Index

Recall that we defined a linear space using the following axioms:

Any near-linear space can be made into a linear space simply by connecting every two points with exactly one line. Thus, all of the definitions, theorems and lemmas associated with near linear spaces also pertain to linear spaces.

Most geometries that we use from this point on will be linear spaces. We call a linear space trivial if b < 1, i.e., all v-many points lie on one or zero lines. A trivial linear space with n points and one line looks like this:

Let's look at all of the distinct linear spaces on 5 points:

We'll now introduce a drawing trick that makes it easier to draw these spaces (especially when we have a large numbers of points). Rather than connect every point to every other, via a line, we will only draw the lines that are incident with 3 or more points. If a line is incident with 2 points we will draw only the points. In addition to simplification, this helps to emphasize the structure of the linear spaces. For example, one could draw two spaces that appear to be different, but by drawing them with this method, it could become apparent that they are isomorphismic to eachother. Let's draw all linear spaces on 5 points using this method.

This is much nicer!

References: Batten, Batten & Beutelspacher