The translation plane of order nine

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

• Translation planes
• The Lenz-Barlotti classification