Origami & Folding
what one sheet can do
Series

Idealisation — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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

    Four things that are not true

    Zero thickness, no stretch, creases that are lines, and perfect memory. Every theorem on this site rests on all four, every one of them is false, and the interesting engineering is exactly where each fails.

    part 1 · material
  2. 5%10%15%15°30°45°60°75°90°strain the rim must takehow much of a sphere the cap coversdry paper1% of stretchreaches 14°damp paper3% of stretchreaches 24°wet-folded6% of stretchreaches 35°a hemisphere needs 36% and nothing made of cellulose is going to supply it

    Paper that stretches on purpose

    Wet-folding breaks the assumption every theorem of flat folding rests on, deliberately. It does not repeal the geometry — it buys a few percent of strain, and a few percent of strain is worth about twenty degrees of sphere.

    part 2 · material
  3. ρ = 0.12 mmarc 0.377 mm, stack advances 0.240 mmlost per crease (π − 2)ρ = 0.1370 mmone crease, at its real radiuson a 150 mm sheet8 × 81.0 mm — 0.6%16 × 162.1 mm — 1.4%24 × 243.2 mm — 2.1%32 × 324.2 mm — 2.8%48 × 486.4 mm — 4.3%lost along every line of the grid, in both directionswhich is why an ambitious grid is folded from thin paper

    The crease has a radius

    A fold does not go through a line. It goes round a small arc, and the arc uses more paper than the stack advances by — a fraction of a millimetre per crease, and several millimetres across a grid, which is why an ambitious tessellation comes out short.

    part 3 · material
  4. 24681012-202halvingsmetres of paper (powers of ten)297 mm — 6 folds, 64 layers1 m — 7 folds, 128 layers10 m — 8 folds, 256 layers100 m — 10 folds, 1024 layers1200 m — 12 folds, 4096 layerspaper 0.1 mm thick · L = (πt/6)(2ⁿ + 4)(2ⁿ − 1)the loss is the paper that goes round the closed end, and it doubles twice per fold

    How many times can it be halved

    The folklore says seven, and the folklore is a statement about one sheet of paper. What actually binds is arithmetic: every halving doubles the layers and the paper spent at the closed end grows as the square of the layer count, so the length needed for twelve folds is nearly a kilometre.

    part 4 · material
  5. computed herenot established herethe curvature a growth field forcesthe growth solverthat any leaf grows that waythat a corrugation with tapered columns folds flatthe pattern librarythat a leaf's creases are where these arethe packed fraction of four folding geometrieswing-packingthe packed fraction of any wingthat one freedom needs one driverdof-censushow an insect actually deploys a wingthe surface a fold count fits in a boxsurface-in-volumethe surface area of any organwhich lattice shapes admit a single routethe routing modulethat any published design used this routeevery row's left side is a computation in this repository; every right side is a claim it does not make

    The organism is not the model

    Every figure in this field draws a fold this repository computed. Not one of them measures a leaf, a wing or a gut. That is the rule the field was built to, and it is worth stating as a table rather than as a preamble — because a field about living things is where a computed geometry is most likely to be read as an observation.

    part 5 · biology
  6. two models, and the sheet that has carried both12 crossings, of which 10 cannot fold flatThe preliminary baseThe hexagon twistthe sheet after bothno crease has moved and none has been added; what is new is where they cross

    The sheet remembers

    Perfect memory is the fourth idealisation, and the least examined of the four. It is usually read as a complaint that paper will not lie flat again; the large half is the opposite. A sheet folded once is no longer blank, so folding a second model into it is folding the union of two patterns — and a union folds flat only where every new crease meets every old one at a right angle.

    part 5 · material
  7. patterncrease length near the grainbest / worstceilingThe preliminary base4 crease directions21% / 21%45°The Miura fold3 crease directions54% / 46%35°The square twist2 crease directions0% / 0%45°The hexagon twist3 crease directions30% / 0%30°The Yoshimura pattern3 crease directions29% / 0%30°Fold and cut — the triangle6 crease directions35% / 11%29°The tapered corrugation3 crease directions42% / 0%33°The waterbomb tessellation3 crease directions21% / 0%45°

    The fifth thing that is not true

    Four idealisations underlie every theorem here and each has had an essay. There is a fifth and it has never been named, because it is invisible in exactly the way the others are not: paper has a grain, no theorem in the subject mentions a direction, and so nothing in the whole apparatus can tell a folder which way up to lay the pattern down.

    part 6 · material
  8. one rectangle, glued four waysa disc, two cylinders and a torus — from one drawing4 edges lefta disc2 edges lefta cylinder, across2 edges lefta cylinder, alongno edges lefta torusthe same rectangle and the same creases in all four, and nothing in the drawing says which is whichmatching arrowheads mean the two edges are one edge of the paper

    The sixth thing that is not true

    Five idealisations underlie every theorem here and each has had an essay: no thickness, no stretch, creases that are lines, perfect memory, no grain. There is a sixth, it is more basic than any of them, and it is the one nobody has ever thought to name — the paper is a disc.

    part 7 · material

All series