Origami & Folding
what one sheet can do
Depth

Series

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
VMMM60°90°120°90°Kawasaki60° + 120° = 180°90° + 90° = 180°both 180° — satisfiedMaekawa3 mountains, 1 valleysdifference 2exactly 2 — satisfiedangles sum to 360°which is what a flat sheet requiresmountainvalley

Flat-foldability

  1. 1 Two conditions at a point
  2. 2 Why the difference is two
  3. 3 The smallest sector decides
  4. 4 Local is not global
  5. 5 Which layer goes on top
  6. +13 more
18 essays · flat-folding the sheeta wedge of 60° marked for removal60°56.4°what it closes intoa cone of half-angle 56.44°83.3% of the turn is left, and the sine of the half-angle is that same fractionthe circles of latitude are the disc's own, arriving shorter than a flat sheet would needno fold can do this: folding moves paper about and never alters how much of it surrounds a point

Kirigami

  1. 1 What one cut buys
  2. 2 Bought with holes
  3. 3 One cut short of falling apart
  4. 4 A cut that removes no paper
  5. 5 A cut is a licence
  6. +9 more
14 essays · material 0204060801001200510152025foldssurface heldpeak at 64 foldsbox of side 1 · sheet thickness 0.01 · most surface at 64 folds · 128 folds fills the box with sheet alone

Surface in a volume

  1. 1 How much surface fits in a body
  2. 2 Nothing grown has a seam
  3. 3 The surface has to be supplied
  4. 4 Two surfaces in one box
  5. 5 A nest pays four a level
  6. +9 more
14 essays · biology 4 corners, all alikesectors 90°, 90°, 90°, 90°two equal pairs, so no sectoris strictly the smallestthe assignment256 of 4096 fold6 mountain, 6 valleythe ring takes two lettersthe panels can be orderedwhat is checked4 interior verticesand not the tilinga twist of radius 0.17 sheet-widths12 creases, 4.70 sheet-widths of foldingmountainvalleyraw edge

Twists

  1. 1 A square that turns
  2. 2 Which polygons twist
  3. 3 Any tiling makes a twist
  4. 4 Fenced at both ends
  5. 5 The dial that decides nothing
  6. +9 more
14 essays · tessellation 1 row0 interior verticespasses every condition2 rows4 interior verticesfails Kawasaki4 rows12 interior verticesfails Kawasakia pattern that folds is not a pattern whose enlargement folds — the conditions arrive with the interior

Boundary

  1. 1 Where the paper stops
  2. 2 The outline is mostly crease
  3. 3 The vertices nobody checks
  4. 4 A ring and a line
  5. 5 The rim lies over less
  6. +8 more
13 essays · flat-folding 6 creases, 7 segments, assignment MVMVMVDoes it fold flat?at most 5,040 orderings, and it may stop earlyyesas far as the first legal oneHow many ways?every one of them, because the last is as likely as the first15,040 orderingsWhat are they?the same search, paying a second time for what it keeps1 stackings, written out5,040 orderings, and the answer as wellCan a machine make it?a different search, over sequences of folds rather than over stackingsno1,275 statesthe four are not four difficulties of one problem — they are four problemsthe cost is work rather than time — a clock reading would differ on every build

Hardness of folding

  1. 1 Four questions about one sheet
  2. 2 Hardness is about the worst one
  3. 3 The answer is bigger than the question
  4. 4 A no costs more than a yes
  5. 5 A short reason to say no
  6. +8 more
13 essays · complexity tilted 15.00°side 1.0352846.41% of the sheetand the same answer twicea corner construction givesside 1.03528from a quadratic, sharing no codethe width maximisation and the closed form agree to nine figures

Optimal constructions

  1. 1 The largest triangle in a square
  2. 2 The biggest one that can also be folded
  3. 3 The square is in the answer
  4. 4 Every even polygon beats every odd one
  5. 5 The crossing is as hard as the polygon
  6. +8 more
13 essays · construction the bar is millimetres of crease at the printed sizea pattern with more creases is not always a pattern with more folding in itThe preliminary base724 mm8 creases · printed at 150 mmThe Miura fold1,049 mm38 creases · printed at 170 mmThe square twist704 mm12 creases · printed at 150 mmThe hexagon twist916 mm18 creases · printed at 150 mmThe Yoshimura pattern2,380 mm86 creases · printed at 170 mmFold and cut — the triangle258 mm6 creases · printed at 150 mmThe tapered corrugation1,057 mm45 creases · printed at 160 mmThe waterbomb tessellation2,290 mm76 creases · printed at 160 mmsix point seven metres of crease on a sheet seventeen centimetres across

Crease density

  1. 1 How much line is on the paper
  2. 2 Where the length sits
  3. 3 The shortest crease is not a crease
  4. 4 A count is not a length
  5. 5 A crease with no vertex to belong to
  6. +7 more
12 essays · material Miura solar array17×Space Flyer Unit, 1995airbag folding25×stored for years, opens in 30 msheart stent6×threaded through an arterystarshade11×26 m disc, 2.5 m launch tubemap fold9×the original problempackeddeployedthe ratio is what is bought; one degree of freedom is what makes it reliable

Deployables

  1. 1 Folding that gets built
  2. 2 From a shell to a solar array
  3. 3 Fourth of eight, and still not chosen for it
  4. 4 The tube that gets built
  5. 5 What a second deployment costs
  6. +7 more
12 essays · rigid 1 × 4161 × 5501 × 61442 × 282 × 3602 × 43203 × 31,3684 × 4300,608filled — counted here, by exhaustive search over stacking ordersopen — Lunnon's published count, quoted rather than computed

Map folding

  1. 1 The oldest open problem
  2. 2 Where the exponent comes from
  3. 3 Two directions that will not separate
  4. 4 The map that is not a rectangle
  5. 5 The count counts labels
  6. +7 more
12 essays · complexity creases at 0.40, 0.50 — assignment MVflat1 layerno sequence of all-layers folds finishes this pattern — the search exhausted 5 states

Simple foldability

  1. 1 The fold a machine can make
  2. 2 The patient machine is the weak one
  3. 3 A machine that can only crimp
  4. 4 The machine that may choose
  5. 5 Where the machine catches up
  6. +7 more
12 essays · complexity 00.20.40.60.8100.511.5radius on the flat sheetgrowth factor Ωhow much each ring grew00.20.40.60.81-2-112radius on the flat sheetcurvature Kclosed formthe curvature that forcesgrown more at the rim · K(0) = −4a = -2.40

Growth as metric

  1. 1 A sheet that grows cannot lie flat
  2. 2 Which way the disc curves
  3. 3 The excess does not choose its waves
  4. 4 A crease carries no curvature
  5. 5 The container picks the member
  6. +6 more
11 essays · biology at every vertexthree of one, one of the other15 interior vertices, all identicalwhat the sheet gainsone degree of freedom, not manyit opens and closes in both directions at oncea negative Poisson's ratio22 mountain and 16 valley creases · 6.2 sheet-widths of foldingmountainvalleyraw edge

Miura

  1. 1 One vertex, repeated
  2. 2 A sheet with one freedom
  3. 3 The corrugation that curves
  4. 4 The Miura folds two ways
  5. 5 The paper a pattern asks for
  6. +6 more
11 essays · tessellation the pattern as it isevery edge keeps its length6.7e-16the same pattern, moved by 0.01and one of them cannot1.7e-21e-181e-161e-141e-121e-101e-81e-61e-41e-21largest change in any edge length, in panel widthswhat an isometry has to do, and what it manages5 × 4 panels, at 50% folded, every edge of both comparedthe moved pattern is fitted the best single panel its own edge lengths allow before being folded at allso the gap is not a bad choice of panel — it is what is left when the best choice has been made

Rigid folding

  1. 1 Panels instead of paper
  2. 2 What the vertex does on the way
  3. 3 A state no motion reaches
  4. 4 The only pattern that moves
  5. 5 The vertex is geared
  6. +6 more
11 essays · rigid heavier means the crease lies on more independent circuits157 panels, 282 arcs, circuit rank 126; circuits run from 4 to 26 arcs

Search order

  1. 1 Which choice the cost lives in
  2. 2 The order that proves nothing exists
  3. 3 What the rim was doing
  4. 4 Pruning on proofs alone
  5. 5 Half the slack
  6. +6 more
11 essays · complexity zero thicknesspanels meet exactly4 layers of real materialeach fold has to clear the ones belowhinge offset to the surfacethe panel rotates about the right linea crease pattern describes a surface with no thicknessand everything anybody builds has some

Thickness

  1. 1 The sheet has a thickness
  2. 2 Getting thickness round a corner
  3. 3 Panels with somewhere to go
  4. 4 Thickness has a sign
  5. 5 The pile, not the panel
  6. +6 more
11 essays · rigid 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

Tolerance

  1. 1 Error is folded too
  2. 2 Nowhere to put the error
  3. 3 Where an error goes
  4. 4 Solved is not built
  5. 5 A tolerance is a direction
  6. +6 more
11 essays · rigid cylinderreachablecurved one way onlyconereachablecurved one way, from a pointsphereunreachablecurved two ways — impossiblesaddleunreachablecurved two ways — impossible

Developability

  1. 1 What a flat sheet can become
  2. 2 A tuck keeps what a gore cuts
  3. 3 A straight tuck is a cone point
  4. 4 Crowd the tucks toward the rim
  5. 5 Crowding outward costs almost nothing
  6. +5 more
10 essays · material 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

Notation

  1. 1 What a dashed line can say
  2. 2 Publishing the pattern instead of the sequence
  3. 3 The half no notation records
  4. 4 A file has no paper
  5. 5 The file records no verdict
  6. +5 more
10 essays · history what the shell produced14 interior vertices, all alike17 mountain, 40 valley57 creases carrying a letterand it folds flatchecked, not asserted11.0 sheet-widths of crease, chosen by a buckling loadmountainvalleyraw edge

Rediscovery

  1. 1 Found before it was designed
  2. 2 The same vertex, found four times
  3. 3 The tail was named somewhere else
  4. 4 The cure was named first
  5. 5 A test imported without its hypothesis
  6. +5 more
10 essays · history

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