Fano plane line

Question

I am confused on this question I know what a fano line is where it contains exactly 3 points.

If we are given 2 of three points how could we find the third one? this question is on my head for so long but cant seem to figure out.

thank you

Answer 1

see this picture:

http://en.wikipedia.org/wiki/File:Fanoperm364.svg

Given any two points you can find the third.

in fact the set of lines are: $$\mathcal B= \{\{1,2,3\},\{1,4,5\},\{1,6,7\},\{3,4,7\},\{2,4,6\},\{2,5,7\},\{3,6,5\}\}$$

if $X=\{1,2,...,7\}$, then $(X,\mathcal B)$ is a 2-(7,3,1)-design (a steiner system in fact).

Answer 2

Note that the Fano plane is an incidence structure in which any two points are contained in exactly one line. And also, each line contains exactly 3 points. Therefore, if you are given any two points there is a unique line containing them and thus you can figure out what the third point is.

A picture might be helpful here, see the wikipedia page for a depiction of the Fano plane.

For example, if you were given points $3$ and $5$. Then by looking at the picture you can see that the unique line containing $3$ and $5$ is $\{3,5,6\}$ and therefore the third point would be $6$.

Answer 3

This probably won't help, but it connects with more general theory. The Fano plane consists of all points $(a,b,c)$, where $a$, $b$ and $c$ are $0$ or $1$, and $(0,0,0)$ is not allowed. Given two points in this notation, we obtain the third point on the line by adding the coordinates modulo $2$ (so $1+1=0$).

Now go to the picture that you were given a link to. The labels there have been chosen to be consistent with the "binary" description given in the previous paragraph. Look for example at the points they call $3$, $5$, and $6$, and that I would call $(0,1,1)$, $(1,0,1)$, and $(1,1,0)$.

Find for example $(0,1,1)+(1,0,1)$ modulo $2$. We get $(1,1,0)$! It is the same with all the others. To find the third point on the line, given that the coordinates of two of the points are $(a,b,c)$ and $(d,e,f)$, add coordinate-wise modulo $2$.