Origami & Folding
what one sheet can do
Concept

Crumpling — where it appears

Folding a sheet without a plan, by pressing it into itself. Every vertex of the resulting pattern folds flat automatically, because each crease was made in a stack that was already folded and so never had to be compatible with anything it did not know about.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

found rather than designedcrushed drink cans, tree bark,deployable boomsat every interior vertexsectors 60°, 60°, 60°, 60°, 60°, 60°two courses and four diagonalstwo of one letter, four of the otherwhy the height is not freea steeper diagonal makes the topsector strictly smallest, flankedby two of the same letter20 interior vertices · 23 mountain and 56 valley creasesmountainvalleyraw edge

Patterns nobody designed

Crush a thin cylinder and it folds into a diamond lattice. Nobody chose the pattern — it is the buckling mode with the lowest energy, and it satisfies the flat-folding theorems because it just folded.

tessellation · Found patterns 50% of the stack, 9 foldsthe whole stack, 9 folds13 interior vertices of odd degree13 creases with a loose end before planarisingrefused by the first condition it is put pastno odd vertex anywhereno loose end anywhereand every vertex satisfies Kawasaki

The crease that stops in the middle

A sheet folded flat at random writes a crease pattern that satisfies every condition in the subject, everywhere. Leave one layer behind on each fold — one layer out of a dozen — and it stops writing crease patterns at all: the creases stop in the middle of the paper, and a crease with a loose end is a thing no flat folded sheet can have.

material · Crumpling 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 letters a folding gives a sheet always agree — these are the ones it might have had insteadfolded from seed 7folded from seed 11folded from seed 2300.2500.5000.750110203040panels in the folded sheetshare of redrawn letterings that agreeeach point is one sheet folded a given number of times, and the horizontal axis is what that produced

The letters a crumple was given

A sheet creased by folding it and folding it again arrives with a mountain-valley labelling that cannot be wrong, because a folding produced it. Nothing about the pattern protects it: reletter the same creases and the share of labellings whose letters agree falls from every one of forty at eight panels to eleven of forty at forty-one. The foldability of a crumple is a fact about its history, not about its drawing.

material · Crumpling the bar is how many nodes the search visitedone sheet crumpled deeper and deeper, its letters rechosen each time4 folds1716 panels · 34 of 40 random letterings agree · 1 backtracks5 folds1918 panels · 34 of 40 random letterings agree · 1 backtracks6 folds3435 panels · 15 of 40 random letterings agree · 0 backtracks7 folds3839 panels · 19 of 40 random letterings agree · 0 backtracks8 folds7271 panels · 11 of 40 random letterings agree · 2 backtracksthe share that agrees falls by more than half along this ladder; the search's cost tracks the panels and nothing else

Rare is not hard

Crumple a sheet deeper and the share of its labellings that agree with themselves falls from thirty-four in forty to eleven. The number of steps a search needs to find one of them does not move at all: it stays at about one per panel, with no backtracking, the whole way down. How often an answer turns up at random and how much work it takes to find one are different quantities, and a crumpled sheet is where they come apart.

material · Crumpling each point is one pattern: panels across, nodes up00202040406060one node per panelnodes visitedpanels3 folds to 8 folds, and not one backtrack anywhere in the family

A crumple has no tail

The least structured crease pattern this collection can produce is a sheet folded at random and flattened. Its consistent letterings get rarer as it deepens — thirty-four of forty down to eleven — and finding one costs one step per panel from beginning to end, with no wrong guess anywhere. Disorder and difficulty turn out to be unrelated quantities.

material · Crumpling

Named alongside it

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

Crease patternAssignmentIdealisationSamplingFlat-foldabilitySearch costThe all-layers simple foldBucklingBuried creaseConstraint propagationEnergy minimisationFacet

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