, Volume 52, Issue 6, pp 1283–1293 | Cite as

Bracing rhombic structure by one-dimensional tensegrities

  • Gyula Nagy Kem


Certain behaviors of some material are characterized by the periodic bar and joint elements. We present a kinematic, geometric and graph theoretic connected model that describes the stability, rigidity property of these rhombic tiling materials in two dimensions. Cables, struts or rods are placed as bracing elements between opposite pairs of diagonal joints, to prevent the rhombic tiling from the rotation of the bars around their common joint. We characterize the rigidity of the finite parts of the rhombic bracing structure in the plane. The results of this paper are based on the theorem of the rigidity of one-dimensional tensegrity framework from Recski and Shai. We have applied our results to describe some auxetic type structures that were mentioned earlier in the scientific literature. We also introduce the model as a possible candidate for the mechanical information processing system in a repetitive bar and joint structure.


Tensegrity Auxetic mechanism Infinitesimal rigidity Tiling Tessellations 



I would like to thank András Recski for many productive discussions. I thank the editors and anonymous reviewers for their careful reading of the manuscript and their many Insightful comments and suggestions. This research was supported by the Hungarian National Science Foundation (Grant Number: 108947).


  1. 1.
    Babaee S, Shim J, Weaver JC, Chen ER, Patel N, Bertoldi K (2013) 3D soft metamaterials with negative poisson’s ratio. Adv Mater 25(36):5044–5049CrossRefGoogle Scholar
  2. 2.
    Baglivo JA, Graver JE (1983) Incidence and symmetry in design and architecture. Cambridge University Press, CambridgeGoogle Scholar
  3. 3.
    Bölcskei A, Molnár E (1998) How to design nice tilings? KoG Sci Prof Inf J Croat Soc Constr Geom Comput Graph 3:21–28MathSciNetMATHGoogle Scholar
  4. 4.
    Bolker ED, Crapo H (1979) Bracing rectangular frameworks. I. SIAM J Appl Math 36(3):473–490MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Borcea CS, Streinu I (2010) Periodic frameworks and flexibility. Proc R Soc A 466:2633–2649ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Böröczky KJ, Réti T, Wintsche G (2006) On the combinatorial characterization of quasichrystals. J Geom Phys 57:39–52ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Campos A, Guenther R, Martins D (2009) Differential kinematics of parallel manipulators using Assur virtual chains. Proc Inst Mech Eng C J Mech 223(7):1697–1711CrossRefGoogle Scholar
  8. 8.
    Frettlöh D, Harriss E (2013) Parallelogram tilings, worms and finite orientations. Discret Comput Geom 4:9531–9539MathSciNetMATHGoogle Scholar
  9. 9.
    Friedman N, Ibrahimbegovic A (2013) Overview of highly flexible, deployable lattice structures used in architecture and civil engineering undergoing large displacements. YBL J Built Env 1:85–103CrossRefGoogle Scholar
  10. 10.
    Frittmann J, Lángi ZS (2015) Decompositions of a polygon into centrally symmetric pieces. arXiv:1504.05418
  11. 11.
    Gáspár Zs, Radics N, Recski A (1998) Square grids with long diagonals. Optim Methods Softw 10:217–231MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gaspar Zs, Radics N, Recski A (1999) Rigidity of square grids with holes. Comput Assist Mech Eng Sci 6:329–335MATHGoogle Scholar
  13. 13.
    Gatt R, Mizzi L, Azzopardi JI, Azzopardi KM, Attard D, Casha A, Briffa J, Grima JN (2013) Hierarchical auxetic mechanical metamaterials. Nature 25(36):5044Google Scholar
  14. 14.
    Grima JN, Evans KE (2000) Auxetic behavior from rotating squares. J Mater Sci Lett 19(17):1563–1565CrossRefGoogle Scholar
  15. 15.
    Guest SD, Hutchinson JW (2003) On the determinacy of repetitive structures. J Mech Phys Solids 51:383–391ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Horváth AG (1997) Dissections of a centrally symmetric hexagon, Bolyai Soc. Math Stud 6:327–334MATHGoogle Scholar
  17. 17.
    Jordán T, Domokos G, Tóth K (2013) Geometric Sensitivity of Rigid Graphs. SIAM J Discret Math 27(4):1710–1726MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jordán T, Recski A, Szabadka Z (2009) Rigid tensegrity labelings of graphs. Eur J Comb 30(8):1887–1895MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kapko V, Treacy MJ, Thorpe MF, Guest SD (2009) On the collapse of locally isostatic networks. Proc R Soc A 465:3517–3530ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Körner C, Liebold-Ribeiro Y (2015) A systematic approach to identify cellular auxetic materials. Smart Mater Struct 24(025013):10Google Scholar
  21. 21.
    Lovász L, Vesztergombi K (2002) Geometric representations of graphs. In: Halász G, Lovász L, Simonovits M, T. Sós V (ed) Paul Erdős and his Mathematics, Bolyai Soc. Math. Stud, János Bolyai Math. Soc., Budapest 11:471–498Google Scholar
  22. 22.
    Lu Guang Hong; Zhang Lei (2012) Connecting microscopic structure and macroscopic mechanical properties of structural materials from first-principles. Sci China Phys Mech Astron 55(12):2305–2315ADSCrossRefGoogle Scholar
  23. 23.
    Mitschke H, Schwerdtfeger J, Schury F, Stingl M, Körner C, Singer RF, Robins V, Mecke K, Schröder-Turk GE (2011) Finding auxetic frameworks in periodic tessellations. Adv Mater 22–23(23):2669–2674CrossRefGoogle Scholar
  24. 24.
    Nagy G (2006) Tessellation-like rod-joint frameworks. Ann Univ 49:3–14MathSciNetMATHGoogle Scholar
  25. 25.
    Nagy G, Katona J (2010) Connectivity for rigidity. In: Studies of the University of Zilina Mathematical Series 24 (1, 6):59–64Google Scholar
  26. 26.
    Nagy Gy (2001) Rigidity of an annex building. Struct Multidiscip Optim 22(1):83–86MathSciNetCrossRefGoogle Scholar
  27. 27.
    Nagy Gy (1994) Diagonal bracing of special cube grids. Acta Tech Acad Sci Hung 106(3–4):256–273MATHGoogle Scholar
  28. 28.
    Owen JC, Power S (2010) Frameworks symmetry and rigidity. Int J Comput Geom Appl 20(6):723–750MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Power SC (2014) Crystal frameworks, symmetry and affinely periodic flexes, New York. J Math 20:1–29MATHGoogle Scholar
  30. 30.
    Radics N, Recski A (2002) Applications of combinatorics to statics—rigidity of grids. Discret Appl Math 123:473–485MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Recski A (1989) Matroid theory and its applications in electric network theory and in statics. Akadémiai Kiadó/Springer, Budapest/BerlinCrossRefMATHGoogle Scholar
  32. 32.
    Recski A, Shai O (2010) Tensegrity frameworks in the one-dimensional space. Eur J Comb 31(4):1072–1079MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Reti T, Böröczky KJ (2004) Topological characterization of cellular structures. Acta Polytech Hung 1:59–85Google Scholar
  34. 34.
    Tanaka H (2013) Bi-stiffness property of motion structures transformed into square cells. Proc R Soc A 469(2156):2013006CrossRefGoogle Scholar
  35. 35.
    Tanaka H, Shibutani Y, Izumi S, Sakai S (2012) Planar mobility modes of eight-barjointed structures with a single degree of freedom. Int J Solids Struct 49:1712–1722CrossRefGoogle Scholar
  36. 36.
    You Z, Pellegrino S (1997) Foldable bar structures. Int J Solids Struct 34:1825–1847CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Ybl Miklós Faculty of Architecture and Civil EngineeringSzent István UniversityBudapestHungary

Personalised recommendations