Although n-fold rotations for n differing from 2, 3, 4, and 6 are forbidden in the strict sense of perfect crystallographic symmetry (these constitute the so-called crystallographic restriction Eric Weisstein's World of Math), there are exotic materials called quasicrystals that display these symmetries. In 1984, D. Shechtman discovered a class of aluminum alloys whose X-ray diffraction patterns display 5-fold symmetry. Since this was long known to be crystallographically forbidden, this came as quite a shock initially, until it became apparent that materials exist which are not exact crystals but very nearly so which display symmetries forbidden by actual crystals.
These materials became known as quasicrystals. The mathematical structure behind quasicrystals had actually been discovered in 1974 by Roger Penrose when he devised Penrose tiles, Eric Weisstein's World of Math which create an aperiodic tiling that displays fivefold symmetry. Many known quasicrystals can be thought of as three-dimensional analogs of the Penrose tiling.
References
Axel, F. Beyond Quasicrystals. New York: Springer-Verlag, 1995.
DiVincenzo, D. P. and Steinhardt, P. J. (Eds.). Quasicrystals: The State of the Art. Singapore: World Scientific, 1991.
Hargittai, I. (Ed.). Quasicrystals, Networks, and Molecules of Fivefold Symmetry. New York: VCH Publishers, 1990.
Janot, C. Quasicrystals: A Primer. 1997.
Kuo, K. H. and Takeuchi, S. (Eds.). Quasicrystals and Imperfectly Ordered Crystals. 1994.
Miekisz, J. Quasicrystals: Microscopic Models of Nonperiodic Structures. Leuven, Belgium: Leuven University Press, 1993.
Patera, J. Quasicrystals and Discrete Geometry. Providence, RI: Amer. Math. Soc., 1998.
Senechal, M. Quasicrystals and Geometry. New York: Cambridge University Press, 1995.
Steinhardt, P. J. and Ostlund, S. (Eds.). The Physics of Quasicrystals. Singapore: World Scientific, 1987.
Weisstein, E. W. "Books about Quasicrystals." http://www.ericweisstein.com/encyclopedias/books/Quasicrystals.html.
