On the incompressibility of cylindrical origami patterns

TL;DR

This paper proves that most cylindrical origami patterns cannot be compressed without stretching or deforming, significantly limiting the design space for origami-based deployable structures and metamaterials.

Contribution

It provides a geometric proof that only finitely many isometric embeddings exist for cylindrical origami, restricting possible designs for deployable structures.

Findings

Most cylindrical origami patterns have limited isometric embeddings.

Compressibility requires material stretching or fold deformation.

Design space for origami deployable structures is significantly restricted.

Abstract

The art and science of folding intricate three-dimensional structures out of paper has occupied artists, designers, engineers, and mathematicians for decades, culminating in the design of deployable structures and mechanical metamaterials. Here we investigate the axial compressibility of origami cylinders, i.e., cylindrical structures folded from rectangular sheets of paper. We prove, using geometric arguments, that a general fold pattern only allows for a finite number of \emph{isometric} cylindrical embeddings. Therefore, compressibility of such structures requires either stretching the material or deforming the folds. Our result considerably restricts the space of constructions that must be searched when designing new types of origami-based rigid-foldable deployable structures and metamaterials.