One crease, one hinge
The atom of origami is a single crease: two rigid panels sharing an edge, free to rotate about it. The only variable is the dihedral angle between them — fold toward you and it's a valley, away and it's a mountain. Everything that follows is just many of these hinges negotiating with each other.
Slide through the fold; drag the space to orbit.
What may fold flat
Where several creases meet, the sheet gets opinionated. For four creases at a vertex, Kawasaki's theorem says it can fold completely flat only if the alternate angles between creases sum to exactly 180° — and Maekawa's adds that mountains and valleys must differ by exactly two. Miss either and the paper physically refuses.
Drag the crease rays and watch the verdict.
The one-dial surface
The Miura-ori is a grid of parallelograms whose vertices all pass Kawasaki's test, and it earns something remarkable: the whole sheet folds rigidly — no panel ever bends or stretches — with exactly one degree of freedom. Set a single angle and every one of its hinges is determined. One dial runs the entire surface, from flat sheet to folded packet.
That dial is the slider. Drag space to orbit the cell.
The sheet that shrinks both ways
Squeeze most materials and they bulge sideways. Fold the Miura and it does the opposite: pull it open along one axis and it opens along the other too; compress it and it contracts both ways at once — a negative Poisson's ratio, straight from geometry. It's why one pull deploys a whole solar array, and one push packs a map into your pocket.
Fold and watch the two footprint bars move together.
The folding studio
The full sheet, yours to run. Fold it by hand or press play and let it breathe; steepen the pattern angle and the geometry turns from gentle corrugation to deep, armour-like pleats; resize the grid; flip on the crease map to see mountains and valleys negotiate — three of one and one of the other at every interior vertex, exactly as Maekawa demands. Every panel here is rigid; the motion is pure hinge.