Optimal recursive decomposition (or DR-planning) is crucial for analyzing, designing, solving or finding realizations of geometric constraint sytems. While the optimal DR-planning problem is NP-hard even for general 2D bar-joint constraint systems, we describe an O(n 3 ) algorithm for a broad class of constraint systems that are isostatic or underconstrained. The algorithm achieves optimality by using the new notion of a canonical DR-plan that also meets various desirable, previously studied criteria. In addition, we leverage recent results on Cayley configuration spaces to show that the indecomposable systems-that are solved at the nodes of the optimal DR-plan by recombining solutions to child systems-can be minimally modified to become decomposable and have a small DR-plan, leading to efficient realization algorithms. We show formal connections to well-known problems such as completion of underconstrained systems. Well suited to these methods are classes of constraint systems that can be used to efficiently model, design and analyze quasi-uniform (aperiodic) and self-similar, layered material structures. We formally illustrate by modeling silica bilayers as body-hyperpin systems and cross-linking microfibrils as pinned line-incidence systems. A software implementation of our algorithms and videos demonstrating the software are publicly available online 1 .

We consider a generalization of the concept of d-flattenability of graphs -introduced for the l2 norm by Belk and Connelly -to general lp norms, with integer P , 1 ≤ p < ∞, though many of our results work for l∞ as well. The following results are shown for graphs G, using notions of genericity, rigidity, and generic d-dimensional rigidity matroid introduced by Kitson for frameworks in general lp norms, as well as the cones of vectors of pairwise l p p distances of a finite point configuration in d-dimensional, lp space: (i) d-flattenability of a graph G is equivalent to the convexity of d-dimensional, inherent Cayley configurations spaces for G, a concept introduced by the first author; (ii) d-flattenability and convexity of Cayley configuration spaces over specified non-edges of a d-dimensional framework are not generic properties of frameworks (in arbitrary dimension); (iii) d-flattenability of G is equivalent to all of G's generic frameworks being d-flattenable; (iv) existence of one generic dflattenable framework for G is equivalent to the independence of the edges of G, a generic property of frameworks; (v) the rank of G equals the dimension of the projection of the d-dimensional stratum of the l p p distance cone. We give stronger results for specific norms for d = 2: we show that (vi) 2-flattenable graphs for the l1-norm (and l∞-norm) are a larger class than 2-flattenable graphs for Euclidean l2-norm case and finally (vii) prove further results towards characterizing 2-flattenability in the l1-norm. A number of conjectures and open problems are posed.

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