What are dimensions?

Answer 1

There are different concepts:

1) At the most general level, "dimensionality" simply refers to the number of variables one needs to specify the basic structure of a system of some kind, whether this be a physical, biological, sociological, linguistic system, etc. Mathematically, a structure is an n-tuple, an ordered set minimally composed composed of elements and relations/functions between those elements. These elements correspond to objects studied, and relations/functions that the relations between these elements.

To give such a structural description is understand the problem space of a system at some level of description, and so at different levels of specificity. The same thing can be studied at different levels of description and as having different numbers of dimensions, depending on the problems we want to solve and study.

For example: To specify the problem-space of a 'traffic light system', for example, we need to consider the functional correlations between a set of possible colors (red, yellow, green) and a set of vehicle rules (stops, slows down, moves), so we need a three dimensional structure with three variables: <color, function, rules). The number of dimensions we need to specify the system is therefore relative to the domain and kinds of problems we want to solve. For example, the physical properties concerning the wavelengths and frequencies that determine color differences are irrelevant to the phenomena being studied for traffic light systems.

2) In fundamental physics, the dimensionality of a physical system refers the number of variables that organize the general structure of entities in space and time. The physical world is conceived of having three spatial dimensions which are determined in relation to a fourth, temporal dimension. These four dimensions specify necessary determinations of all physical systems. So physical reality is composed of different values and relations between those four variables: the variable length, width, depth, and temporal determinations of physical beings. Thus we speak of "four dimensional spacetime" as the general structure of the physical world considered as a system.

So, if you think formally of a system with four variables:

S: <x,y,z,t>

you can think of a physical system as

P: <length, width, depth, tense).

Hopefully that helps!

Answer 2

one dimension is the real number line, measuring distance or length, with linear equations, with no exponent greater than one. It's how polarized political views are, just left and right, far left and far right. Nothing in the middle, but one infinitely small point. Everyone is on one side or the other. No real middle ground.

two dimensions are the cartesian coordinate system, with x and y axes. "Flat land" as Big Bang's Cal Tech's theoretical physicist Professor Sheldon Cooper would call it. Just length & width, measuring area, with quadratic equations, with no exponent greater than 2. In pictures, they're stick figure people, more cartoonish than real. If the discriminate is negative, there is no real solution, just 2 imaginary solutions.

three dimensions are length, width, depth with 3 axes, x,y and z each orthogonal to the other, measuring volume. It's the space we live in, described with cubic equations, and no exponent greater than 3. While solvable, the 3 D equations are more difficult to solve.

four dimensions are length, width, depth and time. It's the world of Einstein's relativity. described with equations of 4th degree, exponents not greater than 4. Graphically it would be a hyper cube, that we have trouble visualizing. While negative changes in length, width and depth are possible, time seems irreversible, going only in the positive direction. Time travel goes forward but cannot go back in time to change the past.

Higher dimensions involve equations that are difficult to solve. Physically they correspond to string theory, with dimensions we cannot visualize. It's an alternative universe beyond space, time and sound, like the Twilight Zone or what goes on in a Black Hole, or imaginary time before the Big Bang, with alternative laws of physics. Energy replaces material objects with uncertainty and Schrodinger cats and other paradoxes.