**Geometry
Simplified
**

(a *projective* space)

The above finite projective
space is the simplest nontrivial example

of a *Galois *geometry (i.e., a finite geometry
with coordinates in a

finite (that is, *Galois*) field.)

The vertical (Euclidean)
line represents a (Galois) point, as does the horizontal line

and also the vertical-and-horizontal cross that represents the first
two points'

binary sum (i.e., symmetric difference, if the lines are regarded as
sets).

Homogeneous coordinates for the points of this line —

(1,0), (0,1), (1,1).

Here 0 and 1 stand for the elements of the two-element Galois field GF(2).

The 3-point line is the *projective*
space corresponding to the *affine* space

(a plane, not a line) with *four* points —

(an *affine* space)

The (Galois) points of this
affine plane are not the single and combined (Euclidean)

*line segments* that play the role
of points in the 3-point projective line,

but rather the four *subsquares* that the line
segments separate.

For further details, see Galois Geometry.

There are, of course, also
the trivial two-point *affine* space

and the corresponding trivial one-point *projective*
space —

Here again, the points of
the affine space are represented by squares,

and the point of the projective space is represented by a line segment

separating the affine-space squares.