Skip to main content
SpringerLink
Log in

Reciprocal Frames Algorithm based on Symmetry Groups

  • Research
  • Published:
Nexus Network Journal Aims and scope Submit manuscript

Abstract

Reciprocal Frames (RF) are a type of structural system that can be used to build temporary structures with large spans. This is possible due their ease of assembly and the use of relatively short, light-weight elements. However, a challenge is presented when this kind of structure is applied to complex surfaces. In this case, the only way to figure out each connection is through an algorithmic geometric model. Only a few algorithms have already been developed for generating complex RF structures, but they still have some limitations. In this paper we present a new approach represent RF geometry, based on symmetry theory, combining pinwheel patterns with wallpaper symmetry groups. In future works, this will be combined with structural performance analysis and the use of digital fabrication methods for producing the structures’ connections.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

Notes

  1. In computational design, the process of flattening is a way to eliminate nested lists (trees), by incorporating each one in a single list. This is done through a recursive process, where the order of the index values is defined by the previous tree. For instance, flattening a tree that contains the lists 1[A, C, Z, Y]; 2[B, K, L]; 3[T, J] and 4[P, Q, R] will result in the new list [A, C, Z, Y, B, K, L, T, J, P, Q, R].

References

  • Adanova, Venera and Sibel Tari. 2019. Analysis of Planar Ornament Patterns via Motif Asymmetry Assumption and Local Connections. Journal of Mathematical Imaging and Vision 61: 269 – 291. https://link.springer.com/article/10.1007%2Fs10851-018-0835-8

  • Asefi, Maziar and Mahnaz Bahremandi-Tolou. 2019. Design challenges of reciprocal frame structures in architecture. Journal of Building Engineering 26: 1 – 16. https://doi.org/10.1016/j.jobe.2019.100867

  • Baverel, O., H. Nooshin, Y. Kuroiwa and G. A. R. Parke. 2000. Nexorades. International Journal of Space Structures 15:2: 155 – 159. https://doi.org/10.1260/0266351001495053

  • Castriotto, Caio, Gabriela Celani and Felipe Tavares da Silva. 2020. Estruturas recíprocas: revisão sistemática da literature e identificação de pontos críticos para projeto e produção. Ambiente Construído 20:4: 397 – 405. https://doi.org/10.1590/s1678-86212020000400479

  • Conway, John H., Heidi Burgiel and Chaim Goodman-Strauss. 2008. The Symmetries of Things. Massachusetts: A. K. Peters Ltd.

  • Coxeter, Harold S. M. 1979. The Non-Euclidean Symmetry of Escher’s Picture “Circle Limit III”. Leonardo 12:1: 19 – 25. https://doi.org/10.2307/1574078

  • Douthe, Cyril and Olivier Baverel. 2009. Design of nexorades or reciprocal frame systems with the dynamic relaxation method. Computers and Structures 87: 1296 – 1307. https://doi.org/10.1016/j.compstruc.2009.06.011

  • Earl, Christopher F. and Iestyn Jowers. 2015. Pinwheel Patterns: from 2D to 3D schemas. Nexus Network Journal: Architecture and Mathematics 17: 899 – 912. https://doi.org/10.1007/s00004-015-0266-4

  • Gheorghe, Andrei and Robert Vierlinger. 2017. DigDesFab 15 Research Pavilion. Frontiers in Digital Humanities 4:18: 1 – 7. https://doi.org/10.3389/fdigh.2017.00018

  • Gotdhelp, Tom S., Andre J. M. Jorissen, Arjan P. H. W. Habraken and Rinus Roelofs. The timber Reciprocal Frame Designer: free form design to production. In IASS Annual Symposia: Structural Morphology, Barcelona, 7 – 10 October 2019, art. 8. p. 1 – 8. Barcelona: International Association for Shell and Spatial Structures.

  • Hargittai, Istvan and Magdolna Hargittai. 1994. Symmetry: A Unifying Concept. Berkeley: PGW.

  • Huylebrouck, Dirk. 2017. A Bike Tour for Rinus Roelofs’ Art in Twente, The Netherlands. The Mathematical Intelligencer 39: 60-65. https://doi.org/https://doi.org/10.1007/s00283-017-9712-3

  • Landwehr, Klaus. 2011. Visual Discrimination of the 17 Plane Symmetry Groups. Symmetry 3:2: 207 – 219. https://doi.org/https://doi.org/10.3390/sym3020207

  • Larsen, Olga Popovic. 2008. Reciprocal Frame Architecture. New York: Architectural Press.

  • Larsen, Olga Popovic. 2014. Reciprocal Frame (RF) Structures: Real and Exploratory. Nexus Network Journal: Architecture and Mathematics 16:1: 119 – 134.

  • March, Lionel and Philip Steadman. 1974. The geometry of environment: an introduction to spatial organization in design. Massachusetts: The M.I.T Press.

  • Martin, George E. 1982. Transformation Geometry: An Introduction to Symmetry. New York: Springer.

  • Mesnil, Romain, Cyril Douthe, Olivier Baverel and Tristan Gobin. 2018. Form finding of nexorades using the translation method. Automation in Construction, 95: 142 – 154.

  • Palumbo, Arianna, Daniele Lancia and Sergio Pone. 2019. Digital Tool for Reciprocal Frame. In IASS Annual Symposia: Structural Morphology, Barcelona, 7 – 10 October 2019, art 9. p. 1 – 8. Barcelona: International Association for Shell and Spatial Structures.

  • Parigi, Dario and Poul Henning Kirkegaard. 2014a. The Reciprocalizer: an Agile Design Tool for Reciprocal Structures. Nexus Network Journal: Architecture and Mathematics 16:1: 61 – 68. https://doi.org/10.1007/s00004-014-0176-x

  • Parigi, Dario and Poul Henning Kirkegaard. 2014b. Design and Fabrication of Free-Form Reciprocal Structures. Nexus Network Journal: Architecture and Mathematics 16:1: 69 – 87. https://doi.org/10.1007/s00004-014-0177-9

  • Pugnale, Alberto and Mario Sassone. 2014. Structural Reciprocity: Critical Overview and Promising Research / Design Issues. Nexus Network Journal: Architecture and Mathematics 16:1: 09 – 35. https://doi.org/10.1007/s00004-014-0174-z

  • Sands, Donald E. 1993. Introduction to Crystallography. New York: Dover Publications.

  • Schattschneider, Doris. 2004. M.C. Escher: Visions of Symmetry. London: Thames & Hudson. p. 353 – 354.

  • Sénéchal, B., C. Douthe and O. Baverel. 2011. Analytical Investigations on Elementary Nexorades. International Journal of Space Structures 26, 4: 313 – 320.

  • Serlio, Sebastiano. 1611. The first [-firft] booke of architecture / made by Sebastian Serly ...; translated out of Italian into Dutch, and out of Dutch into English. London: Printed [by Simon Stafford] for Robert Peake. Rept. ed. The Five Books on Architecture. An Unabridged Reprint of the English Edition of 1611. New York: Dover

  • Song, Peng, Chi-Wing Fu, Prashant Goswami, Jianmin Zheng, Niloy J Mitra, and Daniel Cohen-Or. 2013. Reciprocal Frame Structures Made Easy. ACM Transactions on Graphics 32:4: 94:1 – 94:10. https://doi.org/10.1145/2461912.2461915

  • Thönnissen, Udo. 2014. A Form-Finding Instrument for Reciprocal Structures. Nexus Network Journal: Architecture and Mathematics 16:1: 89 – 107. https://doi.org/10.1007/s00004-014-0172-1.

Download references

Acknowledgements

All images are by the authors.

Funding

The authors gratefully acknowledge the grant #2019/04043–2 of the São Paulo Research Foundation (FAPESP). Also, thanks to the support given by the Laboratory of Automation and Prototyping for Architecture and Construction (LAPAC) from the University of Campinas.

Author information

Authors and Affiliations

  1. School of Civil Engineering, Architecture and Urban Design, University of Campinas, Rua Saturnino de Brito, 224-Cidade Universitária. Campinas, São Paulo, Brazil

    Caio Castriotto & Gabriela Celani

  2. Department of Architecture and Urbanism, Federal University of Paraíba, Rua Castelo Branco, s.n., Cidade Universitária, João Pessoa, Paraíba, Brazil

    Felipe Tavares

Authors
  1. Caio Castriotto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gabriela Celani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Felipe Tavares
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Caio Castriotto.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Castriotto, C., Celani, G. & Tavares, F. Reciprocal Frames Algorithm based on Symmetry Groups. Nexus Netw J 24, 167–186 (2022). https://doi.org/10.1007/s00004-021-00578-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00004-021-00578-5

Keywords

Search

Navigation