# Fano plane line

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

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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).

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i dont get it, so you are saying we can pick any x value from 1 to 7 and one beta number? – MathGeek Feb 24 '13 at 19:20
for example {1,4,5} is a line. so if you're given 1, 4, then you search among the three-point-lines and you find {1,4,5} which contains both 1 and 4. so you find out the other point is 5. no other line contain both 1 and 4. because $(X,\mathcal B)$ is a steiner system. – user59671 Feb 24 '13 at 19:24
... because it's a 2-design. – user59671 Feb 25 '13 at 10:50

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 $2$.

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Actually, I just had a similar question, and this was very helpful! – Newb May 7 '14 at 12:16
This is fascinating! Do you have any recommendation for (undergrad-) accessible material on this or related topics? – The Chaz 2.0 Jun 10 '15 at 13:52
Coxeter's old book Introduction To Geometry has some material. Anyway, t is a very nice book. The usual "homogeneous coordinates" approach to the real projective plane works over any field, in particular the $2$-element field. – André Nicolas Jun 10 '15 at 14:33

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 $6$.

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how would we know its 6 @fidbc, thats what i am failing to understand – MathGeek Feb 24 '13 at 19:17
As @CutieKrait pointed out in his answer, the set of lines is fixed. Once you are given two points you can just look up in that set of lines to see which line contains the two given points. From there you figure out that the only line containing $3$ and $5$ is $\{3,5,6\}$, therefore the third point is $6$. Note that no other point has this property, that is what makes $6$ special in this case. – fidbc Feb 24 '13 at 20:36