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Quoting from Wolfram Mathworld: " It is the shape used in the construction of soccer balls, and it was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in the Fat Man atomic bomb (Rhodes 1996, p. 195)." but further insights are not provided. (http://mathworld.wolfram.com/TruncatedIcosahedron.html)

Why would the Manhattan project scientists choose this geometry for designing the implosive lenses of the Fat Man atomic bomb? Assuming the explosive charges were placed on the faces of the icosahedron, is this the optimal geometry for focusing the shock waves on a single point (presumably a spherical plutonium core), or was this mainly due to practical/engineering considerations?

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I won't make this a answer because I am simply guessing, but presumably they first calculated the maximum variation of pressure they could tolerate and then found the easiest shape to both calculate and manufacture that would fit within those bounds. –  dmckee Jul 17 '11 at 23:36

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Without some additional context, there is no optimal geometry for focusing shock waves. There are only those geometries that produce a spherical shock and those geometries that don't.

I wasn't there, but some considerations that may have influenced the design include:

  1. They wanted a spherical shock, so it is natural to start with an approximately spherical initial geometry, and tweak it into shape. This consideration favors more lenses roughly evenly distributed on the sphere. This is also a good strategy for computational reasons, since the resources necessary to model a propagating shock with low accumulated error at the time were quite expensive. Modeling a bigger lens may lead to greater error. According to Wikipedia, more modern weapons have far fewer detonators and more exotic lens geometries. Presumably, this is because the computational resources became less expensive, and reliability of mass-produced detonators became more of a concern than reliability of mass-produced lenses.
  2. If you have a distribution of lenses, you need to work out the propagation of the shock not only inside a single lens, but also the interface between different lenses. This consideration favors having fewer lenses, or at least fewer lens geometries. In particular, highly symmetric configurations mean fewer computations.

The truncated icosahedron has lots of roughly evenly spaced faces, and is highly symmetric: the symmetry group is transitive on the hexagons, the pentagons, the p-h edges, the h-h edges and the corners. This means that you only need to calculate the shock wave geometry for a pentagon, a hexagon, two kinds of edges between them, and a three-way intersection. One could instead consider dodecahedra or icosahedra (which have more transitive symmetry), but I guess that would make the lenses either substantially larger or have sharp angles that make them more difficult to handle.

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Yes, it was chosen from practical point of view. Given the fact that surface separating 2 types of explosives is curved, it gives you perfect spherical implosion wave.

Main advantage of icosahedron is that you need only one set of forms to make parts of tiles.

Square/tetrahedron for example also needs 1 set of forms, but would require higher curvature of surface separating 2 explosives types which is harder to do with same precision (micron-level CNC would really help back then :-) ), and also would require higher bomb diameter to carry same amount of explosives.

As technology advanced, it became possible to reduce number of tiles while maintaining implosion precision, the last I've heard of was just 2 ignitors/tiles in soviet warheads (going below 2 is not safe, as we still want to survive single accidental ignition).

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