Some nonrealizable line diagrams (Q1842771)

Weaving patterns are planar configurations of lines, no three through a point, together with an ``under'' information given at each crossing. Such a weaving pattern may or may not be realizable, i.e., represent the projection of a three-dimensional configuration of disjoint lines. (To check this for a single instance amounts to a linear programming problem.) In the current article, the author provides a brief survey of work on weaving patterns line diagrams in the last few years, and a review of potentially relevant concepts from oriented matroid theory. His main result shows how, given a suitable planar drawing of a line configuration, one can explicitly construct ``under'' functions of nonrealizable weaving patterns. The corresponding nonrealizable ``under'' functions were much more abstractly given by \textit{W. Whiteley} in Prop. 5.8 of [Rigidity and polarity. II: Weaving lines and tensegrity frameworks, Geom. Dedicata 30, 255-279 (1989; Zbl 0675.51008)].