**Before 1984,** scientists thought they had a complete understanding of all the possible ways atoms and molecules can join together to make a solid. This knowledge, established nearly two centuries ago, is codified in a set of principles known as *the laws of crystallography*, which are essential for understanding and taking advantage of the physical properties of matter, whether it be making steel for a bridge, cleaving the facets of a diamond, or manipulating the electronic properties of silicon for use in integrated circuits. According to these laws, arrangements of atoms in solids are either completely random, as in the case of window glass, or crystalline, as is the case for sugar or table salt. In the case of crystalline materials, the atoms are organized in a symmetrical lattice like the square tiles in a simple bathroom tiling wherein tiles repeat in a periodic pattern (translational symmetry) that also exhibits a discrete rotational symmetry. The two cases are analogous to mosaics in which the tiles are put together either randomly or in an orderly, symmetrical tessellation. A key fact about regular tessellations, known since the ancient Egyptians, is that only certain symmetries can be obtained. The same rules apply to matter. Thus, periodic materials can only exhibit certain rotational symmetries: two-, three-, four-, and sixfold symmetry axes; five-, seven-, eight-, or higher-fold symmetry axes are strictly forbidden. Icosahedral symmetry, which includes six independent fivefold symmetry axes, is super-forbidden.

*Dr. Luca Bindi has been the head of the Division of Mineralogy of the Natural History Museum of the University of Florence for the past five years. He is currently an associate professor of mineralogy in the Department of Earth Sciences at the same university. His research activity is mainly devoted to understanding the complexity of mineral structures, which he accomplishes by combining mineralogy with the most advanced fields of crystallography.*

*Dr. Paul J. Steinhardt is the Albert Einstein Professor in Science and director of the Princeton Center for Theoretical Science at Princeton University. With Dov Levine, he introduced the concept of quasicrystals in 1984; he also developed some of the basic theory describing their physical properties and originated the search for natural quasicrystals. He is also recognized for his contributions to particle physics and cosmology.*