A strip, folded, with its layers solved
strip-folding is one function. Everything below came out of it during this
build, at arguments taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and when the generator changes, this
page changes with it.
At its defaults
sub: "layer-rules", rule: "both"
creases: [0.4, 0.5], assignment: "MV"
creases: [0.2, 0.4, 0.6, 0.8], assignment: "MVMV"
What it checked while it drew
Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.
- the 5 folded segments account for the whole strip, to 0e+0, and each of the 1 layer orders found arranges every one of them ×6
- the 3 folded segments account for the whole strip, to 0e+0, and no layer order exists at all, which the figure reports rather than drawing a plausible one ×2
- the census covers 5 strips, none past the eight-crease ceiling the enumerator can exhaust ×2
- each pair really is an allowed stack beside a forbidden one: the two taco folds nest on the left and interleave on the right, and the flat layer runs outside the fold on the left and through it on the right ×1
Where it is called
Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.
A collision is an order
Paper passing through paper is treated here as a thing that happens during a motion and is caught by watching for it. At the flat state it is not an event at all: it is the absence of an ordering, and it can be proved rather than observed. Four of the six quadrilateral meshes this site solves for rigid folding place perfectly and admit no ordering of their nine panels — so every one of them must pass through itself, and none of them was ever driven to find out.
A machine that can only crimp
Change the atom and the whole picture changes. A machine whose single move folds two adjacent creases at once reaches strips no simple-fold machine reaches, is defeated by strips they handle easily, and cannot fold an odd number of creases at all — for reasons that are pure arithmetic.
A strip is decidable
Take the same problem down one dimension and it stops being hard. The reason is not that strips are small — it is that overlaps on a line form a chain, and chains cannot contain the cycles that make the two-dimensional question intractable.
Four questions about one sheet
Deciding, counting, listing and optimising are not four difficulties of one problem. They are four problems, and folding is the subject that proves it: a ruled map is trivial to decide and unsolved to count, while a general crease pattern is the other way round.
How many assignments fold
The local conditions throw away most of the ways a pattern could be creased. They throw away a smaller and smaller fraction as the pattern grows, and what survives grows faster than what is discarded — which is why a strong filter is not a decision procedure.
More than one way to lie flat
A crease pattern with its mountains and valleys marked is spoken of as though it named a folded object. It does not. The legal stackings can be counted exactly in one dimension, the count is routinely more than one, and its size is a property of the pattern that nobody quotes.
One straight cut
Any drawing made of straight lines can be folded so that the whole drawing lands on a single line, and one cut releases it. The construction is a shrinking process, and it explains itself the moment the shrinking is drawn.
The crease has a radius
A fold does not go through a line. It goes round a small arc, and the arc uses more paper than the stack advances by — a fraction of a millimetre per crease, and several millimetres across a grid, which is why an ambitious tessellation comes out short.
The first thing about layers
A folder is taught four conditions at a vertex, or is taught nothing at all, and neither one says anything about the layers — which is where most of what goes wrong actually goes wrong. There has never been a rule about layer order simple enough to teach, because the question is global and every answer to it was a search. A chain of panels whose arrows all point the same way is the first one that fits on a finger.
The fold a machine can make
A theorem that says a folded state exists says nothing about getting there. A machine that folds every layer at once is stopped by a strip with two creases in it — one that folds flat perfectly well, and that a pair of hands folds in about four seconds.
The gadgets that make it hard
Flat-foldability is NP-hard, and the proof is a construction rather than an obstruction: a machine for turning any satisfiability problem into a sheet of paper that folds exactly when the problem has an answer.
The half no notation records
Every notation this subject has invented writes down the crease pattern or the sequence of folds, and the crease pattern is the half that does not decide the folded object. The field's interchange format has a place for the other half and nothing fills it in — including the files published here, which carry every vertex, edge and letter of a Yoshimura and none of the three hundred bits that would say which of its layer orders the folded object is.
The oldest open problem
In how many ways can a map be folded? The question needs no notation to state, the answer is a small integer for small maps, and after sixty years there is still no formula — only a list of numbers, each one found by searching every possibility.
The order does not name it either
A photograph of a folded model carries an outline and a layer count, and that is not enough to recover the pattern. Hand the observer the layer order as well — everything the object physically is — and most of the ambiguity goes. Most. What is left are pairs of genuinely different crease patterns that fold to the same object, which no better photograph reaches.
The shadow does not name the pattern
A photograph of a folded model carries an outline and a thickness at every point of it, and that is the whole of what it carries. It is not enough. Crease patterns in genuinely different places fold to identical outlines with identical layer counts, and nearly a third of the folded objects a short strip can reach are reached by more than one pattern.
The vertices nobody checks
Every figure on this site is gated on four conditions evaluated at every interior vertex, and the word interior has been carrying the whole sentence. On the printed patterns there are 105 vertices on the edge of the paper against 92 inside it, not one of them has ever been examined, and the condition that decides them has been available since the second phase of the collection.
Two directions that will not separate
A map has rows and columns, and a strip of stamps is a map with one row. The obvious hope is that the two-dimensional count is built from the one-dimensional one — fold the rows, then fold the columns. It is not: a two-by-three map folds 60 ways against a product of 12, and the discrepancy grows from a factor of two to a factor of thirty-eight over the counts anybody has.
Which layer goes on top
The mountain-valley assignment says which way each crease turns. It says nothing at all about which sheet ends up above which, and that second question is a different object with its own rules — and all of the difficulty.
Every generator · The flat-folding field · The patterns a reader can fold