- The
of*points***A**are the points of**P**\. - The
of*lines***A**are the lines of**P**not in . - The
of*t-dimensional subspaces***A**are the t-dimensional subspaces of**P**which are not contained in . - Incidence is inherited.

For fixed t (1td-1), the rank 2 geometry consisting of the points and t-dimensional subspaces of **A** is denoted by **A**_{t} (thus affine space is the geometry **A**_{1}).

Subspaces of an affine space are called ** flats**.

is called the ** hyperplane at infinity** and its points are called

**P** is called the ** projective closure** of

*Examples*:

1.

2.

3. Consider the example of section 1.4. Let be the plane 0001^{T}. The points of this plane (points at infinity) are those with last coordinate 0. The points of the affine space **P**\ are thus: 0001, 0011, 0101, 0111, 1001, 1011, 1101, and 1111. The projective plane 1000^{T} becomes the affine plane whose points are : 0001, 0011, 0101 and 0111. The line labeled a of the projective space becomes the affine line whose points are 0001 and 1001.

**Def**: Let G = (,,I) be a rank 2 geometry. A ** parallelism** of G is an equivalence relation || on the block set satisfying Playfair's axiom (i.e., given a point P and a block B not containing P, there is a unique block B' containing P with B'||B.)

**Theorem 1.6.1**: For t {1,...,d-1}, **A**_{t} has a parallelism.

The parallelism of the above theorem is called the ** natural parallelism**.

For the natural parallelism, any two distinct parallel t-flats span a (t+1)-flat. Two subspaces of arbitrary dimension are parallel if one is parallel to a subspace of the other.
Thus, if is the hyperplane at infinity of **P**, then subspaces U and W are parallel if U W, or vice-versa. In particular, a line g is parallel to a hyperplane H if g is parallel to some line h of H, meaning that gh.

**Lemma 1.6.2**: Let **A** = **P**\ where **P** is a d-dimensional projective space.

- Each line not parallel to a hyperplane H meets H in precisely one point of
**A**. - If d = 2, any two non-parallel lines intersect in a point of
**A**.

- Through any two distinct points there passes exactly one line.
- Playfair's axiom.
- There exist 3 points not on a common line.

**Theorem 1.6.5**: Let **A** be a finite d-dimensional affine space of order q. Then

- There exists an integer q2 such that any line of
**A**is incident with exactly q points. - If U is a t-flat, then |U| = q
^{t}.

**Theorem 1.6.7**: If S is a rank 2 geometry satisfying:

- 2 points determine a unique line,
- there are q
^{2}points in total, and - each line has exactly q points.