What is a Dimension?

Answer

Dimension is a fairly extensive concept. I’ll start with something you’re familiar with: the XY plane
– to define a point unequivocally you need a “base”; You first choose a point as the origin, then choose two axes, usually with equal units and perpendicular, although not necessary. You can also say: I choose a non-zero vector V1, and then a non-zero vector V2. Every other point can now be reached as a sum of multiples of those two vectors. That’s why you call those two a “base”.

∀ P : ∃! (a, b) such that P = a V1 + b V2
The two numbers a and b are then the coordinates of P in that basis; The dimension of that plane is then 2 because two vectors are enough to form a basis, or else because two coordinates are enough to make a point.

Similarly, XYZ space has dimension three

The dimension in these cases is the number of coordinates, the number of freely selectable numbers that you need as a minimum to define an element of the set. So you can say that coordinates are degrees of freedom.

Now a little further: take the surface of the earth: that is a spherical surface. What is the dimension of the Earth’s surface? That is asking, how many numbers do I need to unambiguously record a point?
The dimension is therefore two as you can see from the concepts of (western or eastern) longitude and (northern or southern) latitude.
A straight line in itself is 1-dimensional, because on a straight line you only need a coordinate to determine your position (after choosing a zero point and a unit vector)
So also a circle is in itself 1-dimensional because once you have chosen the position “angle = 0” and the sentence, each point of the circle can be unambiguously determined by 1 single number, namely an angle.

A little further
Take the system
x + y + z = 8
2x – z = 6

the solution is x = 3 + k / 2 ; y = 5 − 3k / 2 ; z = k
the solution space contains 1 freely selectable parameter and is therefore 1-dimensional. Geometrically, these infinitely many solutions lie on a straight line.

Suppose you need to place and aim a spotlight somewhere in the room. What is dimension of the set of all possible positions and directions? The dimension 5 : first the location (x, y and z), and then the direction (you can define this unambiguously by means of two angles). To place and direct a spot, you therefore have 5 degrees of freedom.

Another fun and a bit bizarre: a dimension can also be a non-integer. This is what you call the figure a fractal. Google “sponge mixer”. Menger’s sponge has a dimension between 2 and 3.

But imagine a dimension as independent degrees of freedom.