The geometry obtained by "gluing” a three-dimensional Euclidean space using the group

The space Е32 is defined, which is obtained by "gluing" a Euclidean three-dimensional space using a uniformly discontinuous subgroup of the group of motions of the Euclidean space, which is a direct product of two cyclic groups of parallel translations. The main objects of the new space are determined and their affine and some metric properties are studied.