The Hanging Drape: a Vertex Analogy

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 synchronised at folding of facet pairs. We calculate the folded rotation angles exactly using a spherical image specialised 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 behaviour for the vertex, in analytical- and practical terms.

Calculating the Fold Angles of Any Vertex Roof Using a Spherical Image Technique

The shape of a vertex roof is defined by the geometry of its constituent flat facets and the relative angle of folding across them. The spherical image of the roof, originally from Gauss, expresses these properties simultaneously. We present a method for calculating the image properties and thence the shape of any vertex roof based on subdividing the image into an array of suitable spherical triangles. In particular, we introduce a truss representation of the vertex for choosing viable subdivisions of the image, which allow full calculations to be made. Additionally, this allows us to construct generalized closed-form expressions for the fold angles of vertex roofs, presented here for roofs with up to six facets.