Abstract

Symmetry: Culture and Science
Volume 36, Number 4, pages 397-408 (2025)
https://doi.org/10.26830/symmetry_2025_4_397

FOLDING THE RULE: BEHAVIOR OF TOPOLOGICAL SCALE DISRUPTIONS IN SYMMETRIC YOSHIMURA PATTERNS

Pablo Miguel De Souza Sánchez

Universidad Europea de Canarias. Escuela de Arquitectura, C. Inocencio García, 1, 38300 La Orotava, Santa Cruz de Tenerife, España, Canarias.
Email: pablo.desouza@universidadeuropea.es
Web: http://www.pablodesouza.es
ORCID: 0000-0002-6722-3650

Abstract: This article examines the structural and formal behaviour of hexagonal Yoshimura Origami patterns when topological scale disruptions are applied in plan. Four specific deformation strategies are addressed: (1) progressive deformation along the X-axis, (2) scalar deformation along the X-axis, (3) progressive deformation along the Y-axis, and (4) scalar deformation along the Y-axis. Each variation is projected in 2D and then folded into 3D prototypes. The structural behaviour is analysed according to different degrees of folding (10–30–65%, 5–20–40%, 20–50–80%, and 5–10–20%), as shown in the images accompanying the article. The results demonstrate that geometric symmetries collapse under tension, while topological invariants such as vertex connectivity and crease parity are preserved. The study combines morphogenetic theory, parametric modelling, and project-based learning. It concludes that Yoshimura patterns, once “folded out of rule”, become powerful tools for generating kinetic envelopes and roofs, as well as highly versatile and self-supporting responsive architectural systems.

Keywords: Topological Folding; Yoshimura patterns; Morphogenetic Architecture; Origami tessellations; Folded Plate Structures; Folding Architecture; The Fold in Architecture

References:
Boakes, N. J. (2015). Researching the intersection of origami, mathematics, education and art. Symmetry: Culture and Science, 26(2), 227–242. https://doi.org/10.26830/symmetry_2015_2

Budinski, N. (2015). Origami, symmetry and mathematical education. Symmetry: Culture and Science, 26(2), 243–252. https://doi.org/10.26830/symmetry_2015_2

Connelly, R., Demaine, E. D., & Ziegler, G. M. (2013). Mathematics and modern architecture. Springer. https://link.springer.com/book/10.1007/978-3-0348-0544-5

Darvas, G. (2015). Symmetries in Origami (Editorial). Symmetry: Culture and Science, 26(2), 133. https://doi.org/10.26830/symmetry_2015_2

de Churtichaga, J. M. (2004). La estructura veloz: Trayectorias estructurales a propósito de la obra de Emilio Pérez Piñero y Félix Candela. En M. Seguí (Dir.), Félix Candela y Emilio Pérez Piñero: Un diálogo imaginal. Alcorcón, Madrid: Editorial Rueda. https://chqs.net/archivos/informes/archivo_1_040310_la%2Bestructura%2Bveloz.pdf

De Souza Sánchez, P. M. (2013). Origami folding patterns in the work of F. Ll. Wright. En: New proposals for transformable architecture, engineering and design. In the honor of Emilio Pérez Piñero. Starbooks Structural Architecture. 373-378. https://doi.org/10.5281/zenodo.6365029

De Souza Sánchez, P. M. (2017). El pliegue en la arquitectura = The fold in architecture. Tesis (Doctoral), E.T.S. Arquitectura (UPM). https://doi.org/10.20868/UPM.thesis.47994

De Souza Sánchez, P. M. (2018). Del pliegue conformador y estructural al espacio oblicuo. En P. Arza Garaloces y J. M. Pozo (Coords) La tecnología en la arquitectura moderna (1925-1975): mito y realidad. T6) Ediciones. 213-220. https://doi.org/10.5281/zenodo.6368342

De Souza Sánchez, P. M. (2021). Taller vertical y metodología ABP en la realización de proyectos de arquitectura cinética e interactiva con patrones de pliegue estructurales. In Á. M. Martínez, A. B. B. Martín, M. del Mar Molero Jurado, M. del Carmen Pérez-Fuentes, M. del Mar Simón Márquez y J. J. G. Linares (Eds.), Innovación Docente e Investigación en Ciencias, Ingeniería y Arquitectura: Nuevos Enfoques en la Metodología Docente. (1st, 12/16/21 ed., 49–66).https://doi.org/10.5281/zenodo.6370857

De Souza Sánchez, P. M., Ferrer Román, E. & Naranjo Henríquez, I. (2023). Coordination and methodological innovation of the drawing activities of the degree in Architecture, Human Review, 17(1) 1–27. https://doi.org/10.37819/revhuman.v17i1.1557

De Souza Sánchez, P. M. y Martínez Soto, F. (2023). Proposal of a pilot formative experience for the subject technical systems applied in the design of a rigid origami kinetic pavilion. VIII International Conference on Technological Innovation in Building, 77–80. https://doi.org/10.5281/zenodo.7841961

Demaine, E. D., Demaine, M. L., Koschitz, D., & Tachi, T. (2015). A review on curved creases in art, design and mathematics. Symmetry: Culture and Science, 26(2), 145–161. https://doi.org/10.26830/symmetry_2015_2

Demaine, E. D., & Demaine, M. L. (2007). Geometric folding algorithms: Linkages, origami, polyhedra. Cambridge University Press.

Engel, P. (1994). Breaking symmetry: Origami, architecture, and the forms of nature. Symmetry: Culture and Science, 5(1), 37–68. https://symmetry-us.com/Journals/5-1/engel.pdf

Fujimoto, K. (2004). Introduction to origami tessellations. Origamihouse. URL: https://origamihouse.jp/

García García, R. (2013). Dos décadas de estructuras plegadas de hormigón. Informes de la Construcción, 65(529), 73-89. https://doi.org/10.3989/ic.2013.v65.i529

Gardiner, M. (2015). Folding and unfolding a million times over—Longevity, origami, robotics and biomimetics as material thinking in Oribotics. Symmetry: Culture and Science, 26(2), 189–202. https://doi.org/10.26830/symmetry_2015_2

Ghassaei, A., Demaine, E. D., & Gershenfeld, N. (2018). Fast, interactive origami simulation using GPU computation. In R. J. Lang, M. Bolitho, & Z. You (Eds.), Origami 7: Proceedings of the 7th International Meeting on Origami in Science, Mathematics and Education (Vol. 4, pp. 1151–1166). Tarquin. https://erikdemaine.org/papers/OrigamiSimulator_Origami7/paper.pdf

Ghassaei, A. (2017). Origami Simulator. URL: https://amandaghassaei.com/projects/origami_simulator/

Gibson, C. G. (2011). Topology: A geometric approach. Cambridge University Press. https://doi.org/10.1017/CBO9780511973731

Ida, T., & Ghourabi, F. (2015). Polygonal knot by computational origami. Symmetry: Culture and Science, 26(2), 171–187. https://doi.org/10.26830/symmetry_2015_2

Jackson, P. (2011). Folding techniques for designers and architects. Editorial Promopress.

Kresling, B. (2020). The fifth fold: Complex symmetries in Kresling–origami patterns. Symmetry: Culture and Science, 31(4), 403–416. https://doi.org/10.26830/symmetry_2020_4_403

Osório, F. C., Paio, A., & Oliveira, S. (2023). A kinetic origami surfaces methodology. Architectural Science Review, 67(1), 23–46. https://doi.org/10.1080/00038628.2023.2182270

Martínez Martínez, M. (2018). Proceso de cálculo de las cáscaras cilíndricas largas de cubierta en la obra de Félix Candela. Informes de la Construcción, 70(551), e260. https://doi.org/10.3989/ic.56644

Miura, K., & Tachi, T. (2015). Models of Miura-ori and Tachi–Miura polyhedron. Symmetry: Culture and Science, 26(2), 135–144. https://doi.org/10.26830/symmetry_2015_2

Saito, K. E. et Röhmer Litzmann, M. (2007). Plegado, simetría y software gráfico. Forma y Simetría: Arte y Ciencia. Congreso de Buenos Aires.

Salingaros, N. A. (2020). Symmetry gives meaning to architecture. Symmetry: Culture and Science, 31(3), 231–260. https://doi.org/10.26830/symmetry_2020_3_231

Sanz Balduz, L. J. (1999). El borde libre y Félix Candela. Revista de Obras Públicas, 146(3383), 17–28.

Tachi, T. (5th August 2009). Simulation of Rigid Origami. Origami 4: Proceedings of the Fourth International Conference on Origami in Science, Mathematics, and Education. J. A K Peters/CRC Press. pp. 175-187.

Thompson, D. W. (1917). On growth and form. Cambridge University Press. https://archive.org/details/ongrowthform00thomrich

Torroja, E. (1959). Lámina plegada. Universidad Laboral de Tarragona. Informes de la Construcción, 12(107), 74–81. https://doi.org/10.3989/ic.1959.v12.i107.5449

Torroja Miret, E. (1991). Razón y Ser de los Tipos Estructurales. Editorial CSIC. Madrid.

Trometer, S. and Krupna, M. (2010). Renaissance of paper models and folded plate structures in glass. Int. J. Structural Engineering, Vol. 1, nº3-4: 361–373. https://doi.org/10.1504/IJStructE.2010.033488

Vyzoviti, S. and De Souza Sánchez, P. M. (2012). Origami tessellations in a continuum: Integrating design and fabrication in architectural education. In Voyatzaki, M & Spiridonidis, C. (eds.) Transactions on Architectural Education nº59. International Conference: Scaleless-Seamless: Performing a less fragmented Architecture and Education. ENHSA-Münster School of Architecture, Germany. https://doi.org/10.5281/zenodo.6364968

Vyzoviti, S. (2011). Soft Shells: Porous and Deployable Architectural Screens. Amsterdam: BiS Publishers