Origami & Folding
what one sheet can do
Concept

Self-intersection — where it appears

Paper passing through paper, which no folded state may do. It is what the layer-ordering conditions forbid, and it is the requirement that makes flat-foldability hard rather than merely fiddly.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

6 interior vertices, every one satisfying both theoremswhat the local tests seeangles at each vertexassignment at each vertexwhat they cannot seewhether layer 3 passes through layer 7whether a flap has room to existwhether the order is consistent everywhereBern and Hayes, 1996: NP-hardso this pattern is checked, not proved

Local is not global

Every vertex can satisfy every condition and the sheet still not fold. Deciding whether a whole crease pattern folds flat is NP-hard, which means no figure will settle it and no algorithm will scale.

flat-folding · Flat-foldability taco-tacoallowedforbiddentwo folds at the same place may nest or stand clearthey may not interleavetaco-tortillaallowedforbiddena flat layer may pass outside a foldit may not pass through onea crease pattern can satisfy every vertex condition and still break one of these

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.

flat-folding · Flat-foldability 20° a crease0.50 turns of papernothing touching anything34° a crease0.85 turns of papernothing touching anything36° a crease0.90 turns of paper1 pair through one another50° a crease1.25 turns of paper5 pairs through one anotherone strip of 10 panels, seen end-onit laps itself at 36.0° a crease, which is where its cross-section closesevery panel is the same length in every frame; the only thing changed is how far each crease is turned

Paper through paper

Every test the subject has for rigid folding is a statement about a neighbourhood, and a neighbourhood cannot see the far side of the sheet. So a pattern can satisfy all of them while driving one panel straight through another, and the sharpest witness has no interior vertex in it at all.

rigid · Self-contact 5 of 6 solved meshes are solid at every angle sampledthe bar is the deepest interpenetration found anywhere in the motion, in panel widthsmesh 11closes to 9e-14solid at every anglemesh 17closes to 1e-121.18 — panels 3:0 and 3:2, 2 steps apartmesh 19closes to 6e-14solid at every anglemesh 23closes to 5e-14solid at every anglemesh 27closes to 2e-12solid at every anglemesh 71closes to 4e-12solid at every angle

Closing is not building

A quadrilateral mesh solved so that every loop closes to within a millionth of a radian is a mesh whose fold angles are consistent. It is not necessarily an object. One of the six solved here drives a panel through another at every angle of its motion — there is no part of the fold at which it could be made of solid panels — and the pair that crosses is two steps apart in the sheet, where nothing evaluated at a vertex could see it.

rigid · Self-contact the bar is the share of the population with a folded stateevery pattern in all four passes every condition at every interior vertexthe printed patterns4 of 80 cannot be placed · 0 cannot be ordered · 4 undecidedtwist tessellations2 of 125 cannot be placed · 2 cannot be ordered · 3 undecidedquadrilateral meshes2 of 60 cannot be placed · 4 cannot be ordered · 0 undecidedfold-and-cut patterns5 of 70 cannot be placed · 0 cannot be ordered · 2 undecidedundecided is a real answer here and is not rounded toward either side

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.

rigid · Self-contact the bar is the nodes the ordering search visitedthe letters are consistent on every one of these, so the one-pass test says nothing about any of themmesh 37,4739 panels · 7,473 nodes · no order existsmesh 58,0079 panels · 8,007 nodes · no order existsmesh 89,3469 panels · 9,346 nodes · no order existsmesh 111,0159 panels · 1,015 nodes · an order existsmesh 141449 panels · 144 nodes · an order existsmesh 199,0629 panels · 9,062 nodes · no order existsa red bar is a pattern with no folded state, found only by visiting every ordering it might have had

Two refusals that refuse differently

Four of the six developable quadrilateral meshes this collection solves have no ordering of their nine panels — they must pass through themselves, and a search over every ordering proves it. On all four, the letters agree with themselves perfectly. The linear proof and the exponential search are not a fast test and a slow one: they answer different questions, and neither contains the other.

rigid · Self-contact the bar is how many letterings of the mesh can have their panels stackedout of every labelling of its twelve creases, enumeratedmesh 3016 pass every vertex · 16 agree with themselves · arrived refusedmesh 5032 pass every vertex · 32 agree with themselves · arrived refusedmesh 8832 pass every vertex · 32 agree with themselves · arrived refusedmesh 11832 pass every vertex · 32 agree with themselves · arrived foldablemesh 141416 pass every vertex · 14 agree with themselves · arrived foldablemesh 19416 pass every vertex · 16 agree with themselves · arrived refusedtwo of the meshes have none at all, and two more were refused only at the lettering they came with

Refused at one lettering

Four of six quadrilateral meshes here have no arrangement of their nine panels — established by searching every ordering, at the labelling each mesh arrived with. Enumerate every labelling instead and two of the four fold perfectly well at a different one. What was reported as a fact about four meshes is a fact about two meshes and two labellings.

rigid · Self-contact one period of the square grid's twist tessellationa ring is where a crease leaves and returns on the far side40 crease pieces → 32 creases25 drawn panels → 16 panels16 vertices, every one interiorV − E + F = 0mountainvalleyraw edge

Two panels that are one panel

Paper cannot pass through paper, and every test for it compares pairs of panels. On a glued sheet two pieces of the drawing can be the same piece of paper — so a test that does not know the identification either reports a collision between a panel and itself, or misses one where the sheet meets itself round the loop.

rigid · Self-contact

Named alongside it

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

Layer orderingNon-crossing conditionQuadrilateral meshNecessary conditionPanelEnumerationLayer orderRigid foldingStackingThe taco-taco conditionAssignmentBoundary

All concepts