The usefulness of the 3D Medial Axis (MA)is dependent on both the availability of accurate and stable methods for computing individual MA points and on schemes for deriving the local structure and connectivity among these points. VVc propose a framework which achieves both by combining the advantages of exact bisector computations used in computational geometry: on the one hand, and the local nature of propagation-based algorithms, on the other, but without the computational complexity, connectivity, added dimensionality, and post processing issues commonly found in these approaches. Specifically, the notion of flow of shocks along the MA manifold is used to identify flow along special points and curves which define a shock scaffold. This 1D scaffold is of lower dimensional complexity than the typical geometric locus of medial points which are represented as 2D sheets. The scaffold not only organizes shape information in a hierarchical manner, but is a tool for the efficient recovery of the scaffold itself and can lead to exact reconstruction. VVe present examples of this approach for synthetic data, as well as for sherd data from the domain of digital archaeology.