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2 settembre, 2007

Dall’Articolo dell’American Scientist, consigliatomi sul forum di matematicamente, un passaggio da tenere in considerazione. Si parla del problema di Kelvin, cioè quello di capire qual è il modo migliore di dividere uno spazio tridimensionale in celle di pari volume, volendo minimizzare l’area superficiale delle pareti delle celle.

It is easy to produce a wealth of shapes that do fill space, by building what are known as Voronoi cells. To construct Voronoi cells I must start with an infinite collection of tiny bubbles located at different points in space, then let the bubbles expand until they bump into each other. If the centers of the bubbles are chosen with a little care, the cells produced in this way will be finite polyhedra that fill all of space; if the centers are chosen to form a repeating pattern, the Voronoi cells will also form a repeating pattern. I can do the same thing in two dimensions. (See Figure 10.) In the plane, the Voronoi-cell construction sets up a tight correspondence between the Kepler and Kelvin problems. If I place my “bubbles” at the centers of the pennies in the best penny packing and expand them into Voronoi cells, I get a honeycomb, which is the solution to the Kelvin question. So in the search for the solution to the three-dimensional Kelvin question, it makes sense to start with the Voronoi cells that correspond to the best sphere packing, Kepler’s pyramid arrangement. If I allow the spheres of the pyramid packing to swell until they bump into their neighbors, I get 12-sided figures called rhombic dodecahedra. These form a partition of space with very low surface area, but it is not quite as low as that of a few other configurations, including Kelvin’s truncated octahedra. On the other hand, if I construct Voronoi cells by placing my bubbles not at the centers of the spheres but instead as far away from the spheres as possible, I get truncated octahedra. Thus, in both two and three dimensions, the Kepler packing produces excellent candidates for the answer to the Kelvin question.

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