As I see it, the main contribution of C S Smith is in (a) drawing attention to the possible applications of topological ideas to metallurgical problems; (b) setting the grain growth problem as belonging to the same class as cell structures in biological tissues, soap froths, foams, and problems on space filling; and, (c) using models of soap froths as near exact geometric models of grain growth in metals.
I just want to draw your attention to the reply that Smith gave to the discussions section, in which he expressed a hope and a word of caution:
The discussion by Dr. von Neumann is much appreciated, and his conclusions are as remarkable for their simplicity as they are nonobvious on first consideration of the problem. It is greatly to be hoped that he, or some other mathematician, will be able to deduce similar relations in three dimensions and can combine these with the topological requirements to give the equilibrium distribution of bubbles toward which a froth must tend.
… Metals are not soap bubbles, even though soap bubbles are so useful in illustrating the basic principles, as I hope I have demonstrated in the paper.
Coarsening and growth are not the only problems of interest when it comes to soap bubbles; in the latest issue of This week’s finds in Mathematical Physics, John Baez writes about a closely related problem, namely, the problem of chopping 3D space into equal volume cells with minimal surface area; Baez begins with the most recent ‘Water Cube’ building at Beijing Olympics 2008, and goes on to collect lots of references and web resources on The Kepler conjecture, The Honeycomb conjecture, and the Weaire-Phelan structure; along the way, he also has some interesting historical observations to make:
In his 1887 paper on this subject, Kelvin wrote:
No shading could show satisfactorily the delicate curvature of the hexagonal faces, though it may be fairly well seen on the solid model made as described in Section 12. But it is shown beautifully, and illustrated in great perfection, by making a skeleton model of 36 wire arcs for the 36 edges of the complete figure, and dipping it in soap solution to fill the faces with film, which is easily done for all the faces but one. The curvature of the hexagonal film on the two sides of the plane of its six long diagonals is beautifully shown by reflected light.
I think this is a nice passage. We may remember Kelvin for his profound work on electromagnetism and thermodynamics – or his 1900 lecture on two “dark clouds” hanging over physics: the Michelson-Morley experiment (which foreshadowed special relativity) and black body radiation (which foreshadowed quantum mechanics). We may not imagine him playing around with soap bubbles! But it shows that good science stems from curiosity, and curiosity knows no bounds.You can read Kelvin’s paper here:
10) Lord Kelvin, On the division of space with minimum partitional area, Phil. Mag. 24 (1887), 503. Also available at http://zapatopi.net/kelvin/papers/on_the_division_of_space.html
His lively mind is evident from the selection of papers on this site. For example: “On Vortex Atoms”, where he unsuccessfully tried to build atoms out of knotted electromagnetic field lines, and wound up giving birth to knot theory. Some others I hadn’t heard of: “On the origin of life”, “The sorting demon of Maxwell”, and “Windmills must be the future source of power”.
A piece worth bookmarking and returning to often — for repeated readings and pointers to resources. Take a look!
PS: By the way, the C S Smith special issue of Resonance that I referred to in my grain growth post is also a nice place to read about some of this stuff — for example, it contains a piece by Denis Weaire himself. Have fun!