The translation plane of order nine

Related pages

Small projective planes
Finite projective planes
Finite spreads
There are exactly four projective planes of order nine, and one of these planes is a non-Desarguesian translation plane.

Theorem. Up to isomorphism, there exists exactly one non-Desarguesian translation plane of order 9.

This translation plane is defined by a spreadset in a 2-dimensional vector space over the field GF(3), consisting of the following matrices.

As it turns out, the coordinatizing quasifield is a nearfield. Moreover the non-Desarguesian translation plane of order 9 has Lenz-Barlotti type IVa.3

See also

Contributed by Hauke Klein
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