Origami & Folding
what one sheet can do
Concept

Error propagation — where it appears

How an inaccuracy made at one fold reaches the folds after it. A systematic error grows linearly with the fold count and a random one as a square root, which is why the two need to be told apart before anything is quoted.

Named by 12 essays across 6 fields — each of them below, with the objects they name alongside it.

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

Dividing without measuring

A square can be divided into any whole number of equal parts by folding alone — exactly, with no ruler, and with no error to accumulate. The construction is one fold and a theorem nobody expected.

construction · Exact division 32 × 32 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley

Designing on a grid

Box pleating gives up the efficiency of a free packing and buys creases that land where they are supposed to. For a design with hundreds of folds that is not a compromise — it is the only thing that makes it foldable.

design · Box pleating guessoff by 0.16667fold 1off by 0.08333fold 2off by 0.04167fold 3off by 0.02083fold 4off by 0.01042fold 5off by 0.00521fold 6off by 0.00260solid mark — the fold is aiming at 1/3; dashed — at where 1/3 has gonehalving word R L — period 2, from 2^2 − 1 = 3 × 1every fold halves the error exactly, so 6 folds divide it by 64

Folding a strip into thirds

A third cannot be constructed by the axioms, so it is not constructed. It is guessed, and then halved into place — an algorithm rather than a construction, with an error that falls by exactly half at every fold.

construction · Exact division 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.

material · Idealisation 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.

material · Idealisation 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

Error is folded too

A folded position is a composition of reflections, and a reflection in a line that is slightly off turns everything beyond it by twice as much. So an error does not stay where it was made — and whether it grows with the crease count or with its square root depends on whether it is the same error every time.

rigid · Tolerance 0.5°1%1°2%2°3%5°8%10°16%20°33%how far each sector would have to moveshare that would fold40,000 random four-crease verticesthe median vertex is 31.31° per sector from folding, the mean 33.91°none of them folds, and almost none of them nearly does either

A near miss is nearly as rare

Flat-foldability is a coincidence of measure zero, which is usually where the argument stops. Measure how far a random vertex is from folding rather than whether it does, and the answer is thirty-one degrees a sector — so the tolerance real paper has does not buy back anything at all, and a pattern that nearly folds had to start near one that did.

flat-folding · Genericity the ratio of each column's cosine to the first column'scol 1col 2col 3col 4col 5the sheet that folds1.0000-1.08221.0000-1.08221.00001.0000-1.08221.0000-1.08221.00001.0000-1.08221.0000-1.08221.0000row 1row 2row 3one row moved by 14 per cent1.0000-1.08221.0000-1.08221.00001.0000-1.13971.0000-1.13971.00001.0000-1.08221.0000-1.08221.0000row 1row 2row 3worst disagreement between rows: 4e-16 before, 0.0575 after

Where an error goes

A misplaced crease in a folded sheet has to be paid for somewhere, and this subject has two answers already — the error is folded too, and the hinge is where it ends up. There is a third. In a quadrilateral mesh a mistake in one row has no consequence in that row at all: it is felt by the columns, which is to say by every other row on the sheet.

rigid · Tolerance the bar is the vertices the drawing has and the list does notThe preliminary base09 listed · panels closeThe Miura fold035 listed · panels closeThe square twist016 listed · panels closeThe hexagon twist022 listed · panels closeThe Yoshimura pattern045 listed · panels closeFold and cut — the triangle011 listed · panels closeThe tapered corrugation040 listed · panels closeThe waterbomb tessellation041 listed · panels closethe square grid, assembled064 listed · panels closethe triangular grid, assembled1282 listed · panels 1.73 apartthe honeycomb, assembled1884 listed · panels 2.00 apartthe rhombille tiling, assembled12138 listed · panels 1.86 apartthe elongated triangular tiling, assembled576 listed · panels 1.73 apartevery pattern with a bar has panels that cannot be placed, and every pattern without one places exactly

How deep is a crossing

A crossing is a verdict with no middle: two creases either pass through one another or they do not, and the first makes a pattern unfoldable while the second leaves it untouched. Measured on the patches where they occur, the shallowest crossing runs 0.16 mm past the end of the crease it meets, on a sheet 150 mm across. Five of the forty-seven are under half a millimetre, which is thinner than the line a pencil draws.

rigid · Tolerance guessoff by 0.16667fold 1off by 0.08333fold 2off by 0.04167fold 3off by 0.02083fold 4off by 0.01042fold 5off by 0.00521fold 6off by 0.00260solid mark — the fold is aiming at 1/3; dashed — at where 1/3 has gonehalving word R L — period 2, from 2^2 − 1 = 3 × 1every fold halves the error exactly, so 6 folds divide it by 64

Dividing a loop into n

Fujimoto's method divides a strip into any number of equal parts by folding badly and then folding the error in half, over and over. It converges because each step halves what is left over. On a closed loop there is no edge for the leftover to sit against, and what replaces the edge is the loop's own closure.

construction · Exact division how squarely the folds that fix a reference crossand what a crossing at that angle does to an error in the foldingthe four linear axioms565 references · worst 36.9°with the conic axiom16,890 references · worst 6.3°60° to 90°error × 1.264.6%54.4%45° to 60°error × 1.418.2%21.8%30° to 45°error × 2.017.2%17.1%15° to 30°error × 3.90.0%5.2%8° to 15°error × 7.20.0%1.2%under 8°error × 14.30.0%0.3%the linear axioms bottom out at 36.9° — a multiplier of 1.67 — and the conic axiom reaches 6.3°, a multiplier of 9.12 rounds on a square · a reference priced at 1 ⁄ sin of the widest angle its own folds make · shares, so the two sets are comparable

What buys the reach costs the accuracy

A reference is a crossing of two creases, and a crossing transmits a folding error multiplied by one over the sine of the angle the creases make. Measured across the whole closure on a square, the four linear axioms never produce a crossing shallower than thirty-seven degrees — and the conic axiom, the one that sends a point onto a line and reaches the heptagon, produces crossings under seven.

construction · Constructible polygons the bases at which a crossover is possiblethe backbone turns 360° over a turn's worth of bases, and a crossover needs it facing the neighbourhoneycomb — three neighbours6 exact, 0 near071421283542square — four neighbours2 exact, 4 near4.3°4.3°4.3°4.3°07142128354210.5 bases to a turn · a mark is a base whose backbone faces a neighbour to within 5°, and the number under it is the miss

The helix chooses the lattice

A strand can cross to the next helix only where its backbone faces that helix, and a double helix turns about 34.3° a base. Three neighbours a third of a turn apart are faced exactly every seven bases. Four neighbours a quarter of a turn apart are never faced exactly by any whole number of bases — the nearest miss by 4.3°, and the misses do not average out, they add: 17° over thirty-two bases, 137° over two hundred and fifty-six. The lattice a design is drawn on is decided by the molecule before any shape is chosen.

biology · Molecular folding

Named alongside it

The objects these essays reach for when they reach for this one.

Reference pointToleranceGridIdealisationCrease radiusRational divisionBox pleatingConvergenceFujimoto's methodKawasaki's theoremManufacturingThickness

All concepts