A recent issue of Science announced the discovery of natural quasicrystals, solids whose atomic arrangements have symmetries that are forbidden for periodic crystals. In this case, Bindi et al. (Vol. 324. no. 5932, pp. 1306 - 1309) describe a naturally occurring icosahedral quasicrystal that includes "six distinct fivefold symmetry axes." This is big news! The picture here is their Figure 4A. I like it because it clearly shows 10-fold symmetry very close to the center (which means that 10-fold symmetry extends fairly far in the actual quasicrystal).
Explorations of five-fold symmetry on the plane go back to Persian artists of the twelfth century. I became aware of this when I read Critchow ("Islamic Patterns" 1976), which presents "demonstrations" of "important shapes that enable the five-fold symmetry to harmonize with the necessary repeats of a two-dimensional surface." The matter of Islamic quasicrystalline tilings was taken up in detail by Lu and Steinhardt (who are also authors on this new paper) in 2007 ("Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture"). However, it was only in the 1970s that Penrose tilings, a periodic but infinite tilings with fivefold symmetry, were described.
This is a Penrose tiling showing five-fold symmetry.
More recently, the production of quasicrystals in the laboratory has become something of a cottage industry, but only now have naturally occurring quasicrystals been found.
Most of what I might say is said better in article, so I encourage interested readers to read the original.