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.
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. +5 more
10 essays · complexity 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 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. +4 more
9 essays · biology 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 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. +4 more
9 essays · history 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 the design as it stood20.00 of paper · 5 creases meet the cutthe same design, one strip wider21.80 of paper · the strip is 0.45 acrossthe band is 0.45 × 4 = 1.8000, and that is the entire difference between the two patternsall 44 creases away from the cut keep their length; the 5 that cross it are longer by 0.45 and by nothing elsea design grows by accretion because the arithmetic of growing it is this short

Grafting

  1. 1 Paying in paper
  2. 2 The second term
  3. 3 A graft needs a square line
  4. 4 Six rectangles and one term
  5. 5 A design that keeps its lines clear
  6. +4 more
9 essays · design 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 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. +4 more
9 essays · biology 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. +4 more
9 essays · history 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 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 patternhow much smaller it foldscreasingper sheet-widthpreliminary8.0 layers, 8 at the deepest8.0×4.81.65×footprint × depth = 1.004 of the sheetmiura8.1 layers, 16 at the deepest8.1×6.21.31×footprint × depth = 1.001 of the sheetyoshimura32.0 layers, 36 at the deepest32.0×11.82.71×footprint × depth = 1.000 of the sheetwaterbomb31.6 layers, 32 at the deepest31.8×14.32.22×footprint × depth = 0.992 of the sheettwist3.0 layers, 9 at the deepest3.0×4.70.64×footprint × depth = 0.995 of the sheetthe shrinkage is the pattern's, not the paper's — nothing here knows what the sheet is made of

Shrinkage

  1. 1 What a corrugation costs
  2. 2 A shrink is two numbers
  3. 3 Folding it flat is one similarity
  4. 4 The turn a column costs
  5. 5 The plane the five points were in
  6. +4 more
9 essays · tessellation 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 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 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 the shrink5 intermediate outlines drawneach edge moved inward by thesame distance — computed, not drawnthe tracesstraight, because every edge movesat one rate along its own normalhow far it can go0.5391 sheet-widthsfound by bisection on the outline'sown area, not by inspectionthe construction that always works — which is what universal means here

Molecules

  1. 1 The last free parameter
  2. 2 The molecule that does not exist
  3. 3 The skeleton changes its mind
  4. 4 The corner that splits the shrink
  5. 5 A tree cannot argue
  6. +3 more
8 essays · design the degree of the equation, and what it is made ofnumberdegreemade ofwhere it comes from½11a fold in half√222^1the diagonal of the squareφ22^1the silver rectangle's cousin∛233^1doubling the cube2 cos(2π/7)33^1the regular heptagon∜242^2a square root of a square root∛2 · √262^1 · 3^1a product of two of them2^(1/5)5not twos and threesa fifth root2 cos(2π/11)5not twos and threesthe regular hendecagonchecked by exhaustion: no number here satisfies a rational equation of lower degree with coefficients up to 6

Origami numbers

  1. 1 The numbers a fold reaches
  2. 2 Reachable is not cheap
  3. 3 The field has no edge
  4. 4 Twos and threes run out
  5. 5 Gauss's polygon is the expensive one
  6. +3 more
8 essays · construction

All essays