TITLE: Basic Geometric intuition, context is undergraduate mathematics
QUESTION [10 upvotes]: At some point in your life you were explained how to understand the dimensions of a line, a point, a plane, and a n-dimensional object. 
For me the first instance that comes to memory was in 7th grade in a inner city USA school district. 
Getting to the point, my geometry teacher taught,
"a point has no length width or depth in any dimensions, if you take a string of points and line them up for "x" distance you have a line, the line has "x" length and zero height, when you stack the lines on top of each other for "y" distance you get a plane"
Meanwhile I'm experiencing cognitive dissonance, how can anything with zero length or width be stacked on top of itself and build itself into something with width of length?
I quit math. 
Cut to a few years after high school, I'm deep in the math's. 
I rationalized geometry with my own theory which didn't conflict with any of geometry or trigonometry. 
I theorized that a point in space was infinitely small space in every dimension such that you can add them together to get a line, or add the lines to get a plane.   
Now you can say that the line has infinitely small height approaching zero but not zero.
What really triggered me is a Linear Algebra professor at my school said that lines have zero height and didn't listen to my argument. . . 
I don't know if my intuition is any better than hers . . . if I'm wrong, if she's wrong . . . 
I would very much appreciate some advice on how to deal with these sorts of things.

REPLY [2 votes]: Meanwhile I'm experiencing cognitive dissonance, how can anything with zero length or width be stacked on top of itself and build itself into something with width of length?

That's not what happens, at all, when going from zero to one and then on to n dimensions. You are thinking too literal. A mathematical point, line, or plane does not exist in the physical reality, it is simply a thought construct. 
We have to separate the "dots" we make on paper from mathematical "points". The point your professor is likely talking about probably consists of a vector (a bunch of numbers) together with an associated vector base in some vector space, with no degree of freedom (0 dimensions). A line is the same, with one degree of freedom (1 dimension), which is achieved by adding a second vector, multiplied by a variable. A plane is the same, with two degrees of freedom (2 dimensions), achieved by adding two vectors multiplied by two independent variables and so on. Or, if vector spaces are not used, then the point is simply a collection of numbers describing its location.
At no point whatsoever does even the concept of "thickness" enter the picture. At no point whatsoever do we "stack points with zero length and end up with a line". 
In general, pretty much anything can be a vector space; those are not limited to what we understand as points, lines, planes, volumes, at all. But in each vector space, the concepts are the same - you have objects which are spanned up by vectors, which have no other attributes except those that are explicitely given (a base and something spanning them up) and so on; and usually no easy/intuitive/naive correlation to "reality".

What really triggered me is a Linear Algebra professor at my school said that lines have zero height and didn't listen to my argument. . .

So, did you then listen to his argument?
EDIT: removed some stuff not really belonging here.