Origami & Folding
what one sheet can do
Depth

Series — page 2

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.
Levery point within L is spentthe flapLL = 0.28 of the sheet's side, so the disc costs πL² = 24.6% of itthe circle is not a metaphor — it is the paper the flap consumesso designing a base is packing circles

Circle packing

  1. 1 A flap costs a circle
  2. 2 Packing is the hard part
  3. 3 What joins the flaps
  4. 4 How much paper is wasted
  5. 5 From a packing to a crease pattern
  6. +3 more
8 essays · design geometrypacks tocorrugation8 panels at 0.42 rad40.8% — 2.5× smallerMiura6 × 4, 15 interior vertices16.6% — 6.0× smallerfan8 sectors about one point25.0% — 4.0× smallerroll8 turns12.5% — 8.0× smallerpacked area as a fraction of deployed, computed from each geometry — not measured from any animal

Convergence

  1. 1 The same corrugation in four places
  2. 2 Four finders, one option
  3. 3 The census returns one
  4. 4 Four materials, four optima
  5. 5 The fold count sets the spring
  6. +3 more
8 essays · biology folded 8 times, then unfolded33 interior vertices, all of degree 433 of 33 satisfy Kawasakithe folding is the reason, not the drawing45 creases drawn at random485 interior vertices, all of degree 40 of 485 satisfy Kawasakisame count, same sheet, nothing folded

Crumpling

  1. 1 The creases a sheet gives itself
  2. 2 Every facet is a layer
  3. 3 The crease that stops in the middle
  4. 4 The decision a crumple has taken
  5. 5 The crumple keeps its options
  6. +3 more
8 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. +3 more
8 essays · rigid 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. +3 more
8 essays · biology 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

Idealisation

  1. 1 Four things that are not true
  2. 2 Paper that stretches on purpose
  3. 3 The crease has a radius
  4. 4 How many times can it be halved
  5. 5 The organism is not the model
  6. +3 more
8 essays · material geometrypacks tocorrugation8 panels at 0.42 rad40.8% — 2.5× smallerMiura6 × 4, 15 interior vertices16.6% — 6.0× smallerfan8 sectors about one point25.0% — 4.0× smallerroll8 turns12.5% — 8.0× smallerpacked area as a fraction of deployed, computed from each geometry — not measured from any animal

Insect wings

  1. 1 A wing that folds into nothing
  2. 2 No motor in the fold
  3. 3 Nothing in a body folds on a line
  4. 4 The number is the angle
  5. 5 A corrugation has one resting state
  6. +3 more
8 essays · biology creases at 0.25, 0.50, 0.75, marked MMM3 legal stackings of 4 segments, read from the bottom of the pile up12341: 2 · 1 · 4 · 3assignment12342: 4 · 2 · 1 · 3taco-taco12343: 2 · 4 · 3 · 1taco-tacothe paper lands in the same place every time — only the order through the pile differs

Layer multiplicity

  1. 1 More than one way to lie flat
  2. 2 One marking, many objects
  3. 3 Nothing slides past anything
  4. 4 No height to swap
  5. 5 Consistent is not foldable
  6. +3 more
8 essays · flat-folding a route the search found123456121110987131415161718242322212019helices 24colours 12 : 12a route is not forbiddenscaffold used 21%48 staples of 3224 helices · 1536 bases · 48 staples · colours 12 : 12

Molecular folding

  1. 1 A sheet that routes itself
  2. 2 Two things called folding
  3. 3 Two ceilings
  4. 4 The helix chooses the lattice
  5. 5 A row the route cannot leave
  6. +3 more
8 essays · biology 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. +3 more
8 essays · history 3 equal partsestimated by eyethe left-hand divisions are exact — a consequence of the fold, not of carethe right-hand ones are a guess, and the error compoundsvalleymountain

Pedagogy

  1. 1 The kindergarten was a geometry class
  2. 2 A schoolteacher's theorem
  3. 3 Taught with a wrong reason
  4. 4 The reader decides the junction
  5. 5 The first thing about layers
  6. +3 more
8 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. +3 more
8 essays · history 20° a crease0.50 turns of papernothing touching anything34° a crease0.85 turns of papernothing touching anything36° a crease0.90 turns of paper1 pair through one another50° a crease1.25 turns of paper5 pairs through one anotherone strip of 10 panels, seen end-onit laps itself at 36.0° a crease, which is where its cross-section closesevery panel is the same length in every frame; the only thing changed is how far each crease is turned

Self-contact

  1. 1 Paper through paper
  2. 2 Closing is not building
  3. 3 A collision is an order
  4. 4 Two refusals that refuse differently
  5. 5 Refused at one lettering
  6. +3 more
8 essays · rigid -10010000.20.40.60.81how far the vertex is drivenstored energybranch oneMVMMbranch twoMVVVboth run downhillfrom the flat state,and end at zeroso the energy does notprefer either branch —the noise decides

Self-folding

  1. 1 Paper that folds itself
  2. 2 One crease decides the sheet
  3. 3 Which crease to push
  4. 4 The hardest instant
  5. 5 Only four creases decide a Miura
  6. +3 more
8 essays · rigid axiom 1through two pointslinearaxiom 2point onto pointlinearaxiom 3line onto linelinearaxiom 4through a point, square to a linelinearaxiom 5point onto a line, through a pointquadraticaxiom 6two points onto two linescubicaxiom 7point onto a line, square to a linelinearthe degree each axiom can solve — one of them is why paper beats the compass

The axioms

  1. 1 One fold at a time, and there are exactly seven of them
  2. 2 Folding beats the compass, by exactly one degree
  3. 3 Why the list stops at seven
  4. 4 Where the cubic comes from
  5. 5 What each axiom is worth
  6. +3 more
8 essays · construction 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. +3 more
8 essays · rigid Paper is made in ChinaPaper reaches JapanPaper is made in EuropeFolded paper is used ceremonially in Japan400 yrPaper is folded for amusement in Japan980 yrThe thousand cranes897 yrThe pajarita is folded in Spain293 yrPaper folding is taught as geometryOne fold solves a cubicThe diamond pattern in a crushed cylinderThe conditions at a flat-foldable vertexThe dashed-and-dotted diagram notationThe Miura foldA five-pointed star from one straight cutAny straight-line drawing, from one straight cutyear of the source500100015002000the date generally giventhe oldest source that says somedian overrun 201.5 years

Attribution

  1. 1 Nothing here is as old as it sounds
  2. 2 The name is not the date
  3. 3 Fifty years in the wrong language
  4. 4 A record is not a proof
  5. 5 The patterns nobody owns
  6. +2 more
7 essays · history 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. +2 more
7 essays · material 4×4 — 16 cranes, 9 corner joinsone sheet, and cut2×2 4 cranes 4 sides of slit3×3 9 cranes 12 sides of slit4×4 16 cranes 24 sides of slit5×5 25 cranes 40 sides of slit6×6 36 cranes 60 sides of slitcranes − joins = 2n − 1the rule the subject is usually stated under is one sheet and no cuts; theoldest surviving origami book does not keep it

Folklore of folding

  1. 1 The oldest book cuts the paper
  2. 2 The star that was cut before it was proved
  3. 3 How many wedges the paper allows
  4. 4 Which cranes can stay joined
  5. 5 Nearly every cutting fails at one crane
  6. +2 more
7 essays · history 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. +2 more
7 essays · construction

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