Origami & Folding
what one sheet can do
Concept

Ruling line — where it appears

A straight segment along which a curved fold's surface is flat, running from the crease outward. Every point of such a surface lies on exactly one, which is what makes the surface a family of lines rather than a shape, and what lets its edge be found by asking where two of them meet.

Named by 4 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 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 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 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

Named alongside it

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

Developable surfaceCurved creaseIsometryGeodesic curvatureNormal curvaturePanelReflectionRigid origamiSculptureTorsionTotal turning

All concepts