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How does one construct projective planes of order 5? What are the references of projective geometry that describe the construction of projective planes of order at least up to 5?

Hans
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    If by order you mean the $n$ such that each line has $n+1$ points, then this is just the projective plane on the field $\mathbb F_5\cong \mathbb Z/5\mathbb Z$. – Thomas Andrews Nov 17 '14 at 03:57
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    I'm not sure this answers your question entirely, but the theorems and examples at the beginning of Hartshorne's book Foundations of Projective Geometry might help you find the answer. – Mike Nov 17 '14 at 03:58
  • According to this Wikipedia article http://en.wikipedia.org/wiki/Non-Desarguesian_plane, every projective plane up to order $8$ is Desarguesian. So the only examples are the projective planes over $F_2$, $F_3$, $F_4$ and $F_5$. – Mike Nov 17 '14 at 04:04
  • @ThomasAndrews: I am a novice on this topic. Could you please elaborate on your answer translate into an incidence matrix? – Hans Nov 17 '14 at 04:15

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For a finite projective plane of order $n$, take a finite field $\mathbb F_n$ of order $n$. Then the set of points can be described as

$$\mathbb F_n\mathbb P^2=\frac{\mathbb F_n^3\setminus\left\{{\scriptsize\begin{pmatrix}0\\0\\0\end{pmatrix}}\right\}}{\mathbb F_n\setminus\{0\}}$$

So you take the set of all three-element vectors with elements from this finite field. You exclude the null vector, and you identify scalar multiples. The resulting equivalence classes represent the points of the plane.

The lines can be represented by equivalence classes of the same form. So although they look the same algebraically, they are usually considered two disjoint instances of the same structure. A point is incident with a line if and only if the scalar product between the two is zero.

The above works as the template to construct a projective plane (finite or infinite) over any field. But there are projective planes which don't have an underlying field. In some of them, Desargues' theorem doesn't hold; these are the so-called non-Desarguesian planes. In others, Desargues' theorem does hold, but Pappos' theorem does not. These are planes over skew fields which are not fields. Since there are no finite skew fields which are not also fields, every finite projective plane is either a non-Desarguesian plane or a plane over a finite field.

MvG
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  • Thank you. Could you add how to obtain a incidence matrix from the works you presented? – Hans Nov 17 '14 at 15:06
  • @Hans: Enumerate all points, enumerate all lines, evaluate the scalar product for all. – MvG Nov 17 '14 at 15:15