Origami & Folding
what one sheet can do
Series

Notation — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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

    What a dashed line can say

    Before the Yoshizawa–Randlett symbols a model could not be transmitted, and the subject was not cumulative. The basic notation says exactly one thing — fold this crease, this way, now — which is precisely a simple fold, and the share of flat foldings that simple folds reach collapses from 71% to 13% as a model grows.

    part 1 · history
  2. 32 × 32 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley

    Publishing the pattern instead of the sequence

    A diagram sequence is one picture per step and a crease pattern is one picture. When designers began releasing patterns rather than diagrams, the cost of publishing a model fell by two orders of magnitude and the difficulty moved onto the reader — which is what made the complex era possible and what made most of it unfoldable.

    part 2 · history
  3. 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.

    part 3 · history
  4. 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

    A file has no paper

    The field's interchange format is three arrays — where the vertices are, which pairs of them an edge joins, and a letter for each edge — and that is exactly the object every computation on a crease pattern starts from. A list of edges cannot say that two of them must not cross, because crossing is a property of the drawing and the list has no drawing in it. So a pattern that no paper could carry is a perfectly well-formed file, and four of this collection's own were.

    part 4 · history
  5. 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.

    part 5 · history
  6. 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.

    part 6 · history
  7. 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.

    part 7 · history
  8. 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.

    part 8 · history

All series