  In the current cosmological paradigm, an initially flat three-dimensional
manifold that pervades space (the `dark-matter sheet') folds up to build
concentrations of mass (galaxies), and a cosmic web between them. Galaxies are
nodes, connected by a network of filaments and walls. The folding is in
six-dimensional (3D position, plus 3D velocity) phase space. The positions of
creases, or caustics, mark the edges of structures.
  Here, I introduce an origami approximation to cosmological structure
formation, in which the dark-matter sheet is not allowed to stretch. But it
still produces an idealized cosmic web, with nodes, filaments, walls and voids.
In 2D, nodes form in `polygonal collapse' (a twist-fold in origami),
necessarily generating filaments simultaneously. In 3D, nodes form in
`polyhedral collapse,' simultaneously generating filaments and walls. The
masses, spatial arrangement, and angular momenta of nodes and filaments are
related in the model. I describe some `tetrahedral collapse', or tetrahedral
twist-fold, models.
