What's "erg"? --Rich -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Cordwell, William R Sent: Monday, November 12, 2012 9:39 AM To: math-fun Subject: Re: [math-fun] [EXTERNAL] Optimal Dome Isn't there a shape that has purely compressional tension? The St. Louis arch? It does not have a vertical tangent, as you note. -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Warren Smith Sent: Monday, November 12, 2012 9:19 AM To: math-fun@mailman.xmission.com Subject: [EXTERNAL] [math-fun] Optimal Dome
From: Henry Baker <hbaker1@pipeline.com> 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.
<<chop>>