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With material where parts can locally expand, but cannot change their shape, page 1007 showed that a 2D surface will remain flat if the growth rate is a harmonic function.
As the picture shows, this leads to a simple expanding pattern uniformly filled with black.
The sequence expands by at least one digit every two steps; more rapid expansion is typically correlated with increased randomness.
(Thus it is impossible with ruler and compass to construct π and "square the circle" but it is possible to construct 17-gons or other n -gons for which FunctionExpand[Sin[ π /n]] contains only Plus , Times and Sqrt .)
In plants, cells typically expand—normally through intake of water—only for a limited period, after which the cellulose in their walls crystallizes to make them quite rigid.
For although leaves typically expand significantly after they come out, the basic features of their shapes almost never seem to change.
Some bones in effect just expand by adding material to their outer surface.
But ultimately the asymmetry between emission and absorption is a consequence of the fact that the universe is expanding, rather than contracting, with time.
Writing a Boolean function in DNF is the rough analog of applying Expand to a polynomial.
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] .