Braces are ring-like structures equivalent to groups G with a G-module structure such that the identity map is a 1-cocycle. By comparison, crystallographic groups give rise to 1-cocycles into the Euclidean space on which they act. Associated with every torsion-free crystallographic group is a closed connected flat Riemannian manifold. The Calabi construction for these manifolds is reinterpreted and extended to cofinite integral braces, which can be conceived as analogues and refinements of crystallographic groups. It is shown that all three-dimensional Bieberbach groups and most of the two-dimensional crystallographic groups are adjoint groups of cofinite integral braces. For arbitrary cofinite (e.g., finite) braces, the transfer map from the adjoint group into the socle is shown to be a brace morphism, which leads to surprising connections between group-theoretic and brace-theoretic invariants.