Origami & Folding
what one sheet can do
Concept

Slit — where it appears

A cut of no width along a crease or across a panel, which removes no paper but separates the two sides of the cut. Its end is a slit tip: a full turn of paper whose sectors come in a line, so the vertex theorems stop applying there and the strip's own condition, a big-little-big lemma, takes their place.

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

uncut: 4 vertices inside the paper, 6.3% of letterings admittedcut a crease with an interior vertex at each end25.0% admitted2 vertexes released · 4× the share · 4 such creases, all alikecut a crease that already reaches the edge12.5% admitted1 vertex released · 2× the share · 8 such creases, all alikea released vertex is one the four conditions no longer reach, and each is worth a factor of two

A cut is a licence

What a cut buys is usually described in words — freedom, release, a shape a fold cannot reach. It can be counted, and the unit is vertices. Cutting one crease of a square twist turns two interior vertices into vertices no theorem applies to, and the share of letterings the pattern admits goes up by a factor of two for each vertex released: exactly, on every cut tried.

material · Kirigami every vertex one slit releases, and whether it still refuses anythingeach buried crease of each printed pattern cut in turn; a released vertex is a strip a full turn long, decided exactlypatternvertices releasedstill refusingwhat each refusing one admitsThe Miura fold44126 of 8The square twist80all of its letteringsThe hexagon twist1266 of 8The Yoshimura pattern960all of its letteringsThe tapered corrugation54156 of 8The waterbomb tessellation844824 of 32

A slit tip keeps one condition

A cut was priced at a factor of two in the letterings a pattern admits for every interior vertex it releases — exactly, no more and not less — and the price was measured on the square twist. A released vertex is not free. It is a strip a full turn long, and a strip has its own big-little-big lemma: a middle sector smaller than both its neighbours forces the creases either side of it to differ. Across the printed shelf, 81 of the 298 vertices a single slit releases still refuse some of their letterings. On a Miura every slit along a straight crease multiplies the share by three where the ledger says four, and every slit along a zigzag by four. The square twist, on which the price was set, is one of the two patterns where the strip never bites.

material · Kirigami

Named alongside it

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

The big-little-big lemmaBoundary vertexCrease assignmentKirigamiLocal move

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