Locally isotropic growth

A convenient way to see what happens if elements of a surface grow isotropically is to divide the surface into a collection of very small circles, and then to expand the circle at each point by a factor h[x, y]. If the local curvature of the surface is originally c[x, y], then after such growth, the curvature turns out to be (c[x, y] + Laplacian[Log[h[x, y]]])/h[x, y] where Laplacian[f_] := ∂_{xx}f + ∂_{yy}f. In order for the surface to stay flat its growth rate Log[h[x, y]] must therefore solve Laplace's equation, and hence must be a harmonic function Re[f[x + y]]. This is equivalent to saying that the growth must correspond to a conformal mapping which locally preserves angles. The pictures below show results for several growth rate functions; in the last case, the function is not harmonic, and the surface cannot be drawn in the plane without tearing. Note that if the elements of a surface are allowed to change shape, then the surface can always remain flat, as in the top row of pictures on page 412. Harmonic growth rate functions can potentially be obtained from the large-time effects of a chemical subject to diffusion. And this may perhaps be related to the flatness observed in the growth of leaves. (See also page 1010.)