Origami & Folding
what one sheet can do
Concept

Fold format — where it appears

The interchange format the field uses for crease patterns: lists of vertices, of the edges between them and of a letter for each edge. It has an optional field for the order of the folded layers and almost no file fills it in.

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

what is recorded, against what is left to the folderThe preliminary base8 panels · 15 bits of orderThe Miura fold24 panels · 79 bits of orderThe square twist9 panels · 18 bits of orderThe hexagon twist13 panels · 33 bits of orderThe Yoshimura pattern65 panels · 302 bits of orderFold and cut — the triangle7 panels · 12 bits of orderThe tapered corrugation28 panels · 98 bits of orderThe waterbomb tessellation52 panels · 226 bits of orderthe pattern, as every format records itthe order of the panels, which none of them does

The half no notation records

Every notation this subject has invented writes down the crease pattern or the sequence of folds, and the crease pattern is the half that does not decide the folded object. The field's interchange format has a place for the other half and nothing fills it in — including the files published here, which carry every vertex, edge and letter of a Yoshimura and none of the three hundred bits that would say which of its layer orders the folded object is.

history · Notation what is recorded, against what is left to the folderThe preliminary base8 panels · 15 bits of orderThe Miura fold24 panels · 79 bits of orderThe square twist9 panels · 18 bits of orderThe hexagon twist13 panels · 33 bits of orderThe Yoshimura pattern65 panels · 302 bits of orderFold and cut — the triangle7 panels · 12 bits of orderThe tapered corrugation28 panels · 98 bits of orderThe waterbomb tessellation52 panels · 226 bits of orderthe pattern, as every format records itthe order of the panels, which none of them does

The file records no verdict

A crease pattern file records vertices, edges and letters. Every one of the square twist's two hundred and fifty-six admissible letterings makes a perfectly valid file, and two hundred and forty-eight of them describe an object that does not exist. The format has a field for the layer order — the one thing that would settle it — and nothing fills it in, so a file is a drawing rather than a claim, and the field exchanges them as though they were claims.

history · Notation one rectangle, glued four waysa disc, two cylinders and a torus — from one drawing4 edges lefta disc2 edges lefta cylinder, across2 edges lefta cylinder, alongno edges lefta torusthe same rectangle and the same creases in all four, and nothing in the drawing says which is whichmatching arrowheads mean the two edges are one edge of the paper

The drawing does not say what is glued

One crease pattern, four sheets, four different answers to whether it folds — and nothing in the drawing distinguishes them. The identification is data the picture cannot carry, and the picture is the object this collection has been treating as complete.

flat-folding · As drawn 34560%20%40%60%80%100%creases in the stripreachable by simple folds72%30%17%13%the basic symbolsa dashed line — valleya dotted line — mountainan arrow — fold it nowand what they missreverse, squash, sink,petal — every one of thema move no dashed linecan ask forevery assignment of 68 seeded spacings

No format has a gluing

A crease pattern file records vertices, edges, assignments, faces and layer orders. Every one of those is a feature of the paper's interior, and the boundary appears only as a kind of edge — so there is nowhere in the scheme to say that two boundary edges are the same edge, and the sheet a pattern is on cannot be written down.

history · Notation the layer-order field for the printed patterns, filled in where it can bepairs is how many signs the field holds; varying is how many of them differ between folded statespatternpanelspairsstatesvaryingto recordThe preliminary base82810noneThe Miura fold24228not listed24 panelsThe square twist93610noneThe hexagon twist136610noneThe Yoshimura pattern652055not listed65 panelsFold and cut — the triangle721261.00 bitsThe tapered corrugation28282not listed28 panelsThe waterbomb tessellation52926not listed52 panelsa pattern past eighteen panels is not listed here, and those are the patterns anybody folds

The field is empty where it would say nothing

The interchange format for crease patterns has a field for the layer order and nothing ever fills it in. Filling it in where the folded states can be listed — four of the eight printed patterns, and Miura patches to twelve panels — finds that the preliminary base and both twists have exactly one folded state, so every one of the field's signs follows from the crease pattern and the field would record nothing a reader could not compute. The fold-and-cut triangle has two states. The Miura is different: every patch with three or more columns has several — three, six and eleven on the three-by-two, three-by-three and four-by-three — so on the pattern that gets built the field carries information from six panels up, and the field's size had been measured as log₂ of the panels' orderings, which on the preliminary base is fifteen bits for an object that has zero.

history · Notation 234567024681012columns (rows, for the two-column patches)folded statestwo rows high: 1, 3, 5, 7, 9, 11three rows high: 6, 11two columns wide: 1, 1, 1, 1, 1a strip two rows high has 2c − 3 states; every patch's states are one choice with that many answers

One choice with eleven answers

A folded state was proposed as a short list of free choices — which way a flap lies, where a rim panel sits — with the layer-order field's signs following from them. Listed exhaustively on every Miura patch small enough, the choices are never independent: every sign that varies is tied to every other through a shared panel, so the states are one choice with many answers. And there are more answers than the record said. The overlap test had a blind spot a third of a panel wide, and with it corrected the three-by-three Miura has six folded states, not one, and the four-by-three eleven, not five.

history · Notation

Named alongside it

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

NotationLayer orderingCrease patternFolded stateAttributionBoundaryGluingInformationStackingUnderdeterminationAssignmentCrease assignment

All concepts