Subsets of a v-set are in one-to-one correspondence with vertices of a v-dimensional unit cube, a Delaunay polytope of the lattice Z v . All vertices of the same cardinality k generate a ( v−1)-dimensional root lattice A v−1 and are vertices of the Delaunay polytope P( v, k) of the lattice A v−1 . Hence k-blocks of a t−( v, k, λ) design, being identified with vertices of P( v, k), generate a sub-lattice of A v−1 . We show that 80 Steiner triple systems (STS for short) 2-(15,3,1) are partitioned into 5 families. STSs of the same family generate the same lattice L. Each lattice L is distinguished by a set R( L) of its vectors of norm 2. R( L) is a root system. We find that for the 5 types R( L)=∅, A 1 7, A 2A 3 3, A 6A 7 and A 14. The family with R( L)=∅ contains only one STS, which is the projective space PG(3,2). The family with R( L)= A 1 7 contains also only one STS. Two-graphs related to both the STSs belong to a family of two graphs discovered by T. Spence and described by Seidel (1992, More about two-graphs, Nesetril, J. and Fiedler, M. (Eds.), Elsevier, Amsterdam, pp. 287–308).