**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.