Origami & Folding
what one sheet can do
Series

Curved creases — the series

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

    part 1 · material
  2. 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.

    part 2 · material
  3. 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.

    part 3 · material
  4. 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.

    part 4 · material
  5. 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.

    part 5 · material
  6. 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.

    part 6 · material
  7. 17%7 circles fit; the middle 17 per cent cannot carry anyspacing 0.06, rulings at 0.9 radians to the crease

    The gap between two curves

    The rulings leaving a curved crease are not parallel, so they cross, and the surface exists only as far as the first crossing. That bound is usually read as a limit on how far a design extends outward. It is not: the paper between two curved creases has to be reachable from both, so the bound bites hardest where the circles are smallest, and a concentric pleat has a hole in the middle that no sheet size removes.

    part 7 · material
  8. 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.

    part 8 · material
  9. 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.

    part 9 · material

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