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 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 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. +6 more
11 essays · material 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 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. +5 more
10 essays · biology 32 × 32 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley

Box pleating

  1. 1 Designing on a grid
  2. 2 What the grid settles
  3. 3 Spelling a tree on a grid
  4. 4 The whole alphabet of a grid
  5. 5 The other grid
  6. +4 more
9 essays · design 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. +4 more
9 essays · material the patternconcentric arcs, alternatingwhat the sheet doesa shape with no flat state at allthe curve is in the crease; the saddle is the paper refusing to stretch

Curved creases

  1. 1 A crease that curves
  2. 2 The sculptors got there first
  3. 3 One curve and one number
  4. 4 Where curved creases meet
  5. 5 A curve has no panels
  6. +4 more
9 essays · material the bar is the share of draws whose letters agree among themselvesa draw that disagrees is a proof that the pattern has no flat folded state with those lettersthe preliminary base200 of 2008 panels · 8 creases · 0 contradict themselvesthe square twist198 of 2009 panels · 12 creases · 2 contradict themselvesthe Yoshimura190 of 20065 panels · 86 creases · 10 contradict themselvesthe Miura fold181 of 20024 panels · 38 creases · 19 contradict themselvesa square twist patch26 of 20049 panels · 84 creases · 174 contradict themselvesa hexagonal patch2 of 20077 panels · 142 creases · 198 contradict themselvesa rhombille patch0 of 200157 panels · 282 creases · 200 contradict themselvesthe sampler returns solutions rather than a uniform draw over them, so these are shares of what it found

Forced order

  1. 1 A proof in one pass
  2. 2 The loop a vertex cannot close
  3. 3 The loop is not the tangle
  4. 4 The lettering nobody could draw
  5. 5 Which condition does the refusing
  6. +4 more
9 essays · flat-folding 18 interior vertices26 mountains · 19 valleyscolumns taper 2.44 : 1packs to 11.2% of flatmountainvalleyraw edgethe taper is in the columns, because Kawasaki does not mention their widthtapering the rows instead puts the alternating sums at 186.4° and 173.6°

Leaf folding

  1. 1 A leaf packs by corrugating
  2. 2 The bud chooses the pattern
  3. 3 Opening with nothing to pull
  4. 4 Twice as thick where it is thickest
  5. 5 A leaf ends its pattern
  6. +4 more
9 essays · biology 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. +4 more
9 essays · complexity the bare sheet4 references · 4 linesnothing has been foldedafter 1 fold9 references · 12 lineshalves, and nothing elseafter 2 folds565 references · 92 lineshalves, thirds, fifths — and worsea fold is an alignment, and an alignment needs something already on the paper to align565 references after 2 folds, and the count is finite however many folds are allowed

Reference points

  1. 1 A fold needs something to align
  2. 2 Cheap where it reaches
  3. 3 Closer than a crease is wide
  4. 4 The sheet decides which points exist
  5. 5 A hole is an edge
  6. +4 more
9 essays · construction 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. +4 more
9 essays · complexity 123456760%65%70%75%80%sheet, longer side to shorterfraction of the sheet claimedbest at 7.0 to 17 flaps · every sheet the same areathe sheet's shape is a design variable that origami paper hides by being sold square

Sheet shape

  1. 1 The square is a choice
  2. 2 The corner is worth four times the middle
  3. 3 A hole is cheap paper
  4. 4 Spending the cheap paper
  5. 5 A notch is not a hole
  6. +4 more
9 essays · design 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. +4 more
9 essays · complexity 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. +4 more
9 essays · rigid populationpassfoldhave a gapbranchcut twice at random51 vertices, 13 kinds9.68.020%0%whole multiples of 45°60 vertices, 1 kinds30.019.3100%69%whole multiples of 30°60 vertices, 13 kinds19.713.177%26%a named vertex, jittered60 vertices, 4 kinds8.08.00%0%

Typical instances

  1. 1 Which vertices are the random ones
  2. 2 Four ways to draw a pattern
  3. 3 The patterns a checker is tested on
  4. 4 Drawn by the same hand
  5. 5 A population that cannot fail
  6. +4 more
9 essays · complexity

All essays