Question

Meaning of N-Dimensional

Answer 1

this is a side point to your post, but dimensionality depends on context. if we're looking at where we are spatially in a volume, color isn't a dimension. but, if I'm trying to solve a question about color spaces, it might be useful to consider three dimensions of color like redness / greenness / blueness, or four like cyanness / magentaness / yellowness / darkness. in fact, even spatial dimensions aren't always x/y/z; in polar coordinates, you'd instead use radius, horizontal angle, and vertical angle to identify a point.

it may be easier to think of dimensions just as degrees of freedom instead of trying to make an analogy to spacetime. a dimension is just an axis that represents some variable that can be changed. space is an easy one to understand because we deal with it all the time, but any independent variable can be a dimension in whatever system you're looking at.

Answer 2

Usually dimension refers to the number of basis elements in a vector space. This can be either finite or infinite. The study of finite dimensional vector spaces is linear algebra and the study of infinite is functional analysis.

Answer 3

Most times the number of dimensions just refers to the number of variables in a certain function. For example, to get the surface area of a patch on a function f(x,y,z,w,r) assuming it is smooth and differentiable, we need to integrate 5x to account for the 5 different forces contorting the surface.

Answer 4

Let me introduce a word you may know in a way you might not: Orthogonal.

In school, you might learn "orthogonal means perpendicular," which is true in a sense but it gives you a bad idea of what it means, because it implies that something is perpendicular in physical, 3D space. Orthogonal actually means "this is a new, independent thing that doesn't depend on or influence any other thing in this system." Many non-physical things are orthogonal: my choice of hat to wear is orthogonal

When we talk about colors, we can say that RGB "space" is 3-dimensional because it is created from three orthogonal pieces of information, namely the red "axis," the green "axis" and the blue "axis." Likewise, RGBA "space" is 4-dimensional: it has an additional alpha "axis."

I can talk about this in many different contexts: let's say I wanted to decide what videogame I wanted to play, so I picked the following 5 things: I wanted it to be multiplayer, I wanted it to be medieval themed, I wanted it to be a role-playing game, I wanted it to have a survival mode and I wanted it to have a fishing minigame. For all of the videogames I can buy, I can formulate some way of assigning a 5-dimensional coordinate to each of them, for example Elden Ring could be {20, 100, 100, 0, 0} - quick note, the exact numbers don't actually matter for this example. What I now did was construct a 5-dimensional videogame space in which I can assign any given videogame a point.

Does this make sense? I don't know if I'm making sense or rambling, lol - basically, dimensionality doesn't necessarily mean physical special (or even time) dimensions. We are perfectly capable of understanding much higher dimensional spaces when they have to do with other things like videogames, or buying a house, or watching a movie. We just seem to have trouble picturing geometric shapes and things in higher dimensions than 3, but that is just one specific application of a very general notion in math.

---

You said elsewhere in the post that you haven't taken any linear algebra classes - if I can proselytize for a moment, I would highly suggest you try it if you have the ability! It's an extremely beautiful field of math, and it will completely change the way you see many things. It's the "father" field to many important concepts in things like computer science, and thus is very relevant to many things we rely on these days. I think you'd enjoy it, given what you seem to be interested in learning about by asking this question!

Answer 5

Time is still a dimension in Newtonian physics. The difference is that we don't mix space and time coordinate together, ever, so you can safely think of 3 dimensions of space in one context, and separately 1 dimension of time in another context. But in relativity, space and time are intertwined, so you are forced to deal with all 4 dimensions at the same time.

What I mean when I say the 4 dimensions are mixed together? The same way you can say the 3 dimensions of space are mixed: if you have an object that has horizontal length, you can simply move your eye and turn that length into height or depth.



But in general, dimensions refer to degree of freedom. For example, if you have a physical system in some state, how many possible independent way its state can change? In statistical mechanics/thermodynamics, you consider a "solid" object as being made out of bazillions number of atoms, each of them has 6 numbers associated to them. That's an bazillions dimensions: 6 x numbers of atom - number of equations that relate them. The equations that relate these numbers are a few laws like conservation of energies and momentum, which are only a handful.

The precise technical definition of dimension is more complicated, because there are so many different types of dimensions. They all stem from the same ideas of degree of freedom (n independent variables = n dimensions), but the exact mathematical forms varies based on what you're talking about:

- In vector space, it's about how many vectors are needed to make a basis.

- In geometry setting (including talk about spacetime), it's about number of tangent vector to make up a basis that describes all possible movement.

- In fractal, Hausdorff and packing dimensions talk about how fast their volume grows, when you use "pixel" to measure volume, as the size of the pixel changes. Here, dimension is not even a natural number!

...and there are many more, it's hard to list them all.

Answer 6

There are many different mathematical definitions of dimension, some of which are equivalent or have significant overlap, some not so much. They mostly concern what are generally referred to as "spaces", though this is an informal term broadly denoting a set of elements, often but not always referred to as "points", with additional structure added to the set that relates the points to one another. Some specific examples of "spaces" with dimension are affine spaces, vector spaces, and certain topological spaces.

Broadly speaking, the most familiar definitions of dimension have the same basic idea: in certain spaces, points in that space can be specified by some number of independent values. The dimension of such a space is then how many such values one needs for each point. For example, points in the plane can be described by cartesian or polar coordinates (though these are not the only coordinate systems for the plane), each point requiring two real numbers. Hence, the plane is two dimensional. The independence of the values is really important here. For example, the unit circle in the plane consists of points which can be described by their (x, y) coordinates, but these coordinates are not independent. They must satisfy x² + y² = 1. The circle is, in fact, one dimensional, as you could, for example, specify any point with a single number, e.g. an angle from the positive x-axis.

These notions of dimension are entirely abstract. When describing the dimensionality of real world things, what we're really doing is describing the dimension of some space which we are using as a model for some real world concept. How we interpret the real world meaning of a point or the values used to specify that point is entirely arbitrary. So if you want points to represent physical places and times (events), in most situations four real numbers are sufficient and you have spacetime, which physicists model as a kind of space called a (4D, Lorenzian) manifold. If you want points to represent a color of an RGB light, then you have a 3D color space. If you want points to represent both an event and a color of an RGB light at the same time, then you could use a 7D space. What makes one independent real-world quantity a dimension and another not really just boils down to what you are modelling and how you choose to model it.

Answer 7

Also, we don’t only “know” four dimensions to exist. New fields in physics posit 11 dimensions (string theory), and many of our basic theorems from math work in infinite dimensions by induction (Stokes theorem for example)