wittgenstein said:
So, width is not defined as perpendicular to length? Suppose there are only 2 dimensions (width and height) height would not necessarily be perpendicular to width?
Those aren't dimensions in the sense that the word is being used here.
In a one-dimensional space you can only go forwards and backwards. In a two dimensional space you can go forwards and backwards, but there are place you can't reach by doing that. You need to define another direction - but it need not be perpendicular to the first one. (Indeed, it's possible to define spaces without notions of direction at all but still consider them to have multiple dimensions, although these are typically more interesting to mathematicians than physicists.) That's all that the number of dimensions means.
Mathematically, I can just say "consider all the real numbers" (the reals are all decimals, including whole numbers and infinitely long ones like ##\pi##), and I've defined a one-dimensional space (usually called ##\mathbb{R}^1##). Now I can say consider all
pairs of real numbers, like (1.32, 6.98427) or (1.90, ##e##), and I've defined a 2d space (##\mathbb{R}^2##). But the numbers aren't perpendicular to each other - how could 7 be perpendicular to 3? You need to add more structure to this space to get notions of distance and angle - that's actually what the metric provides.
If I take "all pairs of reals where the second one is between 0 and 1" then I have a 2d space with a finite extent in one dimension. If I add "...and the label (x, 1) refers to the same point as (x, 0)" then I have a space that's infinite in one direction and rolled up in another (called ##\mathbb{R}^1\times\mathbb{S}^1##). But until I define a metric, I don't have any meaning to the angle between "moving so that each point I meet has the same first number" and "moving so that each point I meet has the same second number". If I add a standard Euclidean metric,
then I've defined those two motions as perpendicular - and only then do I have a mathematical model of a drinking straw (more or less - there's some other stuff needed). Carroll covers this a bit more carefully in chapter 2 of
his GR lecture notes.
Getting back to the question you originally asked, you are free to add fifth, sixth, seventh, (etc.) dimensions to your mathematical models if you wish, and make them rolled up or not. You then need to add a ##5\times 5##, ##6\times 6##, ##7\times 7## (etc.) tensor to describe the metric, and then you will always be able to find 5, 6, 7, etc perpendicular directions at each point - but perpendicularity isn't a thing until you add the metric, and which directions are perpendicular will depend on what metric you pick.
Adding those dimensions may or may not help anything. Kaluza-Klein's model naturally describes an electromagnetic field, but IIRC also describes an un-named scalar field that would be easily detectable if it existed. And the jury is still very much out on the utility of string theory.
. That is abbreviation for the number of vector is a basis of x. And not even need to be an orthogonal basis.
