Weavings, sections and projections of spherical polyhedra

Whiteley, W., Weavings, sections and projections of spherical polyhedra, Discrete Applied Mathematics 32 (1991) 275-294. In this paper we give simultaneous answers to three questions: (a) When does a plane picture of lines, weaving over and under in the plane, lift and separate into a configuration of disjoint lines in 3-space? (b) When is a configuration of lines in the plane the cross-section of the extended faces of a spherical polyhedron in 3-space? (c) When is a picture of the edges and vertices of an abstract spherical polyhedron the projection of an actual polyhedron in 3-space? If the lines and intersections of the weaving correspond to the faces and edges of a spherical polyhedron, the first two questions are connected by a polar version of a classical theorem of J. Clerk Maxwell and a theory of vertical statics. The second and third questions are answered by a simultaneous diagram showing the compatible section and projection of the polyhedron. This new projective form of “reciprocal diagram” is necessary and sufficient for correct pictures and for weavings which do not separate into 3-space.