What one sheet can do.
A square of paper, uncut and unstretched, is a surprisingly strict set of rules — and nearly every theorem in this subject is a consequence of refusing to remove material. These are essays about what follows: one idea at a time, illustrated until the argument is visible, and with every crease pattern checked against the theorems rather than drawn to look right.
Nine fields
what the geometry of folding is made of — all of them
Axioms and construction
What a single fold can do, and why folding reaches numbers that a straightedge and compass cannot.
80 essaysFlat-folding
When a crease pattern collapses flat — two local theorems, one global problem, and the gap between them.
51 essaysDesigning a base
Getting from a shape somebody wants to a crease pattern that produces it, by packing circles.
51 essaysTessellations
One vertex repeated until the sheet stops being a sheet and becomes a material.
51 essaysRigid folding
Panels and hinges instead of paper — the version that scales to solar arrays and stents.
51 essaysCurves and material
Curved creases, developable surfaces, and everything the zero-thickness sheet was lying about.
52 essaysWhat it costs to know
Deciding, counting, listing and optimising are four different questions about the same sheet, and folding answers them at four wildly different prices.
52 essaysWho found it, and when
Almost everything repeated about where folding comes from is dated too early, attributed to the wrong person, or both. This field checks the claims against the record — and is honest that a record is not a proof.
52 essaysFolding nobody designed
Leaves, wings and single strands of DNA all fold, and none of them were folded by anybody. Growth changes a sheet's own metric where a crease changes only its shape — the same geometry read in the opposite direction, and every figure here draws a fold this repository computed rather than an organism it did not measure.
The deepest series
an idea, and the essays that argue about it in order — every series
Flat-foldability
Two conditions at a point — and 17 further essays, each with an argument the others do not make.
TessellationsTwists
A square that turns — and 13 further essays, each with an argument the others do not make.
Flat-foldingBoundary
Where the paper stops — and 12 further essays, each with an argument the others do not make.
What it costs to knowHardness of folding
Four questions about one sheet — and 12 further essays, each with an argument the others do not make.
Curves and materialKirigami
What one cut buys — and 10 further essays, each with an argument the others do not make.
TessellationsMiura
One vertex, repeated — and 10 further essays, each with an argument the others do not make.
Recently added
the newest of 492 essays — what's new · all of them, by field
- The cheapest route crosses later — what it costs to know
- The deepest point pays for the paper — who found it, and when
- The wedge belongs to one length — folding nobody designed
- Two faults, not four — rigid folding
- One choice with eleven answers — who found it, and when
- Turning is uphill all the way — axioms and construction
- One sheet down — designing a base
- What the square saves — axioms and construction
All 492 essays · every field · every series · every thread · every object named · every figure, by the generator that drew it · what is taught wrongly · search
Patterns to fold
the evidence, in the hand — every one
The preliminary base
Both diagonals and both midlines of a square: the base under the crane, the lily and half the traditional repertoire, and the one most folders letter wrongly first time.
Miura, 1970 — published as engineeringThe Miura fold
A grid of identical parallelograms, and the pattern behind every deployable this subject reaches. Folded, it opens and closes in both directions at once.
Generated here, from Kawasaki's conditionThe square twist
One twist unit rather than a tessellation, and the cheapest of the family to fold: nine panels, twelve creases, and a middle square that turns as the sheet closes.
Threads running through
themes, not chapters
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.