Origami & Folding
what one sheet can do
How this site is made

The figure library — page 8

Every picture here is generated from code at build time. This is every generator, ordered by how many essays call it, each rendered at its defaults.

what-paper-is-not

5 essays — Four things that are not true · One curve and one number · One cut short of falling apart · The fifth thing that is not true · The sixth thing that is not true · 2 claims checked

Four things the model assumesThe idealisations every crease pattern rests on, and what each one costs when something is actually folded. None of them is a small error at the scale of a complex model, and the engineering versions of this subject are largely about the first one.no thicknesslayers add up; a 64-grid model is millimetres thick at the coreno stretchpaper stretches a little, which is why wet-folding works at allcreases are linesa crease has a radius; sharp folds tear and soft ones springperfect memorypaper relaxes, so a model opens slightly the moment it is put downthe theorems are exact statements about a sheet nobody has ever folded

assignment-census

4 essays — How many assignments fold · Hardness is about the worst one · How little the conditions decide · The lettering that folds nowhere · 8 claims checked

What survives the local conditionsFor each pattern: how many mountain-and-valley assignments there are, and how many of them satisfy every condition at every vertex. The filter is severe and it is not a decision — what passes is still an exponentially large set, and every member of it still has to be checked globally.degree-4 vertex4 of 1625.0% · 4 creasespreliminary base112 of 25643.8% · 8 creasesmiura 2×28 of 1650.0% · 4 creasesmiura 3×232 of 12825.0% · 7 creasesmiura 3×3256 of 4,0966.3% · 12 creasesevery count enumerated, none estimatedthe share falls as the pattern grows, and the count still rises

attribution-table

4 essays — The name is not the date · The patterns nobody owns · Two kinds of claim · A file has no paper · 6 claims checked

The name is not the dateSix results, each drawn from the year of the earliest proof anybody can point at to the year of the publication its name comes from. Nobody was robbed — a field with no journal rediscovers things — but a reader who takes the name for the date acquires a history that is decades too late, every time.the alternating-angle conditionHusimi, 1979Kawasaki, 198910 yrmountains minus valleys is twoHusimi and Maekawa, 1979Justin, 19867 yrthe big-little-big lemmaJustin, 1986the lemma, 19948 yrone fold solves a cubicBeloch, 1936Huzita, 199155 yrthe diamond buckling patternYoshimura, 1951Yoshimura, 196918 yrthe bi-directional foldMiura, 1970Miura-ori, 199525 yr1940196019802000mean lag 21 years · longest 55proof

bud-fit

4 essays — The bud chooses the pattern · A leaf ends its pattern · The plant's pattern is not a hard case · The container picks the member · 3 claims checked

The bud chooses the foldA corrugated leaf packed into a bundle, as a function of how many folds it uses. Too few and the strip is too wide to fit; too many and the stack is too thick. The smallest bundle is in between, and the bud's radius decides which counts are available at all.051015202500.511.522.5number of foldsbundle radiusthe bud, radius 1.248121624tightest at 12 foldsbud radius 1.2 · leaf area 340 · layer thickness 0.12

checker-blindspot

4 essays — What a checker cannot check · Publishing the pattern instead of the sequence · Taught with a wrong reason · Drawn by the same hand · 3 claims checked

What the four tests seeEach of the four conditions this site's checker applies at every interior vertex, run against four patterns. The first three are each caught by exactly one test, which is what makes the tests worth having. The last passes all four at every vertex and is not thereby known to fold.developableKawasakiMaekawabig-little-bigsectors that do not alternatefour creases turning the same waya small sector flanked by one lettera 4×3 Miura, every vertexthe last row passes all four tests at all 6 of its vertices, and passing is not a proofthe tests are conditions at a single vertex; whether the layers can be stacked is a condition on the whole sheetno arrangement of vertex tests decides that, which is what NP-hardness means when it is spelled out

error-growth

4 essays — Error is folded too · Nowhere to put the error · Exact is not accurate · How deep is a crossing · 2 claims checked

Two ways to be a degree outHow far the far end of a folded strip lands from where it belongs, against the number of creases, for the same size of error applied consistently and applied at random. The first line is straight and the second is a square root, and the gap between them is the whole difference between a machine out of calibration and a machine that is merely imprecise.05101520253000.050.10.150.20.250.3creasesdrift, in panel widthsthe same way each timeat randomeach crease 0.5° out · 40 strips averaged for the random casethe drift is a composition of reflections, and would be the same on paper

folding-pieces

4 essays — The pieces without the list · The decision a crumple has taken · A cut is not local · Thirty-two rules, thirty-two pieces · 14 claims checked

How many pieces each printed pattern's letterings fall intoFor every crease pattern this site prints at true scale: the creases with an interior vertex at each end, and the number of mutually unreachable pieces that predicts. Two of the eight have none, and the Yoshimura has forty-eight.the bar is the creases with an interior vertex at each enda folder holding one of these patterns is in one piece of the count on the right, and cannot leave it107 of 862 moves survive across the shelf · 0 touch a buried creaseThe preliminary base0 buried · 1 piecesThe Miura fold22 buried · 4,194,304 piecesThe square twist4 buried · 16 piecesThe hexagon twist6 buried · 64 piecesThe Yoshimura pattern48 buried · 2.81 × 10^14 piecesFold and cut — the triangle0 buried · 1 piecesThe tapered corrugation27 buried · 1.34 × 10^8 piecesThe waterbomb tessellation42 buried · 4.39 × 10^12 pieces

haga-construction

4 essays — A record is not a proof · A schoolteacher's theorem · Two creases at once · The largest triangle in a square · 3 claims checked

One fold, and a thirdThe top-left corner brought down to the midpoint of the bottom edge. The crease is the perpendicular bisector, so it meets the vertical edges at 3/8 and 7/8; and the folded top edge crosses the right-hand edge at exactly two thirds. Nothing is measured and every value is exact.the midpointwhat comes outcrease on the left edge 3/8crease on the right edge 7/8the folded edge crosses at2/3exact, and a trisectionthe corner is placed by folding, not by measuring — which is why theresult is exact

layer-census

4 essays — More than one way to lie flat · One marking, many objects · Nothing slides past anything · No height to swap · 8 claims checked

One assignment, every pile it allowsA strip with its creases marked, drawn once for each way the layers may be stacked. Every one of these is the same crease pattern with the same mountains and valleys; they differ only in which layer lies over which, which the pattern never said.creases at 0.25, 0.50, 0.75, marked VVV3 legal stackings of 4 segments, read from the bottom of the pile up12341: 1 · 3 · 4 · 2taco-taco12342: 3 · 1 · 2 · 4taco-taco12343: 3 · 4 · 1 · 2assignmentthe paper lands in the same place every time — only the order through the pile differs

map-count-table

4 essays — The answer is bigger than the question · What universality costs · The count counts labels · The map counted from the layers · 7 claims checked

Every map anybody has countedThe number of ways a ruled rectangle folds flat. The filled rows were computed while this page was built and each agrees with Lunnon's published value; the hollow rows are the ones this machine will not finish in a build. The table stops where it stops because exhaustive search is the only method there is.mapflat foldingsand what it took2 × 284 cells, computed here2 × 3606 cells, computed here2 × 43208 cells, computed here3 × 31,3689 cells, computed here2 × 51,9800.6 s3 × 415,55254 s4 × 4300,608not reached herethe 1 × n case is the strip, and it is the only row of this table with a fast methodnobody has a formula for any entry, and nobody has proved there is none

mesh-family

4 essays — The family the Miura belongs to · The corrugation that curves · Nothing to average over · Two refusals that refuse differently · 8 claims checked

The same rule, three sheetsThree crease patterns built by one rule: draw a fan of straight lines, then a row that reflects in every one of them. That reflection is Kawasaki at each vertex, so the pattern is flat-foldable before anything has been checked — and the fan's angle is the only thing that differs between them.parallel columnsKawasaki to 3e-14°columns fanning by 5.2°Kawasaki to 5e-14°columns fanning by 9.2°Kawasaki to 5e-14°the mountain-and-valley letters are read off the motion rather than drawn, and then put past Maekawa

proportion-family

4 essays — One member of a family · The sheet decides which points exist · The triangle a strip becomes · The proportion a band asks for · 2 claims checked

The A-series is one member of a familyA rectangle whose sides are in the ratio √n divides into n rectangles of exactly the same shape, and A4 is the case n equals two. The others are just as real and just as foldable: √n is the diagonal of a rectangle one by √(n−1), so the whole family comes off a square one fold at a time.1 : √3 = 1.7321cut into 3, each part is 1.7321 — the same rectanglethe family1 : √2 = 1.4142 → 2 parts, 1 folds to build1 : √3 = 1.7321 → 3 parts, 2 folds to build1 : √4 = 2.0000 → 4 parts, 3 folds to build1 : √5 = 2.2361 → 5 parts, 4 folds to build1 : √6 = 2.4495 → 6 parts, 5 folds to buildA0 is printed at 1.413793and halves into 1.414634, which is a different rectangleevery ratio here is checked against a square root the construction never takes