We present a novel, rigidly folding vertex inspired by the shape of the simplest hanging drape. Fold lines in the vertex correspond to pleats and ridges in the drape and are symmetrically arranged to enable synchronized flat folding of facet pairs. We calculate the folded rotation angles exactly using a spherical image specialized for inextensible vertex folding. We show that the vertex shape is bounded by a pair of conical surfaces whose apex semi-angles directly correspond with fold-line rotations, which expresses a geometrical equivalence between the external shape and internal folding motion of the vertex. We discuss how the vertex, viz. drape, perform as a novel type of conical defect based on its spherical image topography, and we highlight the meaning of bistable behavior for the vertex in analytical and practical terms.