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Published online by Cambridge University Press:  03 May 2025

William F. Baker
Affiliation:
Skidmore Owings and Merrill, Chicago
Allan McRobie
Affiliation:
University of Cambridge
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The Geometry of Equilibrium
James Clerk Maxwell and 21st-Century Structural Mechanics
, pp. 293 - 302
Publisher: Cambridge University Press
Print publication year: 2025

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References

Adriaenssens, S., Block, P., Veenendaal, D., and Williams, C., editors (2014). Shell Structures for Architecture: Form Finding and Optimization. Routledge.CrossRefGoogle Scholar
Aejmelaeus-Lindstrom, P., Mirjan, A., Gramazio, F., Kohler, M., Kernizan, S., Sparrman, B., Laucks, J., and Tibbits, S. (2017). Granular jamming of loadbearing and reversible structures: Rock print and rock wall. Architectural Design, 87(4):8287.CrossRefGoogle Scholar
Ahn, H.-S. (2020). Formation Control: Approaches for Distributed Agents, volume 205 of Studies in Systems, Decision and Control. Springer.CrossRefGoogle Scholar
Airy, G. B. (1863). On the strains in the interior of beams. Phil. Trans. Royal Society of London, 153:4979.Google Scholar
Akbarzadeh, M., Bolhassani, M., Nejur, A., Yost, J. R., Byrnes, C., Schneider, J., Knaack, U., and Costanzi, C. B. (2019). The design of an ultra-transparent funicular glass structure. In Structures Congress 2019, pages 405413. American Society of Civil Engineers.CrossRefGoogle Scholar
Akbarzadeh, M., Van Mele, T., and Block, P. (2015a). On the equilibrium of funicular polyhedral frames and convex polyhedral force diagrams. Computer-Aided Design, 63:118128.Google Scholar
Akbarzadeh, M., Van Mele, T., and Block, P. (2015b). Spatial compression-only form finding through subdivision of external force polyhedron. In Proc. Int. Assoc. Shell & Spatial Structures (IASS) Symp., Amsterdam.Google Scholar
Aldis, W. S. (1865). Elementary Treatise on Solid Geometry. Deighton, Bell, and Co.Google Scholar
Aleksandrov, A. D. (1947). On the works of S. E. Cohn-Vossen. Uspekhi Mat. Nauk (N.S.) 2, 3(19):107141. (Jacob, A., translator).Google Scholar
Aleksandrov, A. D. (1950). Convex Polyhedra. GTI, Moscow (in Russian). English translation: Springer, 2005.Google Scholar
Allen, E. and Zalewski, W. (2010). Form and Forces: Designing Efficient, Expressive Structures. John Wiley & Sons.Google Scholar
Almegaard, H. (2011). On the geometry of discrete Michell trusses. In Taller, Longer, Lighter – Meeting Growing Demand with Limited Resources. IABSE/IASS.Google Scholar
Asimow, L. and Roth, B. (1978). The rigidity of graphs. Trans. Amer. Math. Soc., 245:279289.CrossRefGoogle Scholar
Aspnes, J., Eren, T., Goldenberg, D. K., Morse, A. S., Whiteley, W., Yang, Y. R., Anderson, B. D. O., and Belhumeur, P. N. (2006). A theory of network localization. IEEE Transactions on Mobile Computing, 5(12):16631678.CrossRefGoogle Scholar
Assur, L. V. (1952). Issledovanie ploskih steržnevyh mehanizmov s nizšimi parami s tocki zreniya ih struktury i klassifikacii. Izdat. Akad. Nauk SSSR. Edited by Artobolevskiĭ, I.I..Google Scholar
Baker, W. F. (1992). Energy-based design of lateral systems. Structural Engineering International, 2(2):99102.CrossRefGoogle Scholar
Baker, W. F., Beghini, L. L., Mazurek, A., Carrion, J., and Beghini, A. (2013). Maxwell’s reciprocal diagrams and discrete Michell frames. Struct. & Multidisc. Optimization, 48:267277.Google Scholar
Baker, W. F., Beghini, L. L., Mazurek, A., Carrion, J., and Beghini, A. (2015a). Structural innovation: Combining classic theories with new technologies. American Inst. Steel Constr. Engng. Journal, 52:203217.Google Scholar
Baker, W. F., Mazurek, A., and Hartz, C. (2018). The design of structural “spider webs”. Steel Construction Magazine, 11(2):118124.CrossRefGoogle Scholar
Baker, W. F., McRobie, A., Mitchell, T., and Mazurek, A. (2015b). Mechanisms and states of self-stress of planar trusses using graphic statics, Part I: Introduction and background. In Proc. Int. Assoc. Shell & Spatial Structures Symp., Amsterdam.Google Scholar
Baracs, J., Crapo, H., and Whiteley, W. (1979–1997). Structural Topology magazine. Available at https://upcommons.upc.edu/handle/2099/498.Google Scholar
Barton, M., Shragai, N., and Elber, G. (2009). Kinematic simulation of planar and spatial mechanisms using a polynomial constraints solver. Computer-Aided Design & Applications, 6(1):115123.CrossRefGoogle Scholar
Bauschinger, J. (1871). Elemente der Graphischen Statik. R. Oldenbourg.Google Scholar
Beghini, L. L., Carrion, J., Beghini, A., Mazurek, A., and Baker, W. F. (2014). Structural optimization using graphic statics. Struct. and Multidisc. Optimization, 49(3):351366.CrossRefGoogle Scholar
Beltrami, E. (1892). Osservazioni sulla nota precedente. Atti della Accademia Nazionale dei Lincei Rendiconti, 1:141142.Google Scholar
Bernstein, D. I. (2022). Generic symmetry-forced infinitesimal rigidity: Translations and rotations. SIAM J. Appl. Algebra Geom., 6(2):190215.CrossRefGoogle Scholar
Bertoldi, K., Vitelli, V., Christensen, J., and van Hecke, M. (2017). Flexible mechanical metamaterials. Nature Reviews Materials, 2:111.Google Scholar
Betti, E. (1872). Teoria della elasticita. Il Nuovo Cimento, 7:8797.Google Scholar
Bhooshan, S., Dell’Endice, A., Bhooshan, V., Bouten, S., Megens, J., Casucci, T., Lombois-Burger, H., Steiner, F., Van Mele, T., and Block, P. (2022). The Striatus arched bridge: Computational design and robotic fabrication of an unreinforced, 3D-concrete-printed, masonry bridge. In Proc. 5th Int. Conf. on Structures and Architecture (ICSA), Aalborg.Google Scholar
Block, P. (2005). Equilibrium Systems: Studies in Masonry Structure. Master’s thesis, MIT, Cambridge, MA.Google Scholar
Block, P. (2009). Thrust Network Analysis: Exploring Threedimensional Equilibrium. PhD thesis, MIT, Cambridge, MA.Google Scholar
Block, P. and Lachauer, L. (2014). Three-dimensional (3D) equilibrium analysis of Gothic masonry vaults. International Journal of Architectural Heritage, 8(3):312335.CrossRefGoogle Scholar
Block, P. and Ochsendorf, J. (2007). Thrust network analysis: A new methodology for three-dimensional equilibrium. J. Int. Assoc. Shell & Spatial Structures, 48(155):167173.Google Scholar
Bobenko, A. and Suris, Yu. (2008). Discrete differential geometry: Integrable Structure, volume 98 of Graduate Studies in Mathematics. American Mathematical Society.Google Scholar
Bolker, E. D. and Roth, B. (1980). When is a bipartite graph a rigid framework? Pacific J. Math., 90(1):2744.CrossRefGoogle Scholar
Borcea, C. S. and Streinu, I. (2010). Periodic frameworks and flexibility. Proc. R. Soc. Ser. A, 466(2121):26332649.CrossRefGoogle Scholar
Boulic, L. and Schwartz, J. (2018). Design strategies of hybrid bending-active systems based on graphic statics and constrained force density method. J. Int. Assoc. Shell & Spatial Structures, 59(4):267275.Google Scholar
Bouma, W., Fudos, I., Hoffmann, C., Cai, J., and Paige, R. (1995). Geometric constraint solver. Computer-aided Design., 27(6):487501.CrossRefGoogle Scholar
Bow, R. H. (1873). Economics of Construction in Relation to Framed Structures. E. & F. N. Spon.Google Scholar
Bow, R. H. (1874). A Treatise on Bracing with its Application to Bridges and Other Structures of Wood or Iron. D. Van Nostrand.Google Scholar
Brandenbourger, M., Scheibner, C., Veenstra, J., Vitelli, V., and Coulais, C. (2021). Limit cycles turn active matter into robots. arxiv.org/abs/2108.08837.Google Scholar
Calladine, C. R. (1978). Buckminster Fuller’s tensegrity structures and Clerk Maxwell’s rules for the construction of stiff frames. Int. J. Solids & Structures, 14:161172.CrossRefGoogle Scholar
Campbell, L. and Garnett, W. (1882). The Life of James Clerk Maxwell: With a Selection from his Correspondence and Occasional Writings and a Sketch of his Contributions to Science. Cambridge University Press.Google Scholar
Carlson, D. E. (1966). On the completeness of the Beltrami stress functions in continuum mechanics. J. Math. Anal. & Appl., 15:311315.CrossRefGoogle Scholar
Cauchy, A. L. (1813). Recherche sur les polyèdres – premier mémoire. Journal de l'École Polytechnique, 9:6686.Google Scholar
Chalmers, J. B. (1881). Graphical Determination of Forces in Engineering Structures. Macmillan & Co.CrossRefGoogle Scholar
Charlton, T. M. (1960). A historical note on the reciprocal theorem and theory of statically indeterminate frameworks. Nature, 187:231232.Google Scholar
Charlton, T. M. (1982). A History of the Theory of Structures in the Nineteenth Century. Cambridge University Press.CrossRefGoogle Scholar
Chatzis, K. (2004). La réception de la statique graphique en France, durant le dernier tiers du XIX siècle. Revue d'histoire des mathématiques, 10:743.Google Scholar
Clinch, K., Jackson, B., and Tanigawa, S. (2022a). Abstract 3-rigidity and bivariate -splines I: Whiteley’s maximality conjecture. Discrete Anal., 2022:2,. 50 pp.Google Scholar
Clinch, K., Jackson, B., and Tanigawa, S. (2022b). Abstract 3-rigidity and bivariate -splines II: Combinatorial characterization. Discrete Anal., 2022:3,. 32 pp.Google Scholar
Clinch, K., Nixon, A., Schulze, B., and Whiteley, W. (2020). Pairing symmetries for Euclidean and spherical frameworks. Discrete Comput. Geom., 64(2):483518.CrossRefGoogle Scholar
Cohn-Vossen, S. (1927). Zwei Sätze über die Starrheit der Eiflächen. Nachrichten Ges. d. Wiss zu Göttingen, Math.-Phys. Klasse, pages 125134.Google Scholar
Cohn-Vossen, S. (1929). Unstarre geschlossene Flächen. Math. Ann., 102:1029.CrossRefGoogle Scholar
Connelly, R. (1978). A counterexample to the rigidity conjecture for polyhedra. Inst. Haut. Étud. Sci. Publ. Math., 47:333335.CrossRefGoogle Scholar
Connelly, R. (1982). Rigidity and energy. Invent. Math., 66(1):1133.CrossRefGoogle Scholar
Connelly, R. (2005). Generic global rigidity. Discrete Comput. Geom., 33(4):549563.CrossRefGoogle Scholar
Connelly, R., Fowler, P. W., Guest, S. D., Schulze, B., and Whiteley, W. (2009). When is a symmetric pin-jointed framework isostatic? Int. J. Solids & Structures, 46:762773.CrossRefGoogle Scholar
Connelly, R. and Guest, S. D. (2022). Frameworks, Tensegrities, and Symmetry. Cambridge University Press.CrossRefGoogle Scholar
Connelly, R. and Whiteley, W. (1996). Second-order rigidity and prestress stability for tensegrity frameworks. SIAM J. Discrete Math., 9(3):453491.CrossRefGoogle Scholar
Conzett, J., Reichlin, B., Mostafavi, M., and Hagmann, A. (2006). Structure as Space: Engineering and Architecture in the Works of Jurg Conzett and his Partners. Architectural Association Publications.Google Scholar
Cox, H. L. (1965). The Design of Structures of Least Weight. Pergamon Press Ltd.Google Scholar
Coxeter, H. S. M. (1969). Introduction to Geometry. John Wiley & Sons.Google Scholar
Coxeter, H. S. M. (1974). Projective Geometry. Springer-Verlag. Crapo, H. (1979). Structural rigidity. Structural Topology, 1(19).CrossRefGoogle Scholar
Crapo, H. and Whiteley, W. (1994). 3-stresses in 3-space and projections of polyhedral 3-surfaces: Reciprocals, liftings, and parallel configurations. (Preprint, York University, Ontario).Google Scholar
Cremona, L. (1868). Corso di statica grafica. Regio Istituto tecnico superiore in Milano, anno scolastico 186869.Google Scholar
Cremona, L. (1872). Le Figure Reciproche nella Statica Grafica. Giuseppe Bernardoni.Google Scholar
Cremona, L. (1873). Elementi di Geometria Projettiva. G. B. Paravia E Comp.Google Scholar
Cremona, L. (1885). Elements of Projective Geometry. Clarendon Press, (Leudesdorf, C., translator).Google Scholar
Cremona, L. (1890). Graphical Statics, Two Treatises on the Graphical Calculus and Reciprocal Figures in Graphical Statics. Clarendon Press.Google Scholar
Cromwell, P. R. (1997). Polyhedra. Cambridge University Press.Google Scholar
Cruickshank, J., Kitson, D., and Power, S. C. (2017). The generic rigidity of triangulated spheres with blocks and holes. J. Combin. Theory Ser. B, 122:550577.CrossRefGoogle Scholar
Culmann, K. (1864). Die graphische Statik. Meyer & Zeller.Google Scholar
Culmann, K. and Ritter, W. (1888–1906). Anwendungen der Graphischen Statik, volume 1–4. Meyer & Zeller.Google Scholar
D’Acunto, P., Jasienski, J.-P., Ohlbrock, P. O., and Fivet, C. (2017). Vector-based 3D graphic statics: Transformations of force diagrams. In Proc. Int. Assoc. Shell & Spatial Structures Symp., Hamburg.Google Scholar
D’Acunto, P., Jasienski, J.-P., Ohlbrock, P. O., Fivet, C., Schwartz, J., and Zastavni, D. (2019). Vector-based 3D graphic statics: A framework for the design of spatial structures based on the relation between form and forces. Int. J. Solids & Structures, 167:5870.CrossRefGoogle Scholar
D’Acunto, P. and Shen, Y. (n.d.). VGS Tool – Vector-based Graphic Statics. github.com/pierluigidacunto/VGS.Google Scholar
Darboux, G. (1894). Leçons sur Théorie Générale des Surfaces. Gauthier–Villars.Google Scholar
Dehn, M. (1916). Über die Starreit konvexer Polyeder. Math. Ann., 77:466473.CrossRefGoogle Scholar
Dierichs, K. and Menges, A. (2015). Granular morphologies: Programming material behaviour with designed aggregates. Architectural Design, 85:8691.CrossRefGoogle Scholar
Ding, H. J., Huang, D. J., and Chen, W. Q. (2007). Elasticity solutions for plane anisotropic functionally graded beams. Int. J. Solids & Structures, 44(1):176196.CrossRefGoogle Scholar
Du Bois, A. J. (1875). The Elements of Graphical Statics and their Application to Framed Structures. John Wiley & Sons.Google Scholar
Dunnington, G. W. (1955/2004). Carl Friedrich Gauss: Titan of Science. The Mathematical Association of America.Google Scholar
Dutta, S. (2014). A simple property of isosceles triangles with applications. Forum Geometricorum, 14:237240.Google Scholar
Earnshaw, S. (1858). A Treatise on Statics containing the Theory of the Equilibrium of Forces and numerous examples. Deighton, Bell & Co., 4th edition.Google Scholar
Eddy, H. T. (1878). On the two general reciprocal methods in graphical statics. American Journal of Mathematics, Pure and Applied, 1:322335 & plates IV–V.Google Scholar
Eftekhari, Y. (2017). Geometry of Point-Hyperplane and Spherical Frameworks. PhD thesis, York University, Ontario.Google Scholar
Eftekhari, Y., Jackson, B., Nixon, A., Schulze, B., Tanigawa, S., and Whiteley, W. (2019). Point–hyperplane frameworks, slider joints, and rigidity–preserving transformations. J. Combin. Theory Ser. B, 135:4474.CrossRefGoogle Scholar
Epple, M. (1998). Topology, matter, and space, I: Topological notions in 19th-century natural philosophy. Archive for History of Exact Sciences, 52:297392.CrossRefGoogle Scholar
Eren, T., Whiteley, W., Morse, A. S., Belhumeur, P. N., and Anderson, B. D. O. (2003). Sensor and network topologies of formations with direction, bearing and angle information between agents. In Proc. 42nd IEEE Conf. on Decision and Control, Maui.Google Scholar
Euler, L. (1758). Elementa doctrinae solidorum. Novi Commentarii Academiae Scientarum Petropolitanae, 4:109140.Google Scholar
Favaro, A. (1879). Leçons de Statique Graphique. Gauthier-Villars. Traduites de l’italien par Paul Terrier.Google Scholar
Finbow, W., Ross, E., and Whiteley, W. (2012). The rigidity of spherical frameworks: Swapping blocks and holes. SIAM J. Discrete Math., 26(1):280304.CrossRefGoogle Scholar
Finbow-Singh, W. and Whiteley, W. (2013). Isostatic block and hole frameworks. SIAM J. Discrete Math., 27(2):9911020.CrossRefGoogle Scholar
Fivet, C. (2013). Constraint-Based Graphic Statics. PhD thesis, UCLouvain, Belgium.Google Scholar
Fivet, C. (2016). Projective transformations of structural equilibrium. Int. J. Space Structures, 31:135146.CrossRefGoogle Scholar
Fivet, C. and Ochsendorf, J. (2018). The graphic statics behind the Collier Memorial. In Proc. Int. Assoc. Shell & Spatial Structures Symp., Boston.Google Scholar
Fivet, C. and Zastavni, D. (2012). Robert Maillart’s key methods from the Salginatobel bridge design process (1928). J. Int. Assoc. Shell & Spatial Structures, 53:3947.Google Scholar
Fivet, C. and Zastavni, D. (2015). A fully geometric approach for interactive constraint-based structural equilibrium design. Computer-Aided Design, 61:4257.CrossRefGoogle Scholar
Fogelsanger, A. (1988). The Generic Rigidity of Minimal Cycles. PhD thesis, Cornell University, Ithaca, NY.Google Scholar
Föppl, A. (1892). Das Fachwerk im Raume. BG Teubner.Google Scholar
Fowler, P. W. and Guest, S. D. (2000). A symmetry extension of Maxwell’s rule for rigidity of frames. Int. J. Solids & Structures, 37:17931804.CrossRefGoogle Scholar
Fowler, P. W., Guest, S. D., and Owen, J. C. (2021). Applications of symmetry in point-line-plane frameworks for CAD. Journal of Computational Design and Engineering, 8:615637.CrossRefGoogle Scholar
Fowler, P. W., Guest, S. D., and Tarnai, T. (2008). A symmetry treatment of Danzerian rigidity for circle packing. Proc. R. Soc. Ser. A, 464:32373254.CrossRefGoogle Scholar
Frost, P. and Wolstenholme, J. (1863). A Treatise on Solid Geometry. Macmillan and Co.Google Scholar
Fruchart, M., Zhou, Y. J., and Vitelli, V. (2020). Dualities and non-Abelian mechanics. Nature, 577:636.CrossRefGoogle ScholarPubMed
Gauss, C. F. (1813). Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum, methodo nova tractata. In Schäfer, C., editor, Werke, pages 279286. Teubner.Google Scholar
Gauss, C. F. (1833). Note in Werke, Vol. V, ed. Schäfer, C. (Königliche Gesellschaft der Wissenschaften zu Göttingen), p. 605.Google Scholar
Gluck, H. (1975). Almost all simply-connected surfaces are rigid. In Glaser, L. C. and Rushing, T. B., editors, Geometric Topology, Lecture Notes in Math., volume 438, pages 225239. Springer-Verlag.CrossRefGoogle Scholar
Gogu, G. (2005). Chebychev–Grübler–Kutzbach’s criterion for mobility calculation of multi-loop mechanisms revisited via theory of linear transformations. Eur. J. Mech.–A/Solids, 24:427441.CrossRefGoogle Scholar
Goodrich, C. P., Liu, A. J., and Nagel, S. R. (2015). The principle of independent bond-level response: Tuning by pruning to exploit disorder for global behavior. Physical Review Letters, 114:225501.CrossRefGoogle ScholarPubMed
Gortler, S. J., Healy, A. D., and Thurston, D. P. (2010). Characterizing generic global rigidity. Amer. J. Math., 132(4):897939.CrossRefGoogle Scholar
Graovac, O. (n.d.). 3D Graphic Statics (3DGS). www.food4rhino.com/en/app/3d-graphic-statics. v2.3.0.Google Scholar
Grasegger, G., Guler, H., Jackson, B., and Nixon, A. (2022). Flexible circuits in the d-dimensional rigidity matroid. J. Graph Theory, 100(2):315330.CrossRefGoogle Scholar
Graver, J. (2001). Counting on Frameworks, volume 25 of The Dolciani Mathematical Expositions. The Mathematical Association of America.CrossRefGoogle Scholar
Graver, J., Servatius, B., and Servatius, H. (1993). Combinatorial Rigidity, volume 2 of Graduate Studies in Mathematics. American Mathematical Society.Google Scholar
Gray, J. (2011). History of mathematics and history of science reunited? Isis, 102(3):511517.CrossRefGoogle ScholarPubMed
Guest, S. D. (2000). Tensegrities and rotating rings of tetrahedra: A symmetry viewpoint of structural mechanics. In Science into the Next Millennium: Young Scientists Give their Visions of the Future, Part II, volume 358(1765), pages 229243. The Royal Society of London.Google Scholar
Guest, S. D. and Fowler, P. W. (2007). Symmetry conditions and finite mechanisms. J. Mech. Mat. Struct., 2:293301.CrossRefGoogle Scholar
Guest, S. D. and Fowler, P. W. (2014). Symmetry-extended counting rules for periodic frameworks. Phil. Trans. R. Soc. A., 372(2008):114.CrossRefGoogle ScholarPubMed
Gurtin, M. E. (1972). The linear theory of elasticity. In Flügge, S. and Truesdell, C., editors, Handbuch der Physik, volume VIa/2. Springer-Verlag.Google Scholar
Guy, B. M., Richards, J. A., Hodgson, D. J. M., Blanco, E., and Poon, W. C. K. (2018). Constraint-based approach to granular dispersion rheology. Physical Review Letters, 121:128001.CrossRefGoogle ScholarPubMed
Hablicsek, M., Akbarzadeh, M., and Guo, Y. (2019). Algebraic 3D graphic statics: Reciprocal constructions. Computer-Aided Design, 108:3041.CrossRefGoogle Scholar
Hambly, E. C. (1991). Bridge Deck Behaviour. CRC Press.CrossRefGoogle Scholar
Harman, P. M. (1990). The Scientific Letters and Papers of James Clerk Maxwell, volume 1. Cambridge University Press.Google Scholar
Harman, P. M. (1998). The Natural Philosophy of James Clerk Maxwell. Cambridge University Press.Google Scholar
Harman, P. M. (2002). The Scientific Letters and Papers of James Clerk Maxwell, volume 3. Cambridge University Press.Google Scholar
Harman, P. M. (2004). Maxwell, James Clerk (1831–1879), Physicist. Oxford University Press.Google Scholar
He, Z. and Guest, S. D. (2022). On rigid origami III: Local rigidity analysis. Proc. R. Soc. Ser. A, 478(2258).Google ScholarPubMed
Heisel, F., Lee, J., Schlesier, K., Rippmann, M., Saeidi, N., Javadian, A., Nugroho, A. R., Van Mele, T., Block, P., and Hebel, D. E. (2017). Design, cultivation and application of loadbearing mycelium components: The MycoTree at the 2017 Seoul Biennale of Architecture and Urbanism. Int. J. Sustainable Energy Development (IJSED), 6(1):296303.CrossRefGoogle Scholar
Hendrickson, B. (1992). Conditions for unique graph realizations. SIAM J. Comput., 21(1):6584.CrossRefGoogle Scholar
Henneberg, L. (1903). Die graphische Statik der starren Körper, pages 345434. BG Teubner.Google Scholar
Henneberg, L. (1911). Die graphische Statik der starren Systeme. BG Teubner.Google Scholar
Heyman, J. (1998). Structural Analysis: A Historical Perspective. Cambridge University Press.CrossRefGoogle Scholar
Heyman, J. (1999). Navier’s straitjacket. Architectural Science Review, 42(2):9195.CrossRefGoogle Scholar
Huerta Fernández, S. (2010). Designing by geometry: Rankine’s theorems of transformation of structures. In Cassinello, P., Huerta Fernandez, S., Miguel, de Prada Poole, J., and Sanchez Lampreave, R., editors, Geometry and Proportion in Structural Design - Essays in Ricardo Aroca’s Honour. R. S. Lampreave.Google Scholar
Hughes, J. F., van Dam, A., McGuire, M., Sklar, D. F., Foley, J. D., Feiner, S., and Akeley, K. (2013). Computer Graphics: Principles and Practice. Addison-Wesley, 3rd. edition.Google Scholar
Iannuzzo, A., Dell’Endice, A., Maia Avelino, R., Kao, G., Van Mele, T., and Block, P. (2021). COMPAS Masonry: A computational framework for practical assessment of unreinforced masonry structures. In Proceedings of the SAHC Symp., 2020/21, Barcelona.Google Scholar
Ikeshita, R. and Tanigawa, S. (2018). Count matroids of group-labeled graphs. Combinatorica, 38(5):11011127.CrossRefGoogle Scholar
Inglis, C. E. (1913). Stresses in plates due to the presence of cracks and sharp corners. Trans. Inst. Naval Architects, 55:219241.Google Scholar
Jackson, B. and Jordán, T. (2005a). Connected rigidity matroids and unique realizations of graphs. J. Combin. Theory Ser. B, 94(1):129.CrossRefGoogle Scholar
Jackson, B. and Jordán, T. (2005b). The d-dimensional rigidity matroid of sparse graphs. J. Combin. Theory Ser. B, 95(1):118133.CrossRefGoogle Scholar
Jackson, B., Nixon, A., and Tanigawa, S. (2021). An improved bound for the rigidity of linearly constrained frameworks. SIAM J. Discrete Math., 35(2):928933.CrossRefGoogle Scholar
Jackson, B. and Owen, J. C. (2016). A characterisation of the generic rigidity of 2-dimensional point-line frameworks. J. Combin. Theory Ser. B, 119:96121.CrossRefGoogle Scholar
Jacobs, D. J. and Hendrickson, B. (1997). An algorithm for twodimensional rigidity percolation: the pebble game. J. Comput. Phys., 137(2):346365.CrossRefGoogle Scholar
Jacobs, D. J., Rader, A. J., Kuhn, L. A., and Thorpe, M. F. (2001). Protein flexibility predictions using graph theory. Proteins, 44:150165.CrossRefGoogle ScholarPubMed
Jaeger, H. M. (2015). Toward jamming by design. Soft Matter, 11:1227.CrossRefGoogle ScholarPubMed
James, I. M. (1999). History of Topology. Elsevier.Google Scholar
Jenkin, F. (1869). On the practical application of reciprocal figures to the calculation of strains on framework. Transactions of the Royal Society of Edinburgh, XXV:441447.CrossRefGoogle Scholar
Jordán, T. (2021). A note on generic rigidity of graphs in higher dimension. Discrete Appl. Math., 297:97101.CrossRefGoogle Scholar
Jordán, T., Kaszanitzky, V. E., and Tanigawa, S. (2016). Gainsparsity and symmetry-forced rigidity in the plane. Discrete Comput. Geom., 55(2):314372.CrossRefGoogle ScholarPubMed
Jordán, T. and Whiteley, W. (2017). Global rigidity. In Toth, C., O’Rourke, J., and Goodman, J., editors, Handbook of Discrete and Computational Geometry, chapter 63, pages 16611694. Chapman and Hall/CRC Press, 3 edition.Google Scholar
Kane, C. L. and Lubensky, T. C. (2014). Topological boundary modes in isostatic lattices. Nature Physics, 10:3945.CrossRefGoogle Scholar
Kangwai, R. D. and Guest, S. D. (2000). Symmetry-adapted equilibrium matrices. Int. J. Solids & Structures, 37:15251548.CrossRefGoogle Scholar
Katoh, N. and Tanigawa, S. (2011). A proof of the molecular conjecture. Discrete Comput. Geom., 45(4):647700.CrossRefGoogle Scholar
Katz, V. J. (2009). A History of Mathematics: An Introduction. Addison-Wesley, 3rd edition.Google Scholar
Keller, S. and Jaeger, H. M. (2016). Aleatory architectures. Granular Matter, 18:29.CrossRefGoogle Scholar
Kilian, M., Pellis, D., Wallner, J., and Pottmann, H. (2017). Material-minimizing forms and structures. ACM Trans. Graphics, 36(6):173:112.CrossRefGoogle Scholar
Klein, F. and Wieghardt, K. (1904). Über Spannungsflächen und reziproke Diagramme. Mit besonderer Berücksichtigung der Maxwellschen Arbeiten. Arch. Math. Phys., 3(8):110, 95–119.Google Scholar
Kline, M. (1972). Mathematical Thought from Ancient To Modern Times. Oxford University Press.Google Scholar
Koenderink, I. J. and van Doorn, A. J. (2002). Image processing done right. In Proc. ECCV 2002, LNCS 2350, pages 158172. Springer.Google Scholar
Konstantatou, M., Baker, W. F., Nugent, T., and McRobie, A. (2022). Grid-shell design and analysis via reciprocal discrete Airy stress functions. Int. J. Space Structures, 37(2):150164.CrossRefGoogle Scholar
Konstantatou, M., D’Acunto, P., McRobie, A., and Schwartz, J. (2019). Applications of graphic statics to analysis and design of reinforced concrete: Stress fields and yield lines. In Proc. Int. fib Symp. on Conceptual Design of Structures, pages 185192.Google Scholar
Konstantatou, M., D’Acunto, P., McRobie, A., and Schwartz, J. (2020). Unified geometrical framework for the plastic design of reinforced concrete structures. Structural Concrete, 21:23202338.CrossRefGoogle Scholar
Konstantatou, M., D’Acunto, P., and McRobie, A. (2018). Polarities in structural design and analysis: n-dimensional graphic statics and structural transformations. Int. J. Solids & Structures, 152153:272293.CrossRefGoogle Scholar
Konstantatou, M. and McRobie, A. (2018). Graphic statics for optimal trusses & geometry-based structural optimisation. In Proc. Int. Assoc. Shell & Spatial Structures (IASS) Symp., Boston.Google Scholar
Kurrer, K. E. (2008). The History of the Theory of Structures, From Arch Analysis to Computational Mechanics. Ernst & Sohn.CrossRefGoogle Scholar
Kutzbach, K. (1929). Mechanische Leitungsverzweigung. Maschinenbau, Der Betrieb, 8:710716.Google Scholar
Laman, G. (1970). On graphs and rigidity of plane skeletal structures. Journal of Engineering Mathematics, 4:331340.CrossRefGoogle Scholar
Lamé, G. (1852). Leçons sur la Théorie Mathématique de l'Élasticité des Corps Solides. Bachelier.Google Scholar
Lamé, G. and Clapeyron, B. P. E. (1827). Mémoire sur les polygones funiculaires. Journal des Voies de Communication, 1:4345.Google Scholar
Langhaar, H. L. and Stippes, M. (1954). Three-dimensional stress functions. Journal of the Franklin Institute, 258(5):371382.CrossRefGoogle Scholar
Lee, A. and Streinu, I. (2008). Pebble game algorithms and sparse graphs. Discrete Math., 308(8):14251437.CrossRefGoogle Scholar
Lee, J. (2018). Computational Design Framework for 3D Graphic Statics. PhD thesis, ETH Zurich.Google Scholar
Lee, J. and Kim, K. (1998). A 2-D geometric constraint solver using DOF-based graph reduction. Computer-aided Design., 30(11):883896.CrossRefGoogle Scholar
Lee, J., Mueller, C., and Fivet, C. (2016). Automatic generation of diverse equilibrium structures through shape grammars and graphic statics. Int. J. Space Structures, 31(2–4):146163.CrossRefGoogle Scholar
Lee-St.John, A. and Sidman, J. (2013). Combinatorics and the rigidity of CAD systems. Computer-Aided Design, 45(2):473482.CrossRefGoogle Scholar
Lévy, M. (1874–1888). La Statique Graphique et ses Applications aux Constructions, (4 vols.). Gauthier-Villars.Google Scholar
L’Huilier, G. (1812). Mémoire sur la polyédrométrie. Annales de Mathématiques Pures et Appliquées, 3:169189.Google Scholar
Listing, J. B. (1848). Vorstudien zur Topologie. Vandenhoeck und Ruprecht.Google Scholar
Liu, A. J. and Nagel, S. R. (2010). The jamming transition and the marginally jammed solid. Annual Review of Condensed Matter Physics, 1:347369.CrossRefGoogle Scholar
Liu, Y., Pottmann, H., Wallner, J., Yang, Y.-L., and Wang, W. (2006). Geometric modeling with conical meshes and developable surfaces. ACM Trans. Graphics, 25(3):681689.CrossRefGoogle Scholar
Longair, M. (2016). Maxwell’s Enduring Legacy: A Scientific History of the Cavendish Laboratory. Cambridge University Press.CrossRefGoogle Scholar
Lovász, L. and Yemini, Y. (1982). On generic rigidity in the plane. SIAM J. Algebr. Discrete Methods, 3:9198.CrossRefGoogle Scholar
Love, A. E. H. (1927). A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press.Google Scholar
Lu, Y., Seyedahmadian, A., Chhadeh, P. A., Cregan, M., Bolhassani, M., Schneider, J., Yost, J. R., Brennan, G., and Akbarzadeh, M. (2022). Funicular glass bridge prototype: Design optimization, fabrication, and assembly challenges. Glass Structures & Engineering, 7:319330.CrossRefGoogle Scholar
Lubarda, V. A. and Lubarda, M. V. (2020). On the Kelvin, Boussinesq, and Mindlin problems. Acta Mechanica, 231:155178.CrossRefGoogle Scholar
Lützen, J. (1995). Interactions between mechanics and differential geometry in the 19th century. Archive for History of Exact Sciences, 49:172.CrossRefGoogle Scholar
Mackinlay, J. (1986). Automating the design of graphical presentations of relational information. ACM Trans. Graphics, 5(2):110141.CrossRefGoogle Scholar
Malestein, J. and Theran, L. (2013). Generic combinatorial rigidity of periodic frameworks. Adv. Math., 233:291331.CrossRefGoogle Scholar
Maxwell, J. C. (1850). On the equilibrium of elastic solids. Transactions of the Royal Society of Edinburgh, 1:13346.Google Scholar
Maxwell, J. C. (1855). On Faraday’s lines of force. Transactions of the Cambridge Philosophical Society, X:156229.Google Scholar
Maxwell, J. C. (1864a). On reciprocal figures and diagrams of forces. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, XXVII:250261.Google Scholar
Maxwell, J. C. (1864b). On the calculation of the equilibrium and the stiffness of frames. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, XXVII:294299.Google Scholar
Maxwell, J. C. (1867). On the application of the theory of reciprocal polar figures to the construction of diagrams of forces. The Engineer, II:402.Google Scholar
Maxwell, J. C. (1868). On reciprocal diagrams in space, and their relation to Airy’s function of stress. Proceedings of the London Mathematical Society, II:5860.Google Scholar
Maxwell, J. C. (1870a). On reciprocal figures, frames and diagrams of forces. Transactions of the Royal Society of Edinburgh, XXVI, Pt.1:140.CrossRefGoogle Scholar
Maxwell, J. C. (1870b). On reciprocal figures, frames and diagrams of forces (Communication). Proceedings of the Royal Society of Edinburgh, VII(81):5356.Google Scholar
Maxwell, J. C. (1876). On Bow’s method of drawing diagrams in graphical statics, with illustrations from Peaucellier’s linkage. Transactions of the Cambridge Philosophical Society Proceedings, II:407414.Google Scholar
Mazurek, A., Baker, W. F., and Tort, C. (2011). Geometrical aspects of optimum truss like structures. Struct. and Multidisc. Optimization, 43(2):231242.CrossRefGoogle Scholar
McKenzie, W. M. C. (2013). Examples in Structural Analysis. CRC Press, 2nd edition.CrossRefGoogle Scholar
McRobie, A. (2016). Maxwell and Rankine reciprocal diagrams via Minkowski sums for two-dimensional and threedimensional trusses under load. Int. J. Space Structures, 31(2-4):203216.CrossRefGoogle Scholar
McRobie, A. (2017a). The geometry of structural equilibrium. Royal Society Open Science, 4(160759).Google ScholarPubMed
McRobie, A. (2017b). Graphic analysis of 3D frames: Clifford algebra and Rankine incompleteness. In Bogle, A. and Grohmann, M., editors, Proc. Int. Assoc. Shell & Spatial Structures (IASS) Symp., Hamburg.Google Scholar
McRobie, A. (2017c). Rankine reciprocals with zero bars. At researchgate.net/publication/315027343.Google Scholar
McRobie, A., Baker, W. F., Mitchell, T., and Konstantatou, M. (2015). Mechanisms and states of self-stress of planar trusses using graphic statics, part III: Applications and extensions. In Proc. Int. Assoc. Shell & Spatial Structures (IASS) Symp., Amsterdam.Google Scholar
McRobie, A., Baker, W. F., Mitchell, T., and Konstantatou, M. (2016). Mechanisms and states of self-stress of planar trusses using graphic statics, part II: Applications and extensions. Int. J. Space Structures, 31:102111.CrossRefGoogle Scholar
McRobie, A., Konstantatou, M., Athanasopoulos, G., and Hannigan, L. (2017). Graphic kinematics, visual virtual work and elastographics. Royal Society Open Science, 4(5).CrossRefGoogle ScholarPubMed
McRobie, A., Konstantatou, M., Athanasopoulos, G., Torpiano, G., Millar, C., and Baker, W. F. (2021a). Simple Rankine gridshells. In Proc. Int. Assoc. for Shell & Spatial Structures (IASS), Guilford.Google Scholar
McRobie, A., Millar, C., and Baker, W. F. (2021b). Stability of trusses by graphic statics. Royal Society Open Science, 8(6).CrossRefGoogle ScholarPubMed
McRobie, A. and Williams, C. J. K. (2018). Discontinuous Max well–Rankine stress functions for space frames. Int. J. Space Structures, 33(1):3547.CrossRefGoogle Scholar
Micheletti, A. (2008). On generalized reciprocal diagrams for self-stressed frameworks. Int. J. Space Structures, 23(3):153166.CrossRefGoogle Scholar
Michell, A. G. M. (1904). The limits of economy of material in frame-structures. The Philosophical Magazine and Journal of Science, VIII:589509.Google Scholar
Miki, M., Adiels, E., Baker, W. F., Mitchell, T., Sehlström, A., and Williams, C. J. K. (2022). Form-finding of shells containing both tension and compression using the Airy stress function. Int. J. Space Structures, 37(3):261282.CrossRefGoogle Scholar
Millar, C., Mitchell, T., Mazurek, A., Chhabra, A., Beghini, A., Clelland, J. N., McRobie, A., and Baker, W. F. (2022). On designing plane-faced funicular gridshells. Int. J. Space Structures, 38(1):4063.CrossRefGoogle Scholar
Mitchell, T., Baker, W. F., and McRobie, A. (2015). Mechanisms and states of self-stress of planar trusses using graphic statics, part II: The Airy stress function and the fundamental theorem of linear algebra. In Proc. Int. Assoc. Shell & Spatial Structures (IASS) Symp., Amsterdam.Google Scholar
Mitchell, T., Baker, W. F., McRobie, A., and Mazurek, A. (2016). Mechanisms and states of self-stress of planar trusses using graphic statics, part I: The fundamental theorem of linear algebra and the Airy stress function. Int. J. Space Structures, 31(2-4):85101.CrossRefGoogle Scholar
Möbius, A. F. (1827). Der barycentrische Calcul. Johann Ambrosius Barth.Google Scholar
Mohr, C. O. (1886). Eine Aufgabe der graphischen Statik. In Der Civilingenieur, pages 535538.Google Scholar
Monge, G. (1788). Traité Élémentaire de Statique à l'Usage des Écoles de la Marine. Baudouin.Google Scholar
Monge, G. (1799). Géométrie Descriptive. Gauthier-Villars.Google Scholar
Morera, G. (1892). Atti della reale accademia dei lincei, rendiconti, serie 5. 1:137141 and 233–234.Google Scholar
Mozaffari, S. (2021). Computational Strut-and-Tie Modeling: Explorations of Algebraic Graphic Statics and Layout Optimization. PhD thesis, ETH Zurich.Google Scholar
Mulcahy, J. (1862). Principles of Modern Geometry, with numerous applications to Plane and Spherical Figures. Hodges, Smith & Co., Dublin, 2nd edition.Google Scholar
Müller, A. (2016). Recursive higher-order constraints for linkages with lower kinematic pairs. Mechanism and Machine Theory, 100:3343.CrossRefGoogle Scholar
Müller-Breslau, H. (1881). Elemente der graphischen Statik der Bauconstructionen für Architekten und Ingenieure. Polytechnische Buchhandlung Seydel.Google Scholar
Murphy, K. A., Roth, L. K., Peterman, D., and Jaeger, H. M. (2017). Aleatory construction based on jamming: Stability through self-confinement. Architectural Design. Special issue on Autonomous Assembly: Designing for a New Era of Collective Construction, 187:7481.Google Scholar
Nejur, A. and Akbarzadeh, M. (2021). Polyframe, efficient computation for 3D graphic statics. Computer-Aided Design, 134:103003.CrossRefGoogle Scholar
Nevo, E. (2007). On embeddability and stresses of graphs. Combinatoria, 27(4):465472.CrossRefGoogle Scholar
Niven, W. D. (1890). The Scientific Papers of James Clerk Maxwell, volume 1 & 2. Cambridge University Press.Google Scholar
Nixon, A., Owen, J. C., and Power, S. C. (2012). Rigidity of frameworks supported on surfaces. SIAM J. Discrete Math., 26(4):17331757.CrossRefGoogle Scholar
Nixon, A. and Ross, E. (2014). One brick at a time: A survey of inductive constructions in rigidity theory. In Rigidity and Symmetry, volume 70 of Fields Inst. Commun., pages 303324. Springer.CrossRefGoogle Scholar
Nixon, A., Schulze, B., Tanigawa, S., and Whiteley, W. (2018). Rigidity of frameworks on expanding spheres. SIAM J. Discrete Math., 32(4):25912611.CrossRefGoogle Scholar
Nixon, A., Schulze, B., and Whiteley, W. (2021). Rigidity through a projective lens. Applied Sciences, pages 1105.CrossRefGoogle Scholar
Nixon, A. and Whiteley, W. (2018). Change of metrics in rigidity theory. In Sidman, J., Sitharam, M., and John, A., editors, Handbook of Geometric Constraint Systems Principles. Chapman and Hall/CRC Press.Google Scholar
Oh, K.-K., Park, M.-C., and Ahn, H.-S. (2015). A survey of multi-agent formation control. Automatica J. IFAC, 53:424440.CrossRefGoogle Scholar
O’Hern, C. S., Silbert, L. E., Liu, A. J., and Nagel, S. R. (2003). Jamming at zero temperature and zero applied stress: The epitome of disorder. Physical Review E, 68:011306.CrossRefGoogle ScholarPubMed
Ohlbrock, P. O. and D’Acunto, P. (2020). A computer-aided approach to equilibrium design based on graphic statics and combinatorial variations. Computer-Aided Design, 121:102802.CrossRefGoogle Scholar
Ohlbrock, P. O. and Schwartz, J. (2016). Combinatorial equilibrium modeling. Int. J. Space Structures, 31(2–3):176188.Google Scholar
Owen, J. C. (1991). Algebraic solution for geometry from dimensional constraints. In Proc. First ACM Symp. on Solid Modeling Foundations and CAD/CAM Applications, New York, SMA ’91, pages 397407.CrossRefGoogle Scholar
Owen, J. C. and Power, S. C. (2010). Frameworks symmetry and rigidity. Internat. J. Comput. Geom. Appl., 20(6):723750.CrossRefGoogle Scholar
Paulose, J., Chen, B. G. G., and Vitelli, V. (2015). Topological modes bound to dislocations in mechanical metamaterials. Nature Physics, 11:153156.CrossRefGoogle Scholar
Pellegrino, S. (1993). Structural computations with the singular value decomposition of the equilibrium matrix. Int. J. Solids & Structures, 30:30253035.CrossRefGoogle Scholar
Pellis, D. (2019). Quad Meshes as Optimized Architectural Freeform Structures. PhD thesis, Technische Universität Wien.Google Scholar
Pellis, D., Kilian, M., Dellinger, F., Wallner, J., and Pottmann, H. (2019). Visual smoothness of polyhedral surfaces. ACM Trans. Graphics, 38(4):260:1260:11.CrossRefGoogle Scholar
Phillips, J. C. (1979). Topology of covalent non-crystalline solids I: Short-range order in chalcogenide alloys. Journal of Non-Crystalline Solids, 34:153.CrossRefGoogle Scholar
Phillips, J. C. (1981). Topology of covalent non-crystalline solids II: Medium-range order in chalcogenide alloys and A-Si(Ge). Journal of Non-Crystalline Solids, 43:37.CrossRefGoogle Scholar
Phillips, J. C. and Thorpe, M. F. (1985). Constraint theory, vector percolation and glass formation. Solid State Communications, 53:699702.CrossRefGoogle Scholar
Plücker, J. (1865). On a new geometry of space. Philosophical Transactions of the Royal Society of London, 155:725791.Google Scholar
Pollaczek-Geiringer, H. (1927). Über die Gliederung ebener Fachwerke. ZAMM-J. Appl. Math. Mech. Angew. Math. Mech., 7:5872.CrossRefGoogle Scholar
Poncelet, J.-V. (1822). Traité des Propriétés Projectives des Figures. Bachelier.Google Scholar
Pottmann, H., Asperl, A., Hofer, M., and Kilian, A. (2007a). Architectural Geometry, volume 10. Bentley Institute Press.Google Scholar
Pottmann, H., Eigensatz, M., Vaxman, A., and Wallner, J. (2015). Architectural geometry. Computers and Graphics, 47:145–164.Google Scholar
Pottmann, H., Grohs, P., and Mitra, N. J. (2009). Laguerre minimal surfaces, isotropic geometry and linear elasticity. Adv. Comp. Math, 31:391419.CrossRefGoogle Scholar
Pottmann, H. and Liu, Y. (2007). Discrete surfaces in isotropic geometry. In Sabin, M. and Winkler, J., editors, Mathematics of Surfaces XII, pages 341363. Springer.CrossRefGoogle Scholar
Pottmann, H., Liu, Y., Wallner, J., Bobenko, A., and Wang, W. (2007b). Geometry of multi-layer freeform structures for architecture. ACM Trans. Graphics, 26(3):#65,111.CrossRefGoogle Scholar
Prager, W. (1985). Optimal design of trusses. In Save, M., Prager, W., and Warner, W. H., editors, Structural Optimization. Volume 1: Optimality Criteria, pages 121152. Plenum Press.CrossRefGoogle Scholar
Prandtl, L. (1903). Zur Torsion von prismatischen Stäben. Physikalische Zeitscrift, 4:758770.Google Scholar
Rankine, W. J. M. (1858). A Manual of Applied Mechanics. Richard Griffin and Company.Google Scholar
Rankine, W. J. M. (1863). On the application of barycentric perspective to the transformation of structures. Philosopical Magazine Ser. 4, 27:387388.CrossRefGoogle Scholar
Rankine, W. J. M. (1864). Principle of the equilibrium of polyhedral frames. The Philosophical Magazine and Journal of Science, XXVII(92).Google Scholar
Rankine, W. J. M. (1870). Diagrams of forces in frameworks. Proceedings of the Royal Society of Edinburgh, 7:171172.Google Scholar
Richards, J. L. (1986). Projective geometry and mathematical progress in mid-Victorian Britain. Studies in History and Philosophy of Science Part A, 17(3):297325.CrossRefGoogle Scholar
Richter-Gebert, J. (2011). Perspectives on Projective Geometry: A Guided Tour through Real and Complex Geometry. Springer.CrossRefGoogle Scholar
Rippmann, M., Lachauer, L., and Block, P. (2012). RhinoVAULT – interactive vault design. Int. J. Space Structures, 27(4):219230.CrossRefGoogle Scholar
Rivlin, R. S. (1955). Plane strain of a net formed by inextensible cords. Archives of Rational Mechanics and Analysis, 4:951974.Google Scholar
Rocks, J. W., Pashine, N., Bischofberger, I., Goodrich, C. P., Liu, A. J., and Nagel, S. R. (2017). Designing allostery-inspired response in mechanical networks. Proceedings of the National Academy of Sciences, 114:25202525.CrossRefGoogle ScholarPubMed
Rosenfeld, B. A. (1988). A History of non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. Springer Verlag.CrossRefGoogle Scholar
Ross, E. (2015). Inductive constructions for frameworks on a twodimensional fixed torus. Discrete Comput. Geom., 54(1):78109.CrossRefGoogle Scholar
Rostamian, R. (1979). The completeness of Maxwell’s stress function representation. Journal of Elasticity, 9:349356.CrossRefGoogle Scholar
Roth, B. and Whiteley, W. (1981). Tensegrity frameworks. Trans. Amer. Math. Soc., 265(2):419446.CrossRefGoogle Scholar
Routh, E. J. (1891). A Treatise on Analytical Statics, with Numerous Examples. Cambridge University Press, 1st. edition.Google Scholar
Rozhkova, E. V. (2009). On solutions of the problem in stresses with the use of Maxwell stress functions. Mechanics of Solids, 44:526536.CrossRefGoogle Scholar
Sachs, H. (1987). Ebene isotrope Geometrie. Vieweg.CrossRefGoogle Scholar
Sachs, H. (1990). Isotrope Geometrie des Raumes. Vieweg.CrossRefGoogle Scholar
Sadd, M. H. (2014). Elasticity: Theory, Applications, and Numerics. Academic Press.Google Scholar
Saliola, F. and Whiteley, W. (2005). Constraining plane configurations in CAD: circles, lines and angles in the plane. SIAM J. Discrete Math., 18:246271.CrossRefGoogle Scholar
Saliola, F. and Whiteley, W. (2007). Some notes on the equivalence of first-order rigidity in various geometries. arXiv:0709.3354.Google Scholar
Salmon, G. (1865). A Treatise on the Analytic Geometry of Three Dimensions. Hodges, Smith & Co.Google Scholar
Sauer, R. (1970). Differenzengeometrie. Springer.CrossRefGoogle Scholar
Schaefer, H. (1953). Die Spannungsfunktionen des dreidimensionalen Kontinuums und des elastischen Körpers. Zeitschrift für Angewandte Mathematik und Mechanik, 33:356362.CrossRefGoogle Scholar
Scheibner, C., Souslov, A., Banerjee, D., Surowka, P., Irvine, W. T. M., and Vitelli, V. (2020). Odd elasticity. Nature Physics, 16:475.CrossRefGoogle Scholar
Schek, J. H. (1974). The force density method for form finding and computation of general networks. Computer Methods in Applied Mechanics and Engineering, 3(1):115134.CrossRefGoogle Scholar
Schläfli, L. (1855). Réduction d’une intégrale multiple qui comprend Fare de cieñe et Votre du triangle sphérique comme cas particuliers. Journal de Mathématiques Pures et Appliquées, 20:359394.Google Scholar
Schläfli, L. (1858). On the multiple integral ʃ dx dy...dz whose limits are p1 = a1 x + b1 y + ... + h1z > 0, p2 > 0,..., pn > 0 and x2 + y2 + ... + z2 < 1. The Quarterly Journal of Pure and Applied Mathematics, 2:269301.Google Scholar
Scholz, E. (1994). Graphical statics. In Gratton-Guinness, I., editor, Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, volume 2, pages 978993. Routledge.Google Scholar
Schulze, B. (2010a). Block-diagonalized rigidity matrices of symmetric frameworks and applications. Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 51(2):427466.Google Scholar
Schulze, B. (2010b). Symmetric Laman theorems for the groups C2 and Cs. Electron. J. Combin., 17(1):Research Paper 154, 61.CrossRefGoogle Scholar
Schulze, B. (2010c). Symmetric versions of Laman’s theorem. Discrete Comput. Geom., 44(4):946972.CrossRefGoogle Scholar
Schulze, B. (2010d). Symmetry as a sufficient condition for a finite flex. SIAM J. Discrete Math., 24:12911312.CrossRefGoogle Scholar
Schulze, B., Guest, S. D., and Fowler, P. W. (2014). When is a symmetric body-hinge structure isostatic? International Journal of Solids and Structures, 51:21572166.CrossRefGoogle Scholar
Schulze, B., Millar, C., Mazurek, A., and Baker, W. F. (2022). States of self-stress in symmetric frameworks and applications. Int. J. Solids & Structures, 234-235:111238.Google Scholar
Schulze, B. and Tanigawa, S. (2015). Infinitesimal rigidity of symmetric bar-joint frameworks. SIAM J. Discrete Math., 29(3):12591286.CrossRefGoogle Scholar
Schulze, B. and Whiteley, W. (2011). The orbit rigidity matrix of a symmetric framework. Discrete Comput. Geom., 46:561598.CrossRefGoogle Scholar
Schulze, B. and Whiteley, W. (2017a). Rigidity and scene analysis. In Toth, C. D., O’Rourke, J., and Goodman, J. E., editors, Handbook of Discrete and Computational Geometry, chapter 61. Chapman and Hall/CRC Press, 3rd edition.Google Scholar
Schulze, B. and Whiteley, W. (2017b). Rigidity of symmetric frameworks. In Toth, C., O’Rourke, J., and Goodman, J., editors, Handbook of Discrete and Computational Geometry, chapter 62. Chapman and Hall/CRC Press, 3rd edition.Google Scholar
Schulze, B. and Whiteley, W. (2022). Projective geometry of scene analysis, parallel drawing and reciprocal diagrams. Applied Sciences.Google Scholar
Servatius, B. and Whiteley, W. (1999). Constraining plane configurations in computer-aided design: Combinatorics of directions and lengths. SIAM J. Discrete Math., 12(1):136153.CrossRefGoogle Scholar
Seyrig, T. (1878). Le Pont sur le Douro à Porto. Capiomont & Renault, Paris.Google Scholar
Shai, O., Servatius, B., and Whiteley, W. (2010). Geometric properties of Assur graphs. Eur. J. Combin., 31:11051120.Google Scholar
Shai, O., Sljoka, A., and Whiteley, W. (2013). Directed graphs, decompositions, and spatial linkages. Discrete Appl. Math., 161:30283047.CrossRefGoogle Scholar
Shephard, G. C. (1963). Decompostole convex polyhedra. Mathematika, 10:8995.CrossRefGoogle Scholar
Sidonius, (2009). Der Traversinersteg auf der Via Spluga in der Via Mala-Schlucht, Graubünden. Photograph, commons.wikimedia.org/wiki/File:Traversinersteg1.JPG.Google Scholar
Silbert, L. E., Liu, A. J., and Nagel, S. R. (2005). Vibrations and diverging length scales near the unjamming transition. Physical Review Letters, 95:347369.CrossRefGoogle ScholarPubMed
Simpson, L. B. (2019). Exchange House. Photo, Copyright SOM.Google Scholar
Singh, A., Jackson, G. L., Naald, M. v. d., de Pablo, J. J., and Jaeger, H. M. (2022). Stress-activated constraints in dense suspension rheology. Physical Review Fluids, 7:054302.CrossRefGoogle Scholar
Singh, A., Ness, C., Seto, R., de Pablo, J. J., and Jaeger, H. (2020). Shear thickening and jamming of dense suspensions: The “roll” of friction. Physical Review Letters, 124:248005.CrossRefGoogle ScholarPubMed
Sitharam, M., St. John, A., and Sidman, J., editors (2019). Handbook of Geometric Constraint Systems Principles. Chapman and Hall/CRC Press.Google Scholar
Sljoka, A. (2021). Probing allosteric mechanism with long-range rigidity transmission across protein networks. Methods Mol. Biol., 2253:6175.CrossRefGoogle ScholarPubMed
Smith, C. (1884). Elementary Treatise on Solid Geometry. Macmillan and Co.Google Scholar
Stewart, D. (1965). A platform with six degrees of freedom. Proc. Institution of Mechanical Engineers, 180:371386.CrossRefGoogle Scholar
Stokes, G. G. (1905). Mathematical and Physical Papers. Cambridge University Press.Google Scholar
Strubecker, K. (1941). Differentialgeometrie des isotropen Raumes I: Theorie der Raumkurven. Sitzungsber. Akad. Wiss. Wien, 150:153.Google Scholar
Strubecker, K. (1942a). Differentialgeometrie des isotropen Raumes II: Die Flächen konstanter Relativkrümmung k = rts2. Math. Zeitschrift, 47:743777.CrossRefGoogle Scholar
Strubecker, K. (1942b). Differentialgeometrie des isotropen Raumes III: Flächentheorie. Math. Zeitschrift, 48:369427.CrossRefGoogle Scholar
Strubecker, K. (1962). Airy’sche Spannungsfunktion und isotrope Differentialgeometrie. Math. Zeitschrift, 78:189198.CrossRefGoogle Scholar
Sugihara, K. (1984). An algebraic and combinatorial approach to the analysis of line drawings of polyhedra. Discrete Appl. Math., 9:77104.CrossRefGoogle Scholar
Sugihara, K. (1986). Machine Interpretation of Line Drawings. Artificial Intelligence Series, MIT Press.Google Scholar
Tachi, T. (2012). Design of infinitesimally and finitely flexible origami based on reciprocal figures. J. Geom. Graph., 16:223234.Google Scholar
Tait, P. G. (1880). Clerk-Maxwell’s scientific work. Nature, XXI:317321.CrossRefGoogle Scholar
Tanigawa, S. (2015). Matroids of gain graphs in applied discrete geometry. Trans. Amer. Math. Soc., 367(12):85978641.CrossRefGoogle Scholar
Tarnai, T., Fowler, P. W., Guest, S. D., and Kovács, F. (2022). Equiauxetic hinged Archimedean tilings. Symmetry, 14(232).CrossRefGoogle Scholar
Tay, T. S. (1984). Rigidity of multi-graphs, linking rigid bodies in n-space. J. Comb. Theory, B, 36:95112.CrossRefGoogle Scholar
Tay, T. S. (1989). Linking (n – 2)-dimensional panels in n-space II: (n – 2, 2)-frameworks and body and hinge structures. Graphs Combin., 5(1):245273.CrossRefGoogle Scholar
Tay, T. S. (1991). Linking (n – 2)-dimensional panels in n-space I: (k – 1, k)-graphs and (k - 1, k)-frames. Graphs Combin., 7(3):289304.CrossRefGoogle Scholar
Tay, T. S. and Whiteley, W. (1984). Recent advances in the generic rigidity of structures. Structural Topology, 9:3138.Google Scholar
Tay, T. S. and Whiteley, W. (1985). Generating isostatic frameworks. Structural Topology, 11:2169.Google Scholar
Tellier, X., Douthe, C., Hauswirth, L., and Baverel, O. (2021). Form-finding with isotropic linear Weingarten surfaces. In Advances in Architectural Geometry 2020, pages 1837.Google Scholar
Thomson, W. and Tait, P. G. (1867). Treatise on Natural Philosophy. Oxford University Press.Google Scholar
Timoshenko, S. (1953). History of Strength of Materials. McGraw-Hill.Google Scholar
Todisco, L., Fivet, C., Corres, H., and Mueller, C. T. (2015). Design and exploration of externally post-tensioned structures using graphic statics. J. Int. Assoc. Shell & Spatial Structures, 56(4):249258.Google Scholar
Vaisman, I. (1997). Analytical Geometry, volume 8. World Scientific Publishing.CrossRefGoogle Scholar
Van Mele, T. and Block, P. (2014). Algebraic graph statics. Computer-Aided Design, 53:104116.CrossRefGoogle Scholar
Van Mele, T., Liew, A., Méndez Echenagucia, T., and Rippmann, M. (n.d.). COMPAS. block.arch.ethz.ch/brg/content/tool/.Google Scholar
Van Mele, T., Mehrotra, A., Méndez Echenagucia, T., Frick, U., Ochsendorf, J., DeJong, M., and Block, P. (2016). Form finding and structural analysis of a freeform stone vault. In Proc. Int. Assoc. Shell & Spatial Structures (IASS) Symp., Tokyo.Google Scholar
Van Mele, T., Rippmann, M., Lachauer, L., and Block, P. (2012). Geometry-based understanding of structures. J. Int. Assoc. Shell & Spatial Structures, 53(4):285295.Google Scholar
Varignon, P. (1725). Nouvelle mécanique ou statique, dont le projet fut donné en 1687. In Ouvrage posthume de M. Varignon, volume 1 & 2. Claude Jombert.Google Scholar
Vouga, E., Höbinger, M., Wallner, J., and Pottmann, H. (2012). Design of self-supporting surfaces. ACM Trans. Graphics, 31(4):87:111.CrossRefGoogle Scholar
Wallner, J. and Pottmann, H. (2008). Infinitesimally flexible meshes and discrete minimal surfaces. Monatshefte für Mathematik, 153:347365.CrossRefGoogle Scholar
Westergaard, H. M. (1939). Bearing pressures and cracks. Journal of Applied Mechanics, 6:4953.CrossRefGoogle Scholar
Westergaard, H. M. (1952). Theory of Elasticity and Plasticity. Harvard Monographs in Applied Science.CrossRefGoogle Scholar
White, N. and Whiteley, W. (1987). The algebraic geometry of motions of bar-and-body frameworks. SIAM J. Algebr. Discrete Methods, 8(1):132.CrossRefGoogle Scholar
Whiteley, W. (1982). Motions and stresses of projected polyhedra. Structural Topology, 7:1338.Google Scholar
Whiteley, W. (1984a). Infinitesimal motions of a bipartite framework. Pacific J. Math., 110(1):233255.CrossRefGoogle Scholar
Whiteley, W. (1984b). Infinitesimally rigid polyhedra I: Statics of frameworks. Trans. Amer. Math. Soc., 285:431465.CrossRefGoogle Scholar
Whiteley, W. (1987). Rigidity and polarity I: Statics of sheetworks. Geom. Dedicata, 22:329362.CrossRefGoogle Scholar
Whiteley, W. (1989a). A matroid on hypergraphs, with applications in scene analysis and geometry. Discrete Comput. Geom., 4:7595.CrossRefGoogle Scholar
Whiteley, W. (1989b). Rigidity and polarity II: Weaving lines and plane tensegrity frameworks. Geom. Dedicata, 30:255279.CrossRefGoogle Scholar
Whiteley, W. (1991). Weaving, sections and projections of polyhedra. Discrete Applied Math., 32:275294.CrossRefGoogle Scholar
Whiteley, W. (1996). Some matroids from discrete applied geometry. In Matroid theory (Seattle, WA, 1995), volume 197 of Contemp. Math., pages 171311. American Mathematical Society.Google Scholar
Whiteley, W. (1998). An analogy in geometric homology: Rigidity and cofactors on geometric graphs. In Sagan, B. and Stanley, R., editors, Mathematical Essays in Honor of Gian-Carlo Rota, pages 413437. Birkhauser.CrossRefGoogle Scholar
Whiteley, W. (2005). Counting out to the flexibility of molecules. J. Physical Biology, 2:111.Google Scholar
Williams, C. J. K. and McRobie, A. (2016). Graphic statics using discontinuous Airy stress functions. Int. J. Space Structures, 31(2–4):121134.CrossRefGoogle Scholar
Xiao, Y., Khader, N., Vandenberg, A., Lowke, D., Kloft, H., and Hack, N. (2022). Injection 3D concrete printing (I3DCP) combined with vector-based 3D graphic statics. In Buswell, R., Blanco, A., Cavalaro, S., and Kinnell, P., editors, Third RILEM Int. Conf. on Concrete and Digital Fabrication, pages 4349. Springer.CrossRefGoogle Scholar
Yi, D. K. and Wang, T. C. (2020). On the effect of an out-of-plane constraint on the three-dimensional crack front fields in a thin elastic plate. Acta Mechanica, 231:28952913.CrossRefGoogle Scholar
Zalewski, W. and Allen, E. (1998). Shaping Structures: Statics. John Wiley and Sons.Google Scholar
Zanni, G. and Pennock, G. R. (2009). A unified graphical approach to the static analysis of axially loaded structures. Mechanism and Machine Theory, 44(12):21872203.CrossRefGoogle Scholar
Zastavni, D. (2008). The structural design of Maillart’s Chiasso Shed (1924): A graphic procedure. Structural Engineering International, 18(3):247252.CrossRefGoogle Scholar
Zessin, J., Lau, W., and Ochsendorf, J. (2010). Equilibrium of cracked masonry domes. In Proc. Institution of Civil Engineers - Engineering and Computational Mechanics, number 3 in 163, pages 135145.CrossRefGoogle Scholar
Zhao, S. and Zelazo, D. (2019). Bearing rigidity theory and its applications for control and estimation of network systems: Life beyond distance rigidity. IEEE Control Syst., 39(2):6683.CrossRefGoogle Scholar

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  • References
  • Edited by William F. Baker, Skidmore Owings and Merrill, Chicago, Allan McRobie, University of Cambridge
  • Book: The Geometry of Equilibrium
  • Online publication: 03 May 2025
  • Chapter DOI: https://doi.org/10.1017/9781009397643.029
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  • References
  • Edited by William F. Baker, Skidmore Owings and Merrill, Chicago, Allan McRobie, University of Cambridge
  • Book: The Geometry of Equilibrium
  • Online publication: 03 May 2025
  • Chapter DOI: https://doi.org/10.1017/9781009397643.029
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  • References
  • Edited by William F. Baker, Skidmore Owings and Merrill, Chicago, Allan McRobie, University of Cambridge
  • Book: The Geometry of Equilibrium
  • Online publication: 03 May 2025
  • Chapter DOI: https://doi.org/10.1017/9781009397643.029
Available formats
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