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Sphere packings as stem cells

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Abstract

For Alan Mackay, on his 90th birthday, we recast his notion of crystalloids and crystallites as finite nano-sized sphere packings and discuss current mathematical/computational research on their geometric structures. We suggest that icosahedral clusters are (metaphoric) stem cells for condensed matter.

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Notes

  1. Note that Frank did not define rigidity: he outlawed spheres breaking contact with another one, not with one another.

  2. They also discuss clusters with spheres of different radii, but though these may be physically meaningful, we will not consider them here.

  3. Note their similarity to the Russian mathematician B. N. Delone's axioms for (r, R) systems (now called Delone, or Delaunay, sets).

  4. The Frank-Kasper polyhedra are those whose faces are all triangles and all vertices 5- or 6-valent. The number of 5-valent vertices must, by Euler’s formula, be 12; the number of 6-valent vertices may be 0, 2, 3, or 4. The first of these is the regular icosahedron.

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  1. Department of Mathematics and Statistics, Smith College, Northampton, MA, 01063, USA

    Marjorie Senechal

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  1. Marjorie Senechal
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Correspondence to Marjorie Senechal.

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Senechal, M. Sphere packings as stem cells. Struct Chem 28, 27–31 (2017). https://doi.org/10.1007/s11224-016-0821-y

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