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?


Especially, why If the grid is d-dimensional, $x_t$’s 2d nearest examples are all at the same distance from it? Could anyone please visualize the illustration(if possible)?

The example speaks about a regular grid.
In a regular grid you have for each grid line (on each dimension) two neighbouring nodes one at +1 and one at -1.
For a non-regular distribution the 2d nearest neighbours are not anymore at an exactly similar distance and there is some randomness. But the example with the regular grid shows that there are a lot of directions in which a nearest neighbour can be found (and this makes it easier for noise to overtake the signal).
