Understand the illustration of the curse of dimensionality? I understand how the curse of high dimensionality works when most features are irrelevant in this highly cited article: A Few Useful Things to Know About Machine Learning, but I get stuck in reading the following illustration:

This
is because in high dimensions all
examples look alike. Suppose, for
instance, that examples are laid out
on a regular grid, and consider a test
example $x_t$. If the grid is d-dimensional, $x_t$’s 2d nearest examples are
all at the same distance from it. So as
the dimensionality increases, more
and more examples become nearest
neighbors of $x_t$, until the choice of
nearest neighbor (and therefore of
class) is effectively random

The first sentence is unintuitive. Especially, why If the grid is d-dimensional, $x_t$’s 2d nearest examples are
all at the same distance from it?
To make it concrete, there are two coordinates(A and B) in a 2d plane:

Using geogebra
Suppose that (0, 0) is the test example, and we can see that A is closer to it than B. I wonder if B would be as close to the test example as A in any higher dimensional space? If so, how? If not, how would all samples look alike?

I think @Stephan Kolassa explains well what the authors meant in that paragraph. The theoretical basis for "in high dimensions all samples look alike" is layed out in section 3.4 Instability Result of this paper.
Skipping the proof on page 6, essentially

... all points converge to the same distance from the query point. Thus under these conditions, the concept of nearest neighbor is no longer useful.

I think if the authors cite this paper instead of using that grid example, the presentation would be much clearer. Results in higher dimensions can be unintuitive because we live in a 3-d world.
