This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis. By classifying each skeletal point by its order of contact, we show that generically the medial axis consists of five types of points which are then organized into sheets, curves, and points: (i) sheets (manifolds with boundary) which are the locus of bitangent spheres with regular tangency {\math notation means n distinct k-fold tangency of the sphere of contact, as explained in the text.} \math; two types of curves (ii) the intersection curve of three sheets and the locus of centers of tri-tangent spheres, \math, and (iii) the boundary of sheets which are the locus of centers of spheres whose radius equals the larger principle curvature, i.e., higher order contact A3 points; and two types of points (iv) centers of quad-tangent spheres, \math, and, (v) centers of spheres with one regular tangency and one higher order tangency, A1A3. The geometry of the 3D medial axis thus consists of sheets (\math) bounded by one type of curve (A3) on their free end, which corresponds to ridges on the surface, and attached to two other sheets at another type of curves (\math), which support a generalized cylinder description. The A3 curves can only end in A1A3 points where they must meet an \math curve. The \math curves can either meet one A3 curve or meet three other \math curve at an \math point. This formal result leads to a compact representation for 3D shape, referred to as the medial axis hypergraph representation consisting of nodes (\math and A1 A3 points), links between pairs of nodes (\math and A3 curves) and hyperlinks between groups of links (\math sheets). The description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes, in analogy to 2D. We derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph. Thus, the hypergraph completely characterizes 3D shape and lays the theoretical foundation for its use in recognition, morphing, design and manipulation of shapes.