• Odell, G. M., Oster, G., Alberch, P. & Burnside, B. The mechanical basis of morphogenesis: I. Epithelial folding and invagination. Dev. Biol. 85, 446–462 (1981).


    Google Scholar
     

  • Dawes-Hoang, R. E. et al. folded gastrulation, cell shape change and the control of myosin localization. Development 132, 4165–4178 (2005).


    Google Scholar
     

  • Martin, A. C. & Goldstein, B. Apical constriction: themes and variations on a cellular mechanism driving morphogenesis. Development 141, 1987–1998 (2014).


    Google Scholar
     

  • Tozluoglu, M. et al. Planar differential growth rates initiate precise fold positions in complex epithelia. Dev. Cell 51, 299–3124 (2019).


    Google Scholar
     

  • Savin, T. et al. On the growth and form of the gut. Nature 476, 57–62 (2011).

    ADS 

    Google Scholar
     

  • Collinet, C. & Lecuit, T. Programmed and self-organized flow of information during morphogenesis. Nat. Rev. Mol. Cell Biol. 22, 245–265 (2021).


    Google Scholar
     

  • Inoue, Y., Tateo, I. & Adachi, T. Epithelial tissue folding pattern in confined geometry. Biomech. Model. Mechanobiol. 19, 815–822 (2020).


    Google Scholar
     

  • Münster, S. et al. Attachment of the blastoderm to the vitelline envelope affects gastrulation of insects. Nature 568, 395–399 (2019).

    ADS 

    Google Scholar
     

  • Sui, L. et al. Differential lateral and basal tension drive folding of Drosophila wing discs through two distinct mechanisms. Nat. Commun. 9, 4620 (2018).

    ADS 

    Google Scholar
     

  • Akula, S. K., Exposito-Alonso, D. & Walsh, C. A. Shaping the brain: the emergence of cortical structure and folding. Dev. Cell 58, 2836–2849 (2023).


    Google Scholar
     

  • Nikolopoulou, E., Galea, G. L., Rolo, A., Greene, N. D. E. & Copp, A. J. Neural tube closure: cellular, molecular and biomechanical mechanisms. Development 144, 552–566 (2017).


    Google Scholar
     

  • Schenk, M. et al. Origami folding: a structural engineering approach. Origami 5, 291–304 (2011).


    Google Scholar
     

  • McShane, S. G. et al. Cellular basis of neuroepithelial bending during mouse spinal neural tube closure. Dev. Biol. 404, 113–124 (2015).


    Google Scholar
     

  • Yue, S. A review of origami-based deployable structures in aerospace engineering. J. Phys. Conf. Ser. 2459, 012137 (2023).


    Google Scholar
     

  • Andrejevic, J., Lee, L. M., Rubinstein, S. M. & Rycroft, C. H. A model for the fragmentation kinetics of crumpled thin sheets. Nat. Commun. 12, 1470 (2021).

    ADS 

    Google Scholar
     

  • Aharoni, H. & Sharon, E. Direct observation of the temporal and spatial dynamics during crumpling. Nat. Mater. 9, 993–997 (2010).


    Google Scholar
     

  • Gottesman, O., Andrejevic, J., Rycroft, C. H. & Rubinstein, S. M. A state variable for crumpled thin sheets. Commun. Phys. 1, 70 (2018).


    Google Scholar
     

  • Leal, F. C. B. & Gomes, M. A. F. Unfolding of crumpled thin sheets. Phys. Rev. E 106, 025002 (2022).

    ADS 

    Google Scholar
     

  • Davidovitch, B. & Démery, V. Rucks and folds: delamination from a flat rigid substrate under uniaxial compression. Eur. Phys. J. E 44, 11 (2021).


    Google Scholar
     

  • Box, F. et al. Dynamics of wrinkling in ultrathin elastic sheets. Proc. Natl Acad. Sci. USA 116, 20875–20880 (2019).

    ADS 
    MathSciNet 

    Google Scholar
     

  • Liu, S. et al. Conformability of flexible sheets on spherical surfaces. Sci. Adv. 9, 2709 (2023).


    Google Scholar
     

  • Fouchard, J. et al. Curling of epithelial monolayers reveals coupling between active bending and tissue tension. Proc. Natl Acad. Sci. USA 117, 9377–9383 (2020).

    ADS 

    Google Scholar
     

  • Schmalholz, S. M. & Schmid, D. W. Folding in power-law viscous multi-layers. Phil. Trans. R. Soc. A 370, 1798–1826 (2012).

    ADS 

    Google Scholar
     

  • Peraza Hernandez, H. D., Edwin, A. & Lagoudas, D. Active Origami: Modeling, Designs, and Applications (Springer, 2019).

  • Mu, J. et al. Origami-inspired active graphene-based paper for programmable instant self-folding walking devices. Sci. Adv. 1, 1500533 (2015).

    ADS 

    Google Scholar
     

  • Sato, Y., Terashima, S. & Iwase, E. Origami-type flexible thermoelectric generator fabricated by self-folding. Micromachines https://doi.org/10.3390/mi14010218 (2023).

  • Leanza, S., Wu, S., Sun, X., Qi, H. J. & Zhao, R. R. Active materials for functional origami. Adv. Mater. 36, 2302066 (2024).


    Google Scholar
     

  • Ge, Q., Dunn, C. K., Qi, H. J. & Dunn, M. L. Active origami by 4D printing. Smart Mater. Struct. 23, 094007 (2014).

    ADS 

    Google Scholar
     

  • Felton, S., Tolley, M., Demaine, E., Rus, D. & Wood, R. A method for building self-folding machines. Science 345, 644–646 (2014).

    ADS 

    Google Scholar
     

  • Liu, Y., Shaw, B., Dickey, M. D. & Genzer, J. Sequential self-folding of polymer sheets. Sci. Adv. 3, 1602417 (2017).

    ADS 

    Google Scholar
     

  • Randall, C. L., Gultepe, E. & Gracias, D. H. Self-folding devices and materials for biomedical applications. Trends Biotechnol. 30, 138–146 (2012).


    Google Scholar
     

  • Novelino, L. S., Ze, Q., Wu, S., Paulino, G. H. & Zhao, R. Untethered control of functional origami microrobots with distributed actuation. Proc. Natl Acad. Sci. USA17, 24096–24101 (2020).

    ADS 

    Google Scholar
     

  • Zartman, J. J. & Shvartsman, S. Y. Unit operations of tissue development: epithelial folding. Annu. Rev. Chem. Biomol. Eng. 1, 231–246 (2010).


    Google Scholar
     

  • Brannon, C. M. & Prakash, M. Cilia-driven epithelial folding and unfolding in an early diverging animal. Proc. Natl Acad. Sci. USA 122, 2517741122 (2025).


    Google Scholar
     

  • Smith, C. L. et al. Novel cell types, neurosecretory cells, and body plan of the early-diverging metazoan trichoplax adhaerens. Curr. Biol. 24, 1565–1572 (2014).


    Google Scholar
     

  • Bull, M.S., Prakash, V.N. & Prakash, M. Ciliary flocking and emergent instabilities enable collective agility in a non-neuromuscular animal. Preprint at https://arxiv.org/abs/2107.02934 (2021).

  • Davidescu, M. R., Romanczuk, P., Gregor, T. & Couzin, I. D. Growth produces coordination trade-offs in Trichoplax adhaerens, an animal lacking a central nervous system. Proc. Natl Acad. Sci. USA 120, 2206163120 (2023).


    Google Scholar
     

  • Pieranski, P., Godinho, M.H. Collisions of monopoles, disclinations and dislocations. Eur. Phys. J. Spec. Top. https://doi.org/10.1140/epjs/s11734-024-01253-9 (2024).

  • Bull, M. S., Kroo, L. A. & Prakash, M. Excitable mechanics embodied in a walking cilium. Preprint at https://arxiv.org/abs/2107.02930 (2021).

  • Madhu, G.et al. in Advancements in Optical Methods, Digital Image Correlation & Mechanics of Biological Systems and Materials Vol. 2 (eds Hwang, C.-H., Shaw, G. A., Fujigaki, M., Kasza, K. & McGhee, A.) 83–88 (Springer, 2025).

  • Prakash, V. N., Bull, M. S. & Prakash, M. Motility-induced fracture reveals a ductile-to-brittle crossover in a simple animal’s epithelia. Nat. Phys. 17, 504–511 (2021).


    Google Scholar
     

  • Chopin, J., Démery, V. & Davidovitch, B. Roadmap to the morphological instabilities of a stretched twisted ribbon. J. Elast. 119, 137–189 (2015).

    MathSciNet 

    Google Scholar
     

  • Leria, M.et al. Fast mechanosensitive and Ca2+-dependent reorientation of motile cilia basal bodies in the placozoan, Trichoplax. Curr Biol. https://doi.org/10.1016/j.cub.2026.04.054 (2026).

  • Bouffanais, R. in An Information-Theoretic Approach to Collective Behaviors 75–93 (Springer, 2016); https://doi.org/10.1007/978-981-287-751-2_5.

  • Vella, D., Boudaoud, A. & Adda-Bedia, M. Statics and inertial dynamics of a ruck in a rug. Phys. Rev. Lett. 103, 174301 (2009).

    ADS 

    Google Scholar
     

  • Dey, B.et al. Divergent evolutionary strategies pre-empt tissue collision in gastrulation. Nature 646, 637–646 (2025).

  • Vellutini, B.C.et al. Patterned invagination prevents mechanical instability during gastrulation. Nature 646, 627–636 (2025).

  • Shankar, S., Souslov, A., Bowick, M. J., Marchetti, M. C. & Vitelli, V. Topological active matter. Nat. Rev. Phys. 4, 380–398 (2022).


    Google Scholar
     

  • Hamm, E., Roman, B. & Melo, F. Dynamics of developable cones under shear. Phys. Rev. E 70, 026607 (2004).

    ADS 

    Google Scholar
     

  • Gottesman, O., Efrati, E. & Rubinstein, S. M. Furrows in the wake of propagating d-cones. Nat. Commun. 6, 7232 (2015).

    ADS 

    Google Scholar
     

  • Witten, T. A. Stress focusing in elastic sheets. Rev. Mod. Phys. 79, 643–675 (2007).

    ADS 
    MathSciNet 

    Google Scholar
     

  • Wong, R. H. C. & Chau, K. T. Crack coalescence in a rock-like material containing two cracks. Int. J. Rock Mech. Min. Sci. 35, 147–164 (1998).


    Google Scholar
     

  • Dias, M. A., Dudte, L. H., Mahadevan, L. & Santangelo, C. D. Geometric mechanics of curved crease origami. Phys. Rev. Lett. 109, 114301 (2012).

    ADS 

    Google Scholar
     

  • Dias, M. A. & Audoly, B. A non-linear rod model for folded elastic strips. J. Mech. Phys. Solids 62, 57–80 (2014).

    ADS 
    MathSciNet 

    Google Scholar
     

  • Nasti, G. et al. Patterning of perovskite-polymer films by wrinkling instabilities. Soft Matter 13, 1654–1659 (2017).

    ADS 

    Google Scholar
     

  • Brannon, C. M. & Prakash, M. Creases act as information bottlenecks in active elastic sheets. Dryad https://doi.org/10.5061/dryad.msbcc2gd0 (2026).