The previous post explained how to define a projective plane over a field F. Now let’s look at how we do geometry in a projective plane.
We have a definition of points from the other post: a point is a triple (a, b, c) of elements of F, with not all elements equal to zero, and two points (a, b, c) and (a′, b′, c′) are defined to be equivalent if there is a λ ≠ 0 such that
(a, b, c) = (λa′, λb′, λc′).
OK, so that defines points. How do we define lines? A line is a triple [x, y, z] of elements of F, with not all elements equal to zero, and two lines [x, y, z] and [x′, y′, z′] are defined to be equivalent if there is a λ ≠ 0 such that
[x, y, z] = [λx′, λy′, λz′].
Now you might object that the definitions of point and line are identical. Not at all: one uses parentheses and one uses brackets! :)
A point (a, b, c) is defined to be on the line [x, y, z] if
ax + by + cz = 0.
The definition is symmetric in the way it treats points and lines. A point (a, b, c) lies on the line [x, y, z] if and only if the point (x, y, z) lies on the line [a, b, c].
Points and lines are interchangeable in the sense that if you completely reverse your notion of points and lines all at once, you’d get the same geometry. That doesn’t mean you can ignore the difference between a point and a line; you can’t just swap some points and some lines.
Relation to vector spaces
Points and lines in a projective plane over a field F are defined as equivalence classes. Let’s use double parentheses to denote elements of the vector space F³. So ((a, b, c)) is a point in F³ and not an equivalence class.
The projective point (a, b, c) is on the projective line [x, y, z] if and only if the vector ((a, b, c)) is perpendicular to the vector ((x, y, z)). Points and lines in a projective plane are analogous to vectors and dual vectors (or covectors as a physicist might say).
What about equivalence classes? If you multiply a vector by a scalar, you multiply its inner product with another vector by the same scalar, so the notion of when a point belongs to a line is well defined.
The duality between projective points and lines is analogous to the duality between vectors and planes in F³, as long as vectors are based at the origin and planes go through the origin. You can define a plane as the set of points orthogonal to a vector, or define a vector as the normal to a plane.
You can define projective planes without using fields. You can define a projective plane as a set of points P, a set of lines L, and a set of incidence rules satisfying three axioms:
- Any two distinct points are incident with exactly one line.
- Any two distinct lines are incident with exactly one point.
- There exist four points such that no three of these points are colinear.
Classical finite projective planes are isomorphic to projective planes over a finite field, but there are non-classical possibilities. However, so far all non-classical examples have the same number of points as a classical example.
The Fano plane
The Fano plane is a set of seven points and seven lines as illustrated below.
The projective “points” are the black dots and projective “lines” are the Euclidean lines and the circle in the middle.
By duality, you could reverse these definitions, calling the black dots “lines” and the Euclidean lines and the circle the “points.”
Either way, any two points determine exactly one line, and two lines intersect at exactly one point.
It’s easy to see that the Fano plane satisfies the synthetic definition of a projective plane. It’s also true, but not obvious, that the Fano plane is isomorphic to the projective plane over the field with two elements. It is the unique finite field of its size.
Incidentally, there’s a way to multiply octonions using a Fano plane. See footnote  here. If you label the points with the seven basis elements (besides 1) the right way, and turn the Fano plane into a directed graph, then you can encode the multiplication rules so that the product of two elements is the third element in the line they two elements determine. Direction matters because the order of multiplication matters with octonions.