Themes
One sheet, no cuts
A single square, uncut, unstretched. It is an arbitrary rule that turns out to be a generative one — nearly every theorem in the subject is a consequence of refusing to remove material.
The pattern is the object
A crease pattern is not a picture of a model. It is the model, written down — complete, checkable, and foldable by anyone who has the paper.
Local rules, global behaviour
Two conditions at a single vertex decide whether it folds flat. Whether a whole sheet does is a different question, and a much harder one.
Folding beats the compass
Straightedge and compass solve quadratics. A fold solves cubics, which is why paper trisects an angle and Euclid's tools cannot.
Flat is rare
Almost no crease pattern folds flat. The ones that do are a vanishingly small, highly structured set, and that scarcity is what makes them worth studying.
Paper is not ideal
Zero thickness, no stretch, infinitely sharp creases, perfect memory. Every one of those is false, and the interesting engineering lives in exactly where each fails.
The machine is not the hand
A theorem says a folded state exists. It does not say anybody or anything can get there, and every device that folds paper — a press brake, a laminator, a diagram followed in order — makes one weak kind of move. What those moves can reach is a smaller subject than what folds flat, and a more useful one.
From craft to hardware
The same mathematics that folds a paper crane deploys a solar array, packs an airbag and threads a stent through an artery. The scale changes; the constraints do not.