The Miura fold is used to deploy solar panels on spacecraft. You fold a sheet into a compact rectangle by making a grid of parallelogram-shaped facets that collapse along two diagonal axes simultaneously. To open it, you pull two opposite corners; the entire sheet extends in a single smooth motion. To close it, you push them back. One degree of freedom governs the whole structure. This is not a coincidence or a happy accident of engineering. It is a direct consequence of a mathematical property called rigid foldability: the Miura fold is a mechanism with exactly one degree of freedom, and that is what makes it useful for deployable structures.
Robert Lang, a physicist and origami artist, developed a computer program called TreeMaker that takes a stick figure — a tree-shaped stick diagram — and produces a crease pattern for a base that can be folded into that shape. The algorithm is built on a mathematical insight about how to pack circles on a square. Each appendage in the stick figure corresponds to a circle on the paper; the circles must be packed without overlapping; the crease pattern is determined by the packing geometry. TreeMaker made possible origami designs of biological accuracy that would have been practically impossible to discover by trial and error — insects with articulated legs, crustaceans with six pairs of limbs, human figures with ten fingers. The mathematics is not just describing origami; it is generating it.
Flat-foldability has a cleaner mathematical characterization. A crease pattern can be folded flat if and only if: around every interior vertex, the angles alternate in a way that the even angles sum to 180 degrees and the odd angles sum to 180 degrees; and the number of mountain and valley folds around each vertex differs by exactly two. These are local conditions — they concern each vertex independently. But flat-foldability is a global property; whether the whole sheet can actually be folded flat depends on whether the local conditions at all vertices can be satisfied simultaneously without causing layers to intersect. The local conditions are necessary but not sufficient. The gap between necessary and sufficient is where the interesting mathematics lives.
What origami keeps revealing is that paper folding is constraint satisfaction in a very physical sense. Every fold propagates consequences. A crease that is convenient in one region of the sheet may force an impossible configuration elsewhere. The sheet is a kind of physical constraint propagation system, and your hands are the solver. Experienced folders know this intuitively — they know that certain early choices close off later options, that the order in which you fold matters, that some models have "reversal moves" that unlock otherwise unreachable configurations.
The thing I find philosophically interesting about this is that the constraints live in geometry, not in convention. Paper does not fold badly because of social norms or institutional rules. It folds badly because triangles are rigid and sheets are flat and space has three dimensions. The constraints are features of the territory, not the map. This makes origami a rare case where constraint satisfaction has a direct physical interpretation: you are not reasoning about constraints, you are feeling them in your hands. The paper pushes back when you get it wrong. The mathematics of flat-foldability is, in this sense, a description of what paper already knew.