(some ideas here are mine, and I'm really not an expert about these dimensions, so please, please take them with a grain of salt)
part 1 is here; dimensions (part 1. basic facts)
Part 1 ended with a confusing hypercube
So this time let's just look at what we know best- dots, lines, squares and cubes.
Hypercubes won't come out today(except for the very end, a tiny role.)
This form counts the number of dots, lines, squares and cubes in... a dot, line, square and a cube.
To make explaining easier, I'll call each of the components x-D components (a dot will be 0-D, lines are 1-D, squares are 2-D and so on)
For example, a square is made from four 0-D components (dots), four 1-D components (lines) and one 2-D component (square). Get the idea?
As you can see, the number of 0-D components always gets multiplied by 2 every time there's a new dimension.
This is because, adding a new dimension means you slide the shape from one dimension back to another direction. (e.g. to make a square, you slide a line in the direction of the Y-axis)
So when you think about it, it's natural that the number of dots gets multiplied every time there's a new dimension.
With that info, let's look at the number of dots in hypercube(you know, this one)
So next time, let's look at this hypercube with the number of dots, lines, ...(i hate having to have to keep repeating this)
... well, components, in short.
I think I've got a pretty good idea.