WEAVING PATTERNS OF LINES AND LINE SEGMENTS IN SPACE

A weaving W is a simple arrangement of lines (or line segments) in the plane together with a binary relation specifying which line is ''abov e'' the other. A system of lines (or line segments) in 3-space is call ed a realization of W, if its projection into the plane is Wand the '' above-below'' relations between the lines respect the specifications. Two weavings are equivalent if the underlying arrangements of lines ar e combinatorially equivalent and the ''above below'' relations are the same. An equivalence class of weavings is said to be a weaving patter n. A weaving pattern is realizable if at least one element of the equi valence class has a three-dimensional realization. A weaving (pattern) W is called perfect if, along each line (line segment) of W, the line s intersecting it are alternately ''above'' and ''below.'' We prove th at (i) a perfect weaving pattern of n lines is realizable if and only if n less-than-or-equal-to 3, (ii) a perfect m by n weaving pattern of line segments (in a grid-like fashion) is realizable if and only if m in(m, n) less-than-or-equal-to 3, (iii) if n is sufficiently large, th en almost all weaving patterns of n lines are nonrealizable.