In modern mathematics, the two-dimensional plane is usually defined
as the set of pairs of real numbers; lines and regions are subsets
of this set. The plane is thus treated as an (infinitely dense)
grid of locations. This mathematical point of view does not translate
into a practical convention for finite computational representations,
however. The mathematical correlate of any non-empty shape or line-segment
is a set of uncountably many points, and the coordinates of most
of these points are *irrational numbers* which cannot be represented
in a finite way.

The conception of an image as a uniform *m*
x *n* grid of colored squares is derived from the mathematical
image definition in a simple, direct way. It is the *discrete
*version of the mathematical model, which makes the image representations
finite, and makes the set of image representations enumerable (if
we also assume a similar discretization of the colour space). The
grid therefore inherits the feel of "objectivity" of the
mathematical research tradition. It is also technologically convenient,
since monitors and printers are based on the same idea: *m*
x *n* "pixels".

Note, however, that the grid does *not* inherit
the *mathematical *properties of the mathematical image definition.
Discretization is not an innocent operation. The maxim that "all
directions have equal rights", which is characteristic for
Euclidean space, is no longer valid: two orthogonal directions ("horizontal"
and "vertical") get preferential treatment. As a result,
the notions of distance and neighborhood change beyond recognition,
and the Euclidean notions of translation and rotation get largely
lost.

The grid does not instantiate the mathematical
image conception –– it *denotes* it.

Remko Scha, 2003