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How can the packing density of a set of congruent ellipsoids be calculated? I'm dealing with prolate spheroids so technically I do not need the general answer for ellipsoids, but my abstract mind loves more general answers. If calculating an exact value is too difficult, an estimate which is accurate enough to allow me to compare the relative packing densities of different sizes of spheroids is sufficient.

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    $\begingroup$ Sounds difficult: mathworld.wolfram.com/EllipsoidPacking.html $\endgroup$
    – leonbloy
    Oct 17, 2011 at 20:51
  • $\begingroup$ Would the density be bounded by that of spheres and that of cylinders (which should be equivalent to circles): (74.0%, 90.7%)? $\endgroup$
    – Kazark
    Oct 18, 2011 at 1:23
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    $\begingroup$ Since any ellipsoid is the image of a sphere under an affine transformation, the density is bounded below by the density of sphere packings (take the best sphere packing and transform it such that all the spheres are transformed into copies of the ellipsoid in question). $\endgroup$ Apr 29, 2018 at 19:39

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The densest known packing of ellipsoids of any kind, including prolate spheroids, is given in this paper. From the abstract:

A remarkable maximum density of 0.7707 is achieved for maximal aspect ratios larger than 3, when each ellipsoid has 14 touching neighbors.

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