Higher dimensions are just beginning to be explored. Here we touch on extensions to higher dimensions in all three parts of the book.

PART I

1D (one-dimensional) linkages in higher dimensions have been explored for certain problems. For example, many linkage results that permit crossings generalize to higher dimensions, such as the annulus reachability Lemma 5.1.1 (p. 59) and the results on turning a polygon inside-out (Section 5.1.2, p. 63). Many of the generalizations are straightforward, employing nearly identical proofs. Disallowing crossings can lead to fundamentally different situations, however, as we saw with the lack of locked4Dchains and trees (Section 6.4, p. 92).

What remains largely unexplored here are 2D “linkages” in 4D–and higher-dimensional analogs. One model is 2D polygons hinged together at their edges, which have fewer degrees of freedom than 1D linkages in 3D. For example, hinged polygons can be forced to fold like a planar linkage by extruding the linkage orthogonal to the plane (see Figure 26.1). As we have just seen, Biedl et al. (2005) showed that even hinged chains of rectangles do not have connected configuration spaces, with their orthogonal version of the knitting needles (Figure 25.63). It would be interesting to explore these chains of rectangles in 4D. Another connection is to Frederickson's hinged piano dissections (p. 423), which are also just beginning to be explored.