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Let $T_n(q)$ be the ring of lower triangular matrices of order $n \geq 2$with entries from the finite field $F(q)$ of order $q \geq 2$ and let${^2T_n(q)}$ denote its free left module. For $n=2,3$ it is shown that theprojective line over $T_n(q)$ gives rise to a set of$(q+1)^{(n-1)}q^{\frac{3(n-1)(n-2)}{2}}$ affine planes of order $q$. The pointsof such an affine plane are non-free cyclic submodules of ${^2T_n(q)}$ notcontained in any non-unimodular free cyclic submodule of ${^2T_n(q)}$ and itslines are points of the projective line. Furthermore, it is demonstrated thateach affine plane can be extended to the projective plane of order $q$, withthe `line at infinity' being represented by those free cyclic submodules of${^2T_n(q)}$ that are generated by non-unimodular pairs. Our approach canstraightforwardly be adjusted to address the case of arbitrary $n$.

Affine and Projective Planes Linked with Projective Lines over Certain Rings of Lower Triangular Matrices