Origami & Folding
what one sheet can do
Concept

Repair — where it appears

A change made to a pattern that has been found to fail, small enough to leave the rest of it alone. Which changes count as small depends on what is being preserved: a cut keeps the crease pattern and destroys the folded state.

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

shaded is every panel that lies on some loop49 panels · 1 tangle · biggest 3535 panels on some loop — 71.4% of the patch52 of 84 arcs run inside it, so one cut removes one of them

The loop is not the tangle

A search that finds a contradiction in a pattern's letters reports the first circle it meets, and on a tessellation patch that is eight to twelve panels of forty-nine. It reads as a local fault. Decompose the same arrows a second way and the set of panels that lie on some circle is thirty-five of forty-nine on the square patch and ninety-nine of a hundred and fifty-seven on the rhombille — which is why the smallest available repair does not reach it, and cannot be tried on most of the creases at all.

flat-folding · Forced order 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

Named alongside it

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

AssignmentCrease patternFolded stateLayer orderingBuried creaseDegrees of freedomKirigamiNecessary conditionPatchTessellation

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