# Points in the Fano plane

Problem: Show that any two points in Fano plane are not contained in exactly two lines of the plane and their sum is contained in those two lines in which $$p$$ and $$q$$ are not contained.

My attempt: For the proof we will use Homogeneous coordinates, that is triples with elements of the field $$\mathbb{F}_2$$. We will construct Fano plane in such a way that for any two points $$p$$ and $$q$$ the third point on the line has the label formed by adding the labels of $$p$$ and $$q$$ modulo 2. Take any two arbitrary points $$p$$ and $$q$$, which are not equal. Then $$p,q \in \mathbb{F}_2^{3*}$$. Then $$p,q$$ are both contained in a line where the third point is the sum of $$p$$ and $$q$$. Hence on each line of the Fano plane there are two linearly independent points and third one is formed in a way that it is the sum of the linearly independent points.

I am stuck right now could anyone please explain or give hints? Why each point is contained in exactly three lines?

## 3 Answers

Andreas Caranti already explained why each point belongs to exactly three lines. The other question can be settled as follows.

Let $$p$$ and $$q$$ be the two points in question. The third point on the line determined by $$p$$ and $$q$$ is thus $$p+q$$. We know that $$p+q$$ is on exactly three lines. Call them $$L_1$$, $$L_2$$ and $$L_3$$. Without loss of generality $$L_1=\{p,q,p+q\}$$.

The point $$p$$ cannot belong to either $$L_2$$ or $$L_3$$ for then there would be two lines containing both $$p$$ and $$p+q$$. Similarly we see that $$q$$ cannot be on either $$L_2$$ or $$L_3$$ either.

Recall that there are $$7$$ lines. The point $$p$$ lies on exactly three of them - two others in addition to $$L_1$$. By the above observation these two lines are previously unnamed, so we choose to call them $$L_4$$ and $$L_5$$. Observe that neither of them can contain $$q$$. There the lines containing $$q$$ are $$L_1$$ and the two yet unnamed lines $$L_6$$ and $$L_7$$.

We have covered all the seven lines, and our census gives the conclusion: $$L_2$$ and $$L_3$$ are the only lines not containing either $$p$$ or $$q$$. As required, those were also the two lines passing thru $$p+q$$ (excluding $$L_1$$).

• Again, could you please explain why does the point $a \neq b$ has to lie on line b? – user666225 Apr 21 at 19:32
• @MariyaKav I don't understand. You used $b$ for both a line and a point? – Jyrki Lahtonen Apr 21 at 19:55
• @MariyaKav Does this thread help you? – Jyrki Lahtonen Apr 22 at 3:41

I do not quite understand your Problem (the sum of two points in contained in both of what?).

However, as to your last question, let $$a$$ be any point. Then for each point $$b \ne a$$, $$a$$ is contained in the line $$a, b, a+b$$. Since two distinct points determine a line, any two distinct lines of this form have only $$a$$ in common. Thus each line contribute $$2$$ points besides $$a$$. Since $$7 = 1 + 2 \cdot 3$$, there are exactly three lines containing $$a$$.

• I made an edit. But the problem is the following: – user666225 Apr 21 at 17:32
• We observe from the fano plane that if we take any two points, say $p$ and $q$ then they are not contained in exactly two lines. And the sum of this two points lie on exactly those two lines on which $p$ and $q$ do not lie – user666225 Apr 21 at 17:33
• Why does the point a contained in a line b? Could you please explain using Homogeneous labeling? Since otherwise I am confused – user666225 Apr 21 at 17:48

I'll begin by saying that I have very little knowledge about this area of math. However, regarding the last part of your problem.

Either I'm misunderstanding the question, or it is wrong. Take any two points $$p$$ and $$q$$, then their sum is by definition the third point on the line which both $$p$$ and $$q$$ lie on. Also, as shown by Andreas, any point lies in three distinct lines, so the sum of $$p$$ and $$q$$ can't lie on 'those two lines in which p and q are not contained.'

Maybe someone with more knowledge can shed some light on this.