Origami & Folding
what one sheet can do
Concept

Curved crease — where it appears

A crease that follows a curve rather than a straight line. The paper either side of it is forced into a developable surface nobody creased there, and none of the flat-folding theory applies, because those theorems are about straight creases meeting at points.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

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

A crease that curves

Bend a crease and the paper either side is forced into a shape nobody creased. The flat-folding theorems say nothing about it, because they are statements about straight creases meeting at a point.

material · Curved creases 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 straight creasescurvature 1.4curvature 2.6the tangents are the same in every panelsectors 60.0°, 120.0°, 120.0°, 60.0°they sum to 360.0°, and alternately to 180.0° and 180.0°which is Kawasaki, on tangents rather than on lines

Where curved creases meet

A curved-crease design looks like a smooth object and its constraints are not smooth. They live at the finitely many points where creases cross, and at each of those the conditions are about the creases' tangent directions — the curvature does not appear in them at all.

material · Curved creases the creaseno panels at all128.3° of turning4 segments8 panels128.3° of turning8 segments16 panels128.3° of turning16 segments32 panels128.3° of turningone crease, cut into flat pieceseach panel keeps its width across the crease and loses length along it as the count risesa finer approximation is a better picture and the same total kink, spread over more joints

A curve has no panels

A rigid folding is a finite list of flat pieces joined along lines. A curved crease has no such list, and refining one does not help: the kink at each joint falls as one over the segment count, and the total of the kinks does not fall at all, because it is a constant of the curve.

material · Curved creases the crease, its rulings, and the line where they crossan elliptical crease, rulings at 52° from the tangentthe shortest ruling0.0998the tightest radius0.1274their ratio0.7833nothing on the crease pattern marks this line, and no amount of paper moves it

Where the rulings run out

A curved fold's surface is made of straight lines leaving the crease, and the lines are not parallel, so they cross. Past the first crossing there is no surface: two points of the paper have been sent to one point of space. The boundary is a curve nobody drew, no crease pattern shows it, and it sits at the sine of the ruling angle times the crease's own tightest radius — on every curve tried.

material · Curved creases how far each of a curved fold's two surfaces reaches before its rulings crossas a share of the crease's own tightest radius of curvatureone surfacethe othershare boundeda circular crease at 0.40.389—100% / 0%a circular crease at 0.70.644—100% / 0%a circular crease at 10.841—100% / 0%a circular crease at 1.30.964—100% / 0%an elliptical crease at 0.40.389—100% / 0%an elliptical crease at 0.70.644—100% / 0%an elliptical crease at 10.841—100% / 0%an elliptical crease at 1.30.964—100% / 0%a parabolic crease at 0.40.389—100% / 0%a parabolic crease at 0.70.644—100% / 0%a parabolic crease at 10.841—100% / 0%a parabolic crease at 1.30.964—100% / 0%a wave at 0.40.3890.38950% / 50%a wave at 0.70.6440.64450% / 50%a wave at 10.8410.84150% / 50%a wave at 1.30.9640.96450% / 50%a dash is a surface whose rulings never converge, which is a surface with no boundary of this kind at all

Only one side can run out

A curved fold has two surfaces and every reach ever computed here has been one of them. The closed form's denominator is the crease's curvature plus the rate the ruling angle turns at, and crossing to the other surface negates both — so at any point of any crease at most one of the two surfaces can have its rulings converge. A crease that never changes the way it bends therefore has a surface with no such boundary at all, anywhere along it.

material · Curved creases what happens when the ruling angle is not constantthe share of the crease each surface is bounded over, and how far it reaches thereturning rateone surfaceits reachthe otherits reach0.00100%0.29560%—0.25100%0.23500%—0.50100%0.19090%—0.75100%0.14840%—0.90100%0.11990%—1.00100%0.09930%—1.1086%0.077314%2.95101.2580%0.041320%1.17461.4075%0.000025%0.7250the crease is a circular crease and the angle runs 1.4 plus the rate times a sine, so the rate is how fast it turns against how fast the crease bends

An angle that turns faster than the crease

Which of a curved fold's two surfaces runs out is decided by a sum of two rates — how fast the crease bends and how fast the ruling angle turns — and every measurement so far has set the second to zero. Let it turn and it carries the sign on its own: past a rate of exactly one, a crease of unchanging curvature bounds both of its surfaces, which no constant angle on that crease can do. Below that rate the turning costs reach without changing anything else.

material · Curved creases

Named alongside it

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

Developable surfaceRulingCurvatureEnvelopeIdealisationRuling lineDevelopabilityGeodesic curvatureIsometryKawasaki's theoremNormal curvaturePanel

All concepts