ACM SIGGRAPH 2024, Denver, CO, USA

Modal Folding: Discovering Smooth Folding Patterns for Sheet Materials using Strain-Space Modes

ETH Zürich
Université de Montréal
ETH Zürich
ETH Zürich
University of Haifa
ETH Zürich
ETH Zürich

Fig. 1. Our method automatically generates a diverse set of folding patterns using strain-space modes. Here we show a particular folding transformation for a square sheet of fabric and its physical prototype (bottom right).


Folding can transform mundane objects such as napkins into stunning works of art. However, finding new folding transformations for sheet materials is a challenging problem that requires expertise and real-world experimentation. In this paper, we present Modal Folding---an automated approach for discovering energetically optimal folding transformations, i.e., large deformations that require little mechanical work. For small deformations, minimizing internal energy for fixed displacement magnitudes leads to the well-known elastic eigenmodes. While linear modes provide promising directions for bending, they cannot capture the rotational motion required for folding. To overcome this limitation, we introduce strain-space modes---nonlinear analogues of elastic eigenmodes that operate on per-element curvatures instead of vertices. Using strain-space modes to determine target curvatures for bending elements, we can generate complex nonlinear folding motions by simply minimizing the sheet's internal energy. Our modal folding approach offers a systematic and automated way to create complex designs. We demonstrate the effectiveness of our method with simulation results for a range of shapes and materials, and validate our designs with physical prototypes.


Fig. 2. Periodic folding patterns obtained using Modal Folding (a-e) with reflection boundary conditions and their corresponding physical prototypes (right) manufactured using copper sheets and 3D-printed press dies.

Fig. 6. Folding patterns for a thin sheet cut into the pattern of a life flower.


Supplementary Video


The authors thank the anonymous reviewers for their valuable feedback. This work was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No.866480), and the Swiss National Science Foundation through SNF project grant 200021_200644.