Origami & Folding
what one sheet can do
Concept

The taco-tortilla condition — where it appears

The rule that a panel cannot lie between the two panels a crease joins where that crease's folded image runs across it. The crease is where the paper turns and there is no gap in it to pass through, so the rule is one of the two ways a flat folded state can be refused outright.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

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 the bar is the pairs of panels that lie over one anotherThe preliminary base288 panels · 12 rules · an ordering existsThe Miura fold22824 panels · 228 rules · not decidedThe square twist369 panels · 48 rules · an ordering existsThe hexagon twist6613 panels · 96 rules · an ordering existsThe Yoshimura pattern205565 panels · 1187 rules · not decidedFold and cut — the triangle217 panels · 15 rules · an ordering existsThe tapered corrugation28228 panels · 351 rules · not decidedThe waterbomb tessellation92652 panels · 654 rules · not decideda pattern with no bar has no two panels over one another, and its order is not a question

No height to swap

A folded strip is a permutation of segments, and the smallest change a hand can make to it is a swap of two heights: 672 stackings, 560 of them isolated. A folded sheet has no height. Its layers are ordered by statements about which panels share ground, and on every printed pattern the search can finish, the answer is one stacking and no way out of it.

flat-folding · Layer multiplicity the same tessellation on the same square, cut out of the plane two waysassembled from whole unitsclipped from the plane12 crossings · panels 1.73 apart0 crossings · panels closemountainvalleyraw edge

Cutting a patch out of a plane

A tessellation is infinite and a sheet is not, so every picture of one is a decision about where the paper stops. Assembling whole twist units on a square and running the outstanding pleats to the rim puts 12, 18, 12 and 5 creases across other creases on four of five tilings; generating the pattern over a larger region and clipping it puts none. The panels then place exactly — and what is waiting behind the repair is a different refusal that could not be asked about before.

tessellation · Twists the 10 folded states of a Miura patch three columns by 4 rows, seen edge-onmiddle column grey at the bottom; left column and right column above it, interleaveda short upright joins two panels that cannot be parted: a fold the other column lies acrossleft columnmiddleright columnR RR R LL LLR RR LL R LLR LL RR R LLLL R RR R LLR RR LL LL RR LL RR LL RLL R RR LL RR LL LL RR RLL R LL RR RLL LL R RR R

Two stacks shuffled in glued pairs

A Miura patch three columns wide and r rows high has a central binomial number of folded states — 2, 3, 6, 10, 20 and on to 252 at nine rows — and the count was measured without a reason. The reason is a shuffle. In every state the middle column lies at the bottom and the two end columns lie above it, interleaved like two halves of a deck. Some pairs of an end column's panels can never be parted, because the row crease joining them lies inside the other column's folded footprint, and the two columns take those welds in turn, one at every row crease. That leaves r + 1 unbreakable blocks, and the number of ways to interleave them is the binomial.

history · Notation

Named alongside it

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

Folded stateLayer orderingLayer orderThe taco-taco conditionBoundary vertexConstraint propagationCountingCrease patternCrossingLayer multiplicityMiuraNon-crossing condition

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