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We let $k=\mathbb{Z}_{2}$.

Is the assignment of the homogeneous coordinates $(0:0:1)$, $(0:1:0)$, $(1:0:0)$ to the main equilateral triangle of the Fano plane arbitrary?

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Could we for example start with the base of the triangle and ''name'' the vertex $(0:0:1)$ as $(1:0:0)$ instead?

I know how to obtain the point between the ones that are the vertices of the triangle, but I am affraid I don't get the general idea.

What point in the Cartesian system represents the point given in homogeneous coordinates? $(0:0:1)$?

Thanks

Jeroen Zuiddam
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H.E
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  • As an aside, here's a beautiful article by John Baez, in which he realizes the octonions as a "oriented" Fano plane (see page 7). http://math.ucr.edu/home/baez/octonions/oct.pdf – Fredrik Meyer Feb 12 '14 at 09:04

1 Answers1

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Yes, you can take any 3 linearly independent vectors in $\mathbb{F}_2$ as the vertices of your triangle. Then the rest of the points are determined, they will just be the sum of the other two points on that line. $GL(3,\mathbb{F}_2)$ acts transitively on ordered triples of noncollinear points of the Fano plane, this follows from the fact that it acts transitively on ordered bases of $\mathbb{F}_2^3$.

Nate
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