notions.
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QUESTION: Might it be safe to assume that anything below 3 dimensions doesn't really exist? 1D and 2D as well as straight line in general are simply concepts used to measure things in a simplistic way? However the same the concept of dimensional inquality could be extended into the 4th dimension and up?
ANSWER: It really depends on what you mean by "exist."
First of all, real life is not math. Dimension is a mathematical term, not a real life term. Electrons exist, you exist, but "points" and "lines" and "dimension" do not. They are mathematical terms that are used to represent the world around us.
Now, for basic physics -- which uses math to describe
the world around us. In that model, the world is three dimensional. No objects are two- or one-dimensional.
Oddly, it is often easier to pretend some objects are two-, one-, or even zero-dimensional for the purposes of making certain computations or formulations of laws of physics. However, the objects are (nonetheless) assumed to be three-dimensional according to this type of physics.
In more advanced physics, it is not clear how many dimensions there are -- some of them might be the three we like to use in basic physics, but there may be more (up to 10 or 26 dimensions, perhaps). In those models, objects would have more dimensions.
But if you believe basic physics is "real life" then yes, indeed, all objects are three dimensional. Paper is three-dimensional, even if it is very thin. Electrons are three-dimensional, even if it is very small.
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QUESTION: Although there are many models of the universe or even (multiverse) type ideas. I think string theory gives 11 dimensions that seem to neatly cover all possible events as well as impossible. Whether or not time is the "4th dimension" of space or not, or whether time takes up more than 1 dimension (more complex ideas suggest it could be 2 dimensional) I assume that at least in math 3D space could never really equal anything in 4D space?
That's the short version -- four dimensions, three dimensions, two dimensions -- they are all mathematically different spaces. Sure, you can imagine an infinitely flat square in three dimensions, but that's not a real object -- it has no volume. It's not a three-dimensional object.