Construct a projective plane of order 7

Question

I am meant to construct a projective plane of order 7. Where the points are the one-dimensional subspaces of $\mathbb{Z_7^3}$. And the lines the two-dimensional subspaces. Incidence is given by $\in$ relation. I started writing them out like first point = {(k,k,k)| k in $\mathbb{Z_7}$},2nd point = {(k,k,2k)| k in $\mathbb{Z_7}$} and so on. I read that there should be 57 points like this.

My question is am I meant to write them all out and show a picture?

Answer 1

[…] am I meant to write them all out and show a picture?

This looks like a task from some course, in which case only your instructor can tell you with certainty what they expect you to do.

Here are a few things you can do. Some of them may be too obvious to even try. Some of them may be too tedious to complete. The key point is that you should understand how each of them works, and then decide on whether you will learn something by actually doing it.

1. In your question you start very basic, writing down actual sets. It is very useful to know that you are in fact dealing with such sets, so this is a good starting point.
2. Instead of dealing with linear subspaces, it is often easier to work with equivalence classes. Since the zero vector is a member of all linear spaces, it needs to be excluded from all equivalence classes to make them disjoint. In your notation you want $k\in\mathbb Z_7\setminus\{0\}$. For planes $\{(x,y,z)\mid ax+by+cz=0\}$ you would could use the normal vector $(a,b,c)$ and multiples thereof as the equivalence class, making points and lines represented by the same object namely a three-dimensional homogeneous coordinate vector.
3. After a while this verbose set-based notation becomes tedious, so you might want to switch to a more concise notation. It is common to write equivalence classes by denoting a single representative, often using square brackets. $$[v]:=\bigl\{kv\mid k\in\mathbb Z_7\setminus\{0\}\bigr\}$$ So now you'd have classes such as $$\left[\begin{pmatrix}1\\1\\1\end{pmatrix}\right], \left[\begin{pmatrix}1\\1\\2\end{pmatrix}\right], \dots$$
4. While the above notation allows you to pick any member of the equivalence class, picking a canonical one will make enumerating them a lot easier. A common convention is to require either the first or the last non-zero entry to be one. At this point, listing all $7^2+7+1=57$ points and lines in a systematic fashion becomes reasonable.
5. Next, you might want to represent incidence. You could start with a bipartite graph. Have one row of graph vertices representing points, and a second row of graph vertices representing lines. Draw a graph edge between two vertices if they are incident. Note that you will have $7+1=8$ incidences per point (or line), for a total of $(7^2+7+1)\times(7+1)=456$ incidences. So being able to draw that bipartite graph is a useful skill, but being able to do so in a manner that still allows all edges to be clearly distinguished might be far from trivial. If you do create such a graph, be very clear about what it depicts. In particular that half the vertices of the graph represent lines not points, and that the edges represent incidences not lines. Otherwise this will be very confusing.
6. Ideally you'd create a nice picture where all points of the finite plane are seen as points, and all lines are seen as lines, or at least some line-like curves. Affine Plane of Order 4 Picture? does something like this for an order $4$ affine plane.
7. You can see that different answers to that question have different aesthetics, and can highlight different symmetry properties. That plane only has $16$ points, far less than your $57$, and it's already fairly hard to create a beautiful static image of it. So I wouldn't be surprised if for your case it would be even harder. The wrap-around semantics of the modulo arithmetic make drawing lines of constant slope a bit of a pain. A collection of images, one for each slope, would probably work better in keeping things neat. There might be other options for finding a pleasing visual representation for this plane, making this an open-ended question.