Origami & Folding
what one sheet can do
Concept

Isometry — where it appears

A map that preserves every distance. A folding is one, which is why no panel changes shape, why the paper's area survives, and why every theorem in this subject can be stated about the flat pattern rather than about the folded object.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

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

A flap costs a circle

A flap of a given length uses up every point of the sheet within that distance of it. Two flaps whose circles overlap are asking for the same paper twice — and that one observation turned origami design from an art into an algorithm.

design · Circle packing the patternwhat the sheet doesconcentric arcs with their rulings drawn as radii; the metric matches the flat sheet to 5e-7so nothing here is stretching — every point of the surface is where folding alone can put itthe flat-folding theorems say nothing about any of this: they are about straight creases meeting at a point

The sculptors got there first

Curved-crease folding produced its best objects decades before anybody could compute one. The surfaces were made by hand, the ruling lines that determine them were not calculated until much later, and the mathematics has been catching up ever since.

material · Curved creases cylinderreachablecurved one way onlyconereachablecurved one way, from a pointsphereunreachablecurved two ways — impossiblesaddleunreachablecurved two ways — impossible

What a flat sheet can become

A sheet that cannot stretch cannot become a sphere. That much belongs to differential geometry; what belongs to folding is the three ways round it — seams, curved creases, and a few percent of stretch — and what each one costs.

material · Developability 2030405060708000.511.522.5cone half-angle β, degrees — 90° is the unfolded sheetcurvature, in units of 1/rκ, in space1/(r sin β) — unboundedκ_g, in the surface1/r — flat, at every anglethe isometry, as a lineκ_n, out of the surfacecot β / r — all of the gainκ² = κ_g² + κ_n² to 3e-15true at every sample,not only at the ends

One curve and one number

Folding cannot change how curved a crease is within the surface — that is what an isometry means. Everything a curved fold produces is the curvature it adds out of the surface, and one number controls all of it.

material · Curved creases the patternthe panels, foldedsheet 12.000footprint 1.966 · 6.11 layers on average · 12 at the deepest1.966 × 6.11 = 12.007, which is the sheet

The paper is all still there

A folded sheet is smaller than it was and none of it has gone anywhere. How much smaller it is and how many layers deep it is are not two properties of a pattern — they are one number, and their product is the sheet.

flat-folding · Conservation 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

Paper through paper

Every test the subject has for rigid folding is a statement about a neighbourhood, and a neighbourhood cannot see the far side of the sheet. So a pattern can satisfy all of them while driving one panel straight through another, and the sharpest witness has no interior vertex in it at all.

rigid · Self-contact 1.7e-6 at 1e-61e-61e-51e-41e-31e-21e-61e-51e-41e-31e-2how far each vertex of the flat pattern was moved, in panel widthslargest edge-length errorfitted slope0.9999over four decadesat a displacement ofexactly zero the erroris 6.7e-16, which iswhere the arithmeticstops and not wherethe geometry does5 × 4 panels at 50% folded, with one set of displacement directions scaled across the decadesa slope of one is the claim: the failure is first order, so no displacement is small enough to be free

The only pattern that moves

A rigid motion is not a generic property of a folded pattern. Move one vertex of a Miura by a thousandth of a panel and the sheet has no isometric folded position of that kind at all — and the amount by which it fails is first order in the displacement, so no move is small enough to be free.

rigid · Rigid folding the composition, and what it has to equal3 reflections, in order[ 1.000 0 ][ 0 -1.000 ]+ ( -1.732, 1.000 )=?the gluing map of a Möbius band[ 1.000 0 ][ 0 -1.000 ]+ ( -1.732, 1.000 )they agree to rounding, so the band foldsand both turn the paper the same way, so the parity is righton a disc the right-hand side is the identity, which is why nobody writes it down

Closure is not the identity

Walk a folded state from panel to panel, composing a reflection at every crease, and come back to where the walk started: the composition has to be the identity. That is the rule everybody states, and it is a special case. On a sheet whose edges are glued the walk does not come back to where it started, and what the composition has to equal is the gluing map.

flat-folding · Flat-foldability 3 creases on a Möbius bandthe panels take two coloursseamthe same seam123the right edge onto the left, turned over3 creases, 3 panelsinterior vertices: 0two-coloursthe reflections closeand turn the paper the right waymountainvalleyraw edge

The triangle a strip becomes

A Möbius band of paper folds flat into an equilateral triangle, and the shortest strip that will do it is √3 times its own width. The number is not put in: the crease angles come out of a condition on their alternating sum, the positions come out of two linear equations, and the length is where the drawing stops fitting.

flat-folding · Flat-foldability the same arithmetic three waysm divisions leave a residual of f ⁄ m inside each piece, whatever the divisions are made ofthe material givesdivisions neededas goresas tucksas curved creases2.0%1919 cuts, 59.7 of seam19 tucks, 3 sheets deep19 creases, no cut and no pile5.0%88 cuts, 25.1 of seam8 tucks, 3 sheets deep8 creases, no cut and no pile10.0%44 cuts, 12.6 of seam4 tucks, 3 sheets deep4 creases, no cut and no pile20.0%22 cuts, 6.28 of seam2 tucks, 3 sheets deep2 creases, no cut and no pilecap of 90°, rim excess 36.3% · the count is ⌈f ⁄ ε⌉ in every column; only the cost of a division changes

Three answers, one count

Seams, curved creases and a few per cent of stretch are the three ways round the sphere, and a tuck is a fourth. All four dispose of one quantity — the excess circumference a flat disc has over the sphere's circle — and all four dispose of it by dividing the circle. So the number of divisions needed is the same whichever answer is chosen: nineteen for a hemisphere in a material that gives two per cent, eight at five, four at ten. What differs is what a division costs, and one of the four runs out.

material · Developability 00.20.40.60.811.21.41.600.10.20.30.4arc from the pole (radians)excess, as a share of the circle7 rings5.0% stretchfirst at 0.35last at 0.98of the way to the rima ring goes in wherever the residual excess would otherwise pass what the material takes

Where a ring of divisions belongs

A pattern that divides the circle everywhere as finely as its rim requires is over-divided for most of its radius, because the excess grows from nothing. Putting a ring of new divisions in wherever the residual would otherwise pass what the material takes gives seven rings on a hemisphere at five per cent of stretch, at 0.35, 0.50, 0.62, 0.72, 0.81, 0.90 and 0.98 of the way out — and the first of those sits where a completely different criterion put its first tuck start.

material · Developability

Named alongside it

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

Developable surfaceGaussian curvatureClosureConservationFlat-foldabilityFolded stateGluingGoreLayer countThe Möbius bandOrientabilityPleat

All concepts