Origami & Folding
what one sheet can do
Concept

Buried crease — where it appears

A crease with an interior vertex at each end. Flipping it disturbs two vertices at once and the smallest legal change has only one other crease to spend, so its letter is settled when the pattern is drawn — and the number of such creases is the number of mutually unreachable pieces the pattern's letterings fall into, as a power of two.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

the bar is the creases with an interior vertex at each enda folder holding one of these patterns is in one piece of the count on the right, and cannot leave it107 of 862 moves survive across the shelf · 0 touch a buried creaseThe preliminary base0 buried · 1 piecesThe Miura fold22 buried · 4,194,304 piecesThe square twist4 buried · 16 piecesThe hexagon twist6 buried · 64 piecesThe Yoshimura pattern48 buried · 2.81 × 10^14 piecesFold and cut — the triangle0 buried · 1 piecesThe tapered corrugation27 buried · 1.34 × 10^8 piecesThe waterbomb tessellation42 buried · 4.39 × 10^12 pieces

The pieces without the list

The letterings a pattern folds in fall into pieces no folder can cross, and the count was found by writing every lettering down — which stops at eighteen creases. The Miura has thirty-eight, the Yoshimura eighty-six, and the number of pieces can be read off the drawing without listing anything: four million and two hundred and eighty-one million million.

flat-folding · Local moves the bar is the buried creases a crumple of that depth writestwo sheets at each depth, from two streams2 folds3 creases · 1 pieces3 folds8 creases · 2 pieces4 folds17 creases · 128 pieces5 folds22 creases · 512 pieces6 folds45 creases · 2.68 × 10^8 pieces7 folds61 creases · 1.09 × 10^12 pieces

The decision a crumple has taken

A sheet crumpled at random satisfies every condition in the subject, because it just folded. It also wrote itself a lettering — one of very many the pattern admits — and it is now in a piece of that space it cannot leave: at six folds a crumpled sheet carries thirty-nine buried creases, which is half a million million million pieces, and every change it admits stays inside one of them.

material · Crumpling The square twist, one crease at a timethe bar is the number of pieces, and a cut anywhere reduces itbefore any cut16 pieces · 0 vertices releasedcutting a crease that reaches the edge4 pieces · 1 vertices releasedcutting a buried crease2 pieces · 2 vertices released

A cut is not local

Cutting one crease of a square twist takes its letterings from sixteen mutually unreachable pieces to two. The cut crease is one of the four that were settled when the pattern was drawn — and it takes two others with it, because the vertices it releases were the far ends of those. Even a cut along a crease that was never settled quarters the count.

material · Kirigami a 3 × 3 patch: 42 creases, 18 of them buriedthe bar is on a log scale, because the two numbers differ by four orders of magnitudepieces the 3×3 patch has262,144pieces containing a repeating rule32, one eachevery one of the 32 rules is in a piece no other rule is in

Thirty-two rules, thirty-two pieces

The waterbomb tessellation's surviving repeating rules fold to one object — same panels, same places, same areas. Put them in the space of letterings the patch admits and they occupy thirty-two different pieces of a quarter of a million, so no two of them can be reached from one another without unfolding the sheet.

tessellation · Waterbomb the bar is the cuts after which the panels still place at allnone of them clears the contradiction, and no cut of a buried crease leaves a sheet that placessquare0 of 8484 cut, one at a time · 0 still place · 60 are buried and none of those doeselongated0 of 106106 cut, one at a time · 0 still place · 74 are buried and none of those doeshexagonal14 of 142142 cut, one at a time · 14 still place · 100 are buried and none of those doestriangular2 of 142142 cut, one at a time · 2 still place · 100 are buried and none of those doesa cut along a crease removes no paper — the two panels are still there and are no longer joined

One cut removes one arc

A crease pattern whose letters contradict themselves has, in principle, an obvious smallest repair: cut one crease and the statement it was making goes away. Cut every crease of four tessellation patches in turn — four hundred and seventy-four cuts — and sixteen of them leave a sheet whose panels still land anywhere at all. A cut gives the paper a freedom, and a sheet with a freedom in it has no folded state to order.

material · Kirigami the bar is how many creases the found lettering writes differentlymeasured against the lettering the pattern's own construction producedthe square patch4545 of 84 creases · 31 of them buriedthe elongated patch6666 of 106 creases · 42 of them buriedthe hexagonal patch6767 of 142 creases · 45 of them buriedthe triangular patch8787 of 142 creases · 65 of them buriedthe rhombille patch155155 of 282 creases · 117 of them burieda buried crease has an interior vertex at each end, and no legal move ever changes one

One solution of a search nobody ran

A crease pattern arrives with its letters already on it, and they look like part of the drawing. They are not. Every construction here ends in a propagation, a propagation ends wherever its first guess took it, and the lettering that comes out differs from the one a search finds on between a half and three-fifths of the creases — on patterns whose own letters are perfectly good.

flat-folding · As drawn the bar is how many moves survive the conditions at a vertexa move flips two creases meeting at one point, which is what pushing a vertex through doesthe square patch0216 pairs tried at each of two letterings · 0 legal · 0 leave the verdict alonethe elongated patch6270 pairs tried at each of two letterings · 6 legal · 6 leave the verdict alonethe hexagonal patch8360 pairs tried at each of two letterings · 8 legal · 8 leave the verdict alonethe triangular patch8360 pairs tried at each of two letterings · 8 legal · 8 leave the verdict alonethe rhombille patch16756 pairs tried at each of two letterings · 16 legal · 16 leave the verdict aloneevery one of them leaves the lettering on the side of the question it was already on

Every move leaves the verdict

The only change a folder can make to a lettering without breaking it is to push one vertex through, flipping two creases at once. Try every such move on five tessellation patches, from two different letterings each: nineteen of two thousand nine hundred and sixty-four survive the conditions, and not one of the nineteen turns a lettering that agrees with itself into one that does not, or the other way about.

flat-folding · Local moves

Named alongside it

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

AssignmentLocal moveFlat-foldabilityCrease patternKirigamiBoundary vertexConfiguration spaceConnectivityConstraint propagationConstructionCrease assignmentCrimp

All concepts