In this paper, we mainly consider four infinite towers of locally finite diagram geometries, and show that they are characterized by their diagram and the property of having a flag-transitive automorphism group. They form polar towers; that is, they are successive circle extensions of classical generalized quadrangles (series {Gdk;;>l of (ck. Cz)-geometries, Gk being a point residue of Gk + l ) , and possess flag-transitive automorphism groups. (The name goes back to a remark in [6]). Their residues of type (cl . Cz) can be viewed as locally polar spaces (see [6, Remarks (1.5)(3) and (4), p. 369]). We classify infinite towers of (c • Cz)-geometries with prescribed (c' . Cz)-residues, i = 1, resp. 2, resp. 3. The results of this paper, together with the classification of locally polar spaces [6,9,22] and results from [20] and [14], yield a classification of all flag-transitive (c • Cn)-geometries, including geometries related to the sporadic groups C03 , MeL, HS, Suz, COl and the Fischer groups, up to three cases of small rank, which will be dealt with in subsequent work. Here, a (c • Cz)-geometry is a residually connected geometry (of rank k + 2), with diagram