1002: Flat-folding Curve-following Parabolic Corrugation

I had been exploring corrugations that followed curved lines, as you do, and sort of worked out that you needed a quadrilateral face with equidistant gutters either side, but my rough approximations were foldable but not pretty:

Flat-folding Curve-following Parabolic Corrugation

Then I saw a published paper, about the same thing, that suggested square/rhombi arranged diagonally to follow the line, organised diagonal-based accordion pleats, and a scale factor bigger of the same shape for the gutter creases and bingo, problem solved.

Flat-foldability is a thing, there is lots of maths in it, but it is so satisfying to have manually derived something that was subsequently proven (*flex*).

Flat-folding Curve-following Parabolic Corrugation CP

I love the geometry of this fold, a parabola. This same technique can be used for most curves (arbitrary complexity), but there are limitations if the accordion folds intersect – regular geometry lets these intersections co-exist in pretty ways, but when the curve is too concave/convex then the distortion means you have to intervene (using scissors to remove the area of argument).

Flat-folding Curve-following Parabolic Corrugation development

I have to explore this more – I imagine sine waves would look amazing, and using nice paper might make frame-worthy art. We shall see.

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