I've always wondered about the shape of the ideal self-supported masonry dome -- e.g., the Roman Pantheon, Hagia Sophia in Istanbul, St. Peter's Rome, etc. "Self-supported" here means compression forces only, as masonry has very little strength in tension or shear. Here's the solution, which involves the erf & erg functions, instead of (exp(x)+exp(-x))/2=cosh(x) (2D catenary). z_r = 1/2 r_0 [erg(r/r_0) - sqrt(pi)/2 erf(r/r_0)] (equation (9) in [Heyman1998] below) Note the obvious analogy to cosh/sinh functions. The Roman Pantheon is hemispherical, which is too narrow near the top and too wide near the bottom. All domes since the Pantheon incorporate one or more circular iron/steel chains horizontally around the lower portions of the dome to counteract the spreading forces; the Pantheon avoids the need for these chains with an absolutely massive circular drum structure which acts like a continuous circular buttress. Hagia Sophia was originally built without chains, but after several collapses, chains were finally added to stabilize the dome. The Pantheon is a living example of Archimedes' Hat Box Theorem: the right cylindrical drum encloses a perfect sphere touching the center of the Pantheon floor and the hemispherical interior of the dome. From the outside, the dome looks a lot more like a cone truncated at the eye; the space between the cone and the hemispherical internal surface is more-or-less filled with material, but with coffering. As the paper below demonstrates, Hooke's guess at the appropriate curve (z_r ~ r^3) (used in Wren's St. Paul's Cathedral, London) is the first term in the Taylor series for the correct erg/erf function. The most interesting feature of this equation is the fact that at r=0 the radius of curvature is infinite (i.e., a plane), unlike a spherical done, which has a finite radius of curvature. But an infinite radius of curvature implies that the compressive forces at r=0 are also infinite (in order to keep the load in the very center from falling down). But the Romans guessed/intuited this: the Pantheon cleverly cuts out the center & leaves the famous "eye" hole; Wren's St. Paul's also includes a hole in the center. Heyman, Jacques. "Hooke's Cubico-Parabolical Coniod". Notes Rec. R. Soc. Lond. 52(1), 39-50 (1998). http://rsnr.royalsocietypublishing.org/content/52/1/39.full.pdf