Characterization of Curved Creases and Rulings: Design and Analysis of Lens Tessellations

TL;DR

This paper introduces a broad family of curved-crease tessellations with lens-shaped motifs, demonstrating that the curves can be any smooth convex shape and providing a geometric proof of foldability without extra creases.

Contribution

It generalizes the design of lens tessellations by allowing arbitrary smooth convex curves and establishes their foldability using differential geometry.

Findings

Curves can be any smooth convex shape without inflection points.

Folded forms exist without additional creases.

Identifies ruling configurations for curved foldings.

Abstract

We describe a general family of curved-crease folding tessellations consisting of a repeating "lens" motif formed by two convex curved arcs. The third author invented the first such design in 1992, when he made both a sketch of the crease pattern and a vinyl model (pictured below). Curve fitting suggests that this initial design used circular arcs. We show that in fact the curve can be chosen to be any smooth convex curve without inflection point. We identify the ruling configuration through qualitative properties that a curved folding satisfies, and prove that the folded form exists with no additional creases, through the use of differential geometry.