Origami & Folding
what one sheet can do
Concept

Exhaustive search — where it appears

A search that visits every candidate the constraints admit, so that finding nothing is a proof rather than a report. Its cost is the size of the tree it builds, which depends on the order it takes its decisions in — fifteen steps or half a million, on patterns refused for the same reason.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

each cell is one patch, searched to a verdictgreen: a lettering exists · magenta: none exists, by exhaustion0.150.250.350.50.70.91.11.3turn angle, in radianssquare2626262626262626elongated1515323231313232hexagonal1515394545464545triangular1515393939373737the number in a cell is the nodes the search visited; 6 of 32 patches have no lettering at all

The order that proves nothing exists

Twelve crease patterns with no consistent lettering at all. Proving it takes fifteen steps under one rule and half a million under another — and on three of the twelve the two rules swap places, so neither is the good one. The cost of a negative is two to the power of how many free choices sit above the contradiction.

complexity · Search order each cell is one patch, searched to a verdictgreen: a lettering exists · magenta: none exists, by exhaustion0.150.250.350.50.70.91.11.3turn angle, in radianssquare2626262626262626elongated1515323231313232hexagonal1515394545464545triangular1515393939373737the number in a cell is the nodes the search visited; 6 of 32 patches have no lettering at all

A region with no lettering

One turn angle at which a tessellation patch has no consistent lettering was found by sweeping a dial. Sweeping two dials finds nine patches with none, across three tilings, filling a corner of the parameter space — and never touching the square tiling, whose sectors have no sixty degrees to cross.

flat-folding · Genericity 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

A search with nothing to reorder

One search on a crease pattern costs eighty steps or fifteen thousand depending on the order it takes its decisions in. The other search on the same crease pattern costs 1,188,571 steps whatever order it is given — twelve permutations of the panels, twelve identical counts. The difference between them is one line of code that neither has and one has.

rigid · Self-contact the same 2×2 glued cell, searched under two rulesa cycle is a contradictiona cycle whose steps add to zero isand what the loops dothe square gridnothing, in 359 nodesevery loop travels (2 directions)the triangular gridnothing, in 12,143455 nodesevery loop travels (2 directions)the honeycombnothing, in 9,6191,043 nodesevery loop travels (3 directions)the elongated triangular tilingnothing, in 9,123162 nodesevery loop travels (5 directions)the rhombille tilingunfinished at 200,000unfinished at 200,000—“nothing, in n” is an exhausted search: a proof that the pattern has no consistent lettering, which is false

A loop that goes somewhere

Every crease says which of its two panels lies above the other, and a loop in those statements is a proof that the pattern has no flat folded state. On a sheet with no edge that sentence is false. The loops of a periodic pattern carry a lattice step each, and a loop that ends one cell to the right is not a contradiction — it is a stack of paper with no bottom layer.

flat-folding · Layer multiplicity the impossible lettering, on ordinary patchessquare ×140 creases16 vertices · every condition holds · no forced loopsquare ×2144 creases64 vertices · every condition holds · no forced loopsquare ×3312 creases144 vertices · every condition holds · no forced looptriangular ×1116 creases48 vertices · every condition holds · no forced looptriangular ×2424 creases192 vertices · every condition holds · no forced loophexagonal ×1116 creases48 vertices · every condition holds · no forced loophexagonal ×2424 creases192 vertices · every condition holds · no forced loophexagonal ×3924 creases432 vertices · every condition holds · no forced loopelongated ×1184 creases80 vertices · every condition holds · no forced loopelongated ×2688 creases320 vertices · every condition holds · no forced loopthe bar is the crease count; the note is what the ordinary checks said

The lettering that was proved impossible

A search closed its whole tree on a glued square tessellation and reported that no mountain-and-valley assignment of it is consistent. Written onto ordinary patches of one, four and nine periods and handed to the four vertex theorems and a folded sheet rebuilt from scratch, the assignment it says cannot exist passes every check, on four tilings, up to fifteen hundred creases.

flat-folding · Flat-foldability proving the glued square cell has no lettering1×1, 4 panels3proved there is none · the other test found one in 32×2, 16 panels35proved there is none · the other test found one in 93×3, 36 panels3,455proved there is none · the other test found one in 6254×4, 64 panels200,000still running at the budgeta bar at the budget is a search still running, not a proof

Pruning on proofs alone

A search that discards a branch it cannot prove wrong is not a search. Deciding whether a periodic pattern's layer relations really contradict themselves is far dearer than the disc's one-pass test, so the cheap test is asked first — it is sufficient, so it settles almost everything — and the expensive one runs only on what the cheap one rejects. Five of nine steps on a small cell, fifty thousand of fifty-seven on a large one.

complexity · Search order proving the glued square cell has no lettering1×1, 4 panels3proved there is none · the other test found one in 32×2, 16 panels35proved there is none · the other test found one in 93×3, 36 panels3,455proved there is none · the other test found one in 6254×4, 64 panels200,000still running at the budgeta bar at the budget is a search still running, not a proof

The cost of proving something false

A search closing its whole tree is the strongest result this collection can produce, and on a glued tessellation it produces one that is wrong. What it costs to reach is three steps at one period, thirty-five at four, three thousand four hundred and fifty-five at nine, and more than two hundred thousand at sixteen — growing far faster than the cost of finding the lettering it says does not exist.

complexity · Hardness of folding panels with nothing below them, and where they aresquare ×1125 panels, 16 of them touching the edge · all 1 at the edgesquare ×2281 panels, 32 of them touching the edge · all 2 at the edgesquare ×33169 panels, 48 of them touching the edge · all 3 at the edgetriangular ×1369 panels, 39 of them touching the edge · all 3 at the edgetriangular ×25233 panels, 79 of them touching the edge · all 5 at the edgehexagonal ×1469 panels, 39 of them touching the edge · all 4 at the edgehexagonal ×27233 panels, 79 of them touching the edge · all 7 at the edgehexagonal ×310493 panels, 119 of them touching the edge · all 10 at the edgeelongated ×12105 panels, 48 of them touching the edge · all 2 at the edgeelongated ×23369 panels, 96 of them touching the edge · all 3 at the edgethe sheet these letters belong to has no such panel at all

An order with no least element

Enumerating every way a folded pattern can be stacked works by building upward from a panel with nothing below it. The smallest square twist patch has exactly one such stacking and takes eleven thousand steps to find it. The pattern that patch was cut from has no panel with nothing below it at all, so the enumeration has nothing to start from — and the sheet is perfectly well stacked anyway.

rigid · Self-contact the same 2×2 glued cell, searched under two rulesa cycle is a contradictiona cycle whose steps add to zero isand what the loops dothe square gridnothing, in 359 nodesevery loop travels (2 directions)the triangular gridnothing, in 12,143455 nodesevery loop travels (2 directions)the honeycombnothing, in 9,6191,043 nodesevery loop travels (3 directions)the elongated triangular tilingnothing, in 9,123162 nodesevery loop travels (5 directions)the rhombille tilingunfinished at 200,000unfinished at 200,000—“nothing, in n” is an exhausted search: a proof that the pattern has no consistent lettering, which is false

A test imported without its hypothesis

The rule that a loop in a folded sheet's layer relations proves the pattern cannot fold arrives from the layer-ordering literature, where the sheet is a disc and the panels are finitely many. This collection took the rule and not the sentence that says which sheets it is about, then applied it for years to patterns whose whole interest is that they repeat.

history · Rediscovery which bands foldcreases across the strip123456nofoldsnofoldsnofoldsfoldsnofoldsnofoldsnocylinderMöbius bandthe gluing map of a cylinder is a slide and of a Möbius band a slide with a flipand a composition of k reflections turns the paper over exactly when k is odd

A proof in no nodes at all

A parity refuses a sheet before any search begins. It costs one addition, it is certain, and it says nothing about why — while a search that exhausts on the same sheet costs thousands of nodes and produces a proof of the same fact. Two proofs of one thing, and the cheap one is available only where somebody has noticed the invariant.

complexity · Hardness of folding the same 2×2 glued cell, searched under two rulesa cycle is a contradictiona cycle whose steps add to zero isand what the loops dothe square gridnothing, in 359 nodesevery loop travels (2 directions)the triangular gridnothing, in 12,143455 nodesevery loop travels (2 directions)the honeycombnothing, in 9,6191,043 nodesevery loop travels (3 directions)the elongated triangular tilingnothing, in 9,123162 nodesevery loop travels (5 directions)the rhombille tilingunfinished at 200,000unfinished at 200,000—“nothing, in n” is an exhausted search: a proof that the pattern has no consistent lettering, which is false

The cost of asking the wrong sheet

A test written for a sheet with an edge, run on a sheet without one, does not fail. It exhausts — proving, at three, thirty-five and three thousand four hundred and fifty-five nodes, that no lettering exists — and the letterings it proved impossible fold, on the collection's own machinery, at every size they were tried at.

complexity · Hardness of folding

Named alongside it

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

AssignmentTessellationLayer orderPanelPeriodicitySearch costBoundaryConstraintLayer orderingSearchConstraint propagationInterior vertex

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