Origami & Folding
what one sheet can do
Concept

Typical instance — where it appears

A pattern drawn from a stated population rather than picked, so that a measurement says something about the family. It exists to keep a result from being about the examples somebody happened to find interesting, so every population here is generated from a rule quoted alongside the answer.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

the bar is the mean share of redraws that agree with themselvesas the populations stand, every member is consistent and the refusal fires on none of themthe printed patterns96.7%8 of 8 could be asked · worst member 90%twist tessellations55.0%7 of 12 could be asked · worst member 7%quadrilateral meshes96.9%6 of 6 could be asked · worst member 82%fold-and-cut patterns100.0%7 of 7 could be asked · worst member 100%a member with no folded state has no letters to redraw and is counted as not asked rather than as passing

A population that cannot fail

Thirty-three crease patterns are kept here to run the checkers over, and every one of them has letters that agree with themselves. That is not a property of the patterns. It is a property of how they were made: each came from a construction that returns a lettering, so a test looking for letters that contradict themselves has nothing to fire on. Reletter the same thirty-three and the failure is available at once — on one member, four of sixty redraws.

complexity · Typical instances sliding the cut across one period of the square tessellation36 vertices at every position, and a different set of creases divided at each0102030cut at the start of a periodone period alongnodes; the axis starts at zero, and the whole spread is inside a factor of 1.32

Where you cut hardly matters

Slide the same rectangle across one whole period of the same tessellation and every position gives a different patch: different creases divided, different half-panels round the edge, panel counts from forty-nine to sixty-one. The cost of lettering them runs from twenty-five steps to thirty-three. Whether a cut is made changes the answer by three orders of magnitude; where it falls changes it by a third.

complexity · Typical instances the angles that admit a thirdφ₁ − φ₂ + φ₃ a multiple of a straight angle30°30°60°60°90°90°120°120°150°150°60°, 120°the first crease's angle, against the secondevery other pair of angles folds nothing,at any length and any positions

How rare a band that folds is

Almost every crease pattern fails to fold flat, and the usual way of saying so is a count over discrete choices. A glued band fails for a reason that no count can reach: its crease angles have to satisfy an equation, and a set defined by an equation has no volume in the space it sits in.

flat-folding · Genericity the grid, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices4444free letters1210108panels9664V − E + F1000the vertex row is the control: identifying edges can neither make nor destroy a vertexand Euler's number is the cheapest check that the gluing did what it says

One population, four sheets

A population of patterns is a way of asking what is typical, and it has always been a population of drawings. Put the same drawings on four different sheets and the verdicts move — not because the drawings changed but because the sheet did, which means a population has two halves and only one of them was ever chosen.

complexity · Typical instances

Named alongside it

The objects these essays reach for when they reach for this one.

SamplingBoundaryCrease patternDistributionGluingSearch costAssignmentClosureConstraintCountingDecision procedureEnumeration

All concepts