What Is the Shape of a Cupola?

References

• Alexandrov, A. D. (1962). Uniqueness theorems for surfaces in the large I. Amer. Math. Soc. Transl. 21: 341–354.
   Google Scholar
• Beltrami, E. (1882). Sull equilibrio delle superficie flessibili ed inestensibili. Memorie della Academia delle Scienze dell Istituto di Bologna, Series 4. 3: 217–265.
   Google Scholar
• Bemelmans, J., Dierkes, U. (1987). On a singular variational integral with linear growth, I: Existence and regularity of minimizers. Arch. Ration. Mech. Anal. 100: 83–103. DOI: 10.1007/BF00281248.
   Web of Science ®Google Scholar
• Böhme, R., Hildebrandt, S., Taush, E. (1980). The two-dimensional analogue of the catenary. Pacific J. Math. 88: 247–278. doi.org/pjm/1102779515 DOI: 10.2140/pjm.1980.88.247.
   Web of Science ®Google Scholar
• Chilton, J. (2000). The Engineer’s Contribution to Contemporary Architecture: Heinz Isler. London: Thomas Telford Press.
   Google Scholar
• Dierkes, U. (1988). A geometric maximum principle, Plateau’s problem for surfaces of prescribed mean curvature, and the two-dimensional analogue of the catenary. In: Hildebrandt, S., Leis, R., eds. Partial Differential Equations and Calculus of Variations. Springer Lecture Notes in Mathematics, Vol. 1357. Berlin: Springer, pp. 116–141. doi. 10.1007/BFb0082864
   Google Scholar
• Dierkes, U. (2003). Singular minimal surfaces. In: Hildebrandt, S., Karcher, H., eds. Geometric Analysis and Nonlinear Partial Differential Equations. Berlin: Springer, pp. 177–193.
   Google Scholar
• Dierkes, U., Huisken, G. (1990). The n-dimensional analogue of the catenary: existence and nonexistence. Pacific J. Math. 141: 47–54. DOI: 10.2140/pjm.1990.141.47.
   Web of Science ®Google Scholar
• Dunn, W. (1908). The principles of dome construction: I and II. J. Royal Institute of British Architects. 23: 401–412.
   Google Scholar
• Fouladgar, K., Simon, L. (2020). The symmetric minimal surface equation. Indiana Univ. Math. J. 69: 331–366. DOI: 10.1512/iumj.2020.69.8412.
   Web of Science ®Google Scholar
• Germain, S. (1821). Recherches sur la théorie des surfaces elastiques. Paris: Veuve Courcier.
   Google Scholar
• de Gresy, C. (1818). Considération sur l’équilibre des surfaces flexibles et inextensibles. Mem. Reale Accad. Sci. Torino. 21: 259–294.
   Google Scholar
• Heyman, J. (1977). Equilibrium of Shell Structures. Oxford: Oxford University Press.
   Google Scholar
• Hooke, R. (1675). A description of helioscopes, and some other instruments. London: Royal Society.
   Google Scholar
• Jellet, J. H. (1853). On the properties of inextensible surfaces. Transactions of the Royal Irish Academy. 22: 343–378.
   Google Scholar
• Lagrange, J. L. (1788). Mécanique Analytique. Tome XI., 4th ed. (1888). Paris: Gauthier-Villars.
   Google Scholar
• López, R. (2018). Invariant singular minimal surfaces. Ann. Global Anal. Geom. 53: 521–541. DOI: 10.1007/s10455-017-9586-9.
   Web of Science ®Google Scholar
• López, R. (2019). The Dirichlet problem for the α-singular minimal surface equation. Arch. Math. (Basel). 112: 213–222. DOI: 10.1007/s00013-018-1255-0.
   Web of Science ®Google Scholar
• López, R. (2019). Uniqueness of critical points and maximum principles of the singular minimal surface equation. J. Differ. Equ. 266: 3927–3941. DOI: 10.1016/j.jde.2018.09.024.
   Web of Science ®Google Scholar
• López, R. (2020). Compact singular minimal surfaces with boundary. Amer. J. Math. 142: 1771–1795. DOI: 10.1353/ajm.2020.0044.
   Web of Science ®Google Scholar
• Nitsche, J. C. C. (1986). A nonexistence theorem for the two-dimensional analogue of the catenary. Analysis. 6: 143–156. DOI: 10.1524/anly.1986.6.23.143.
   Google Scholar
• Oppenheim, I. J., Gunaratnam, D. J., Allen, R. H. (1989). Limit state analysis of masonry domes. J. Struct. Eng. 115: 868–882. DOI: 10.1061/(ASCE)0733-9445(1989)115:4(868).
   Web of Science ®Google Scholar
• Osserman, R. (2010). Mathematics of the gateway arch. Not. Am. Math. Soc. 57: 220–229.
   Google Scholar
• Otto, F. (1962/1966). Zugbeanspruchte Konstruktionen. Bd. I, II. Berlin, Frankfurt, Wien: Ullstein.
   Google Scholar
• Frei Otto: Spanning the Future. (2020). Documentary film. Dir. Joshua Hassel. https://www.youtube.com/watch?v=P5hKnOyg43k
   Google Scholar
• Paradiso, M., Rapallini, M., Tempesta, G. (2003). Masonry domes. Comparison between some solutions under no-tension hypothesis. Proceedings of the First International Congress on Construction History, Madrid: Instituto Juan de Herrera, SEdHC, ETSAM, A. E. Benvenuto, COAM, F. Dragados. pp. 1571–1581.
   Google Scholar
• Poisson, S. D. (1812). Mémoire sur les surfaces élastiques. Mémoires de l’Institut de France, 1814/1816. Vol. 9, pp. 167–226.
   Google Scholar
• Polyanin, A. D., Zaitsev, V. F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations. Boca Raton: Chapman & Hall/CRC.
   Google Scholar
• Pottmann, H., Asperl, A., Hofer, M., Kilian, A. (2007). Architectural Geometry. Exton: Bentley Institute Press.
   Google Scholar
• Todhunter, I., Pearson, K. (1986). A History of Elasticity and Strength of Materials, Vol. 1. Cambridge: Cambridge Univ. Press.
   Google Scholar
• Volterra, V. (1884/1885). Sulla deformazione delle superficie flessibili ed inestensibili. Atti della R. Accad. Dei Lincei, Rendiconti, Series 4, Vol.1: 274–278.
   Google Scholar