A popup card is a sheet of paper with cuts and folds. Flat when closed. Three-dimensional when opened. The standard understanding is that the cut-fold pattern determines the deployed shape — the geometry at fabrication specifies the geometry at deployment. One pattern, one shape.
Chavda and Ganga Prasath found that this is wrong for patterns with splay. When the rectangular unit cell is given an angular offset, the same cut-fold pattern takes multiple shapes along the trajectory of deployment. At one angle of opening, the surface has negative Gaussian curvature — saddle-shaped, curving in opposite directions. At a different angle, the same surface has positive Gaussian curvature — dome-shaped, curving in the same direction. The transition happens smoothly as the card opens. No reconfiguration. No additional cuts. The curvature sign reverses through the act of opening alone.
The flat pattern does not encode a shape. It encodes a trajectory through shape space. The identity of the structure is not determined at fabrication. It is selected during deployment, by how far the card is opened. The same sheet of paper, with the same cuts and the same folds, is simultaneously a saddle and a dome — both shapes coexist as accessible states along a single kinematic path.
The design pipeline runs in reverse: given a desired 3D shape, compute the cut-fold parameters that produce it at a specific deployment angle. The framework uses discrete Gaussian curvature defined at fold vertices — the curvature lives at the hinges, not on the faces. The mathematics is clear: three parameters per unit cell (cut length, cut height, fold width) plus splay angle determine the full trajectory.
What's structurally interesting is not the engineering (deployable structures are well studied) but the ontological shift. The structure's identity is not a state but a path. A popup card with splay doesn't have a shape. It has a shape function — a mapping from deployment angle to Gaussian curvature. The card is not a thing that opens. It is the opening.