This reminds me of a related question I've long had about measuring curvature of a given physical object. In the University of Goettingen's math museum, there's a sculpture of Apollo with lines inked on the surface to separate regions of positive curvature from regions of negative curvature. The book Geometry and the Imagination by Hilbert and Cohn-Vossen shows this object as Fig. 204 and says Felix Klein had the lines drawn on it to test his theory of facial beauty based on curvature. I'm curious what his theory was, but what I'm mainly wondering is how were the lines drawn? What technique or physical device does one apply to a given physical surface to locate the "parabolic lines" that separate areas of positive and negative curvature? In case you don't have your Hilbert and Cohn-Vossen handy, I put a photo of Klein's drawn-on Apollo in its Goettingen display cabinet here: http://georgehart.com/private/Apollo-Gottingen.jpg There have been all kinds of mechanical devices of a mathematical nature for constructing ellipses, trisecting angles, and such. Is there a mechanical device that identifies points of zero curvature on a physically given surface? George http://georgehart.com/ On 4/10/2014 6:54 AM, Adam P. Goucher wrote:
I propose the following experiment:
For varying values of r, cut discs of radius r out of the kale. Then measure the area A (or, equivalently, weigh them, assuming kale is of uniform thickness and density). You are looking for a relationship of the following form:
A = 2 pi s^2 (cosh(r/s) - 1)
where s is the characteristic length-scale of kale (and, I believe, the reciprocal of the absolute value of the curvature).
Sincerely,
Adam P. Goucher
----- Original Message ----- From: Fred Lunnon Sent: 04/10/14 04:04 AM To: math-fun Subject: Re: [math-fun] Kale
Jim, stop playing with your food, and eat up your greens! WFL
On 4/10/14, James Propp <jamespropp@gmail.com> wrote:
Does kale have constant (negative) curvature?
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