From 3-Point-Constellations to Self-organizing Folded/Bent Spatial Configurations

  • Günther H. Filz
  • Stefan Kainzwaldner


Our investigation is based on a seemingly simple phenomenon that we all know from kinking sheets of paper. In this sense it is related to self-organizing processes and forms. The controlled elastic deformation of thin, planar sheet elements into 3 dimensional objects describe above mentioned phenomenon by predefining 3 points (MAB) on the surface with M fixed in space and identified shape characteristics of the short self-organizing curved ridge-line, which is smoothly fading to an elastically bent surface. This paper presents the geometrical and kinematic simulation of controlled elastic deformation of planar sheet elements of any shape into self-organizing folded/bent spatial configurations by predefining any 3-point-constellation (M, A, B) with A, B equidistant and symmetrical to M (Fig. 1). The geometrical description of spatial configurations in all transition phases and its kinematic simulation is based on a microscopic analysis of physical models. From this we receive a line-pattern, which can be applied to the sheet-element as a basis for simulating spatial movements of surface-points. At present simulation and physical models show maximum 5 % deviation only. Several tests and various 3-point-constellations verified that our simulation can be a powerful and reliable tool in exploring design freedoms as well as limits, which will be extended to more complex arrangements.


  1. Demaine E, Demaine M, Koschitz D (2011) Reconstructing David Huffman’s legacy in curved-crease folding. In: Yim M (eds) Origami 5, fifth international meeting of origami science, mathematics, and education. CRC Press, Boca RatonGoogle Scholar
  2. Filz GH, Kainzwaldner S (2014) Between bending and folding—buckling from plane to object by 3 predefined points. Paper presented at the VI. Latin-American symposium on tension structures (IASS-SLTE). Shells, membranes and spatial structures: ‘Footprints’, Brasilia, 2014Google Scholar
  3. Filz GH, Kainzwaldner S (2015) Freedoms and limits in the arrangement of 3 point constellation on planar sheet elements, generating spatial, selforganized folded/bent forms. Paper presented at the international association for shell and spatial structures (IASS), Amsterdam, 2015 (Abstract accepted, full paper in review)Google Scholar
  4. Fuchs D, Tabachnikov S (2007) Mathematical omnibus. American Mathematical Society, USA, pp 185–197, 207–215 Google Scholar
  5. Jackson P (2011) Folding techniques for designers—form sheet to form. Laurence King Publishing Ltd, London, pp 176–187Google Scholar
  6. Kilian M, Flöry S, Chen Z, Mitra JN, Sheffer A, Pottmann H (2008) Curved folding. In: Proceedings of ACM transactions on graphicsGoogle Scholar
  7. Leopoldseder S, Pottmann H (2003) Approximation of developable surfaces with cone spline surfaces. Institut für Geometrie, WienGoogle Scholar
  8. Lienhard J, Schleicher S, Knippers J (2011) Bending-active structures—research pavilion ICD/ITKE. Paper presented at the international symposium of the IABSE-IASS Symposium, Taller Longer Lighter, London, 2011Google Scholar
  9. Lienhard J, Alpermann H, Gengnagel C, Knippers J (2012) Active bending, a review on structures where bending is used as a self formation process. In: Kim (SD): From Spatial structures to space structures, IASS-APCS 2012. Abstract book. Seoul: Korean association for spatial structuresGoogle Scholar
  10. Nettelbladt M (2013) The geometry of bending. Mårten Nettelbladt, Stockholm, pp 9–11, 17Google Scholar
  11. Peternell M (2009) Unterteilungskurven und—flächen. In: Proceedings of Fortbildungstagung für Geometrie, Wien, 2009Google Scholar
  12. Pottmann H, Asperl A, Hofer M, Kilian A (2010) Architekturgeometrie. Springer, Wien, pp 221, 222, 247–257Google Scholar
  13. Tirapegui E, Martinez J, Tiemann R (2000) Insatbilities and nonequilibrium structures VI. Kluwer Academic Publishers, Dordrecht, pp 269–274Google Scholar
  14. Zhu L, Igarashi T, Mitani J (2013) Soft folding; computer graphics forum, vol 32(7). In: Proceedings of the 21st Pacific conference on computer graphics and applications. Pacific Graphics, SingaporeGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Faculty of Architecture, Institute of Design/unit koge. Structure and DesignUniversity of InnsbruckInnsbruckAustria

Personalised recommendations