There are applications of the Johnson solids to characterizing patterns that occur in nature at the nanoscale. You wouldn't be able to see them with the naked eye, but they are very clearly there!

For example, we observed nested J27 "shells" in the structure of the Au_146(p-MBA)_57 nanoparticle [1]. In particular, take a look at the attached .mpg video to get a clear picture of the "shells" inside this particular nanoparticle [2]. We observed three nested (two complete, one outer incomplete, corrupted by the surface protectant p-MBA) J-27 shells.

Nanoparticles like this exhibit interesting surface plasmonic effects. For smaller particles, a long standing theory was that they behave as "super-atoms", with gold atoms taking the place of neutrons and protons, and metals in the protectant shell taking the place of electrons.

While I don't subscribe to that theory, this particle in particular occupies a sort of partial transition point between the regime in which it was previously hypothesized and the regime of bulk gold where it clearly does not hold.

Disclaimer: I am a first author on this paper and produced this visualization, as well as many of the figures shown in the paper. I think the video in particular is quite neat :).

[1] https://pubs.acs.org/doi/abs/10.1021/acs.jpclett.7b02621

[2] https://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.7b02621/s...