paul_j_steinhardt's picture
Albert Einstein Professor in Science, Departments of Physics and Astrophysical Sciences, Princeton University; Coauthor, Endless Universe


My first exposure to true elegance in science was through a short semi-popular book entitled SYMMETRY written by the renowned mathematician Hermann Weyl. I discovered the book in the fourth grade and have returned to reread passages every few years. The book begins with the intuitive aesthetic notion of symmetry for the general reader, drawing interesting examples from art, architecture, biological forms, and ornamental design. By the fourth and final chapter, though, Weyl turns from vagary to precise science as he introduces elements of group theory, the mathematics that transforms symmetry into a powerful tool.

To demonstrate its power, Weyl spends his final chapter outlining how group theory can be used to explain the shapes of crystals. Crystals have fascinated humans throughout history because of the beautiful faceted shapes they form. Most rocks contain an amalgam of different minerals, each of which is crystalline, but which have grown together or crunched together or weathered to the point that facets are unobservable. Occasionally, though, the same minerals form individual large faceted crystals. That is when we find them most aesthetically appealing. "Aluminum oxide" may not sound like something of value, but add a little chromium, give nature sufficient time and you have a ruby worthy of kings.

If you have not done so recently, I urge you to visit the mineral collection in a museum to observe the remarkable variety and beauty of crystal forms. You will discover for yourself a basic mineralogical fact that the crystal facets found in nature meet at only certain angles corresponding to one of a small set of symmetries. But why does matter take some shapes and not others? What scientific information do the shapes convey? Weyl explains how these questions can be answered seemingly unrelated abstract mathematics aimed at answering a different question: namely, what shapes can be used to tessellate a plane or fill space if the shapes are identical, meet edge-to-edge and leave no spaces? Squares, rectangles, triangles, parallelograms and hexagons can do the job. Perhaps you imagine that many other polygons would work as well; however, try and you will discover there are no more possibilities. Pentagons, heptagons, octagons and all other regular polygons cannot fit together without leaving spaces. Weyl's little book describes the mathematics that allows a full classification of possibilities including as distinct patterns with decorated tiles and including reflections, glides and screw axes,. The final tally is only 17 distinct possibilities in two dimensions (the so-called wallpaper patterns) and 230 distinct symmetry possibilities in three dimensions.

The stunning fact about the mathematicians' list was that it precisely matched the list observed for crystals shapes found in nature. The inference is that crystalline matter is like a tessellation made of indivisible, identical building blocks that repeat to make the entire solid. Of course, we know today that these building blocks are clusters of atoms or molecules. However, bear in mind that the connection between the mathematics and real crystals was made in 19th century when the atomic theory was still in doubt. It is amazing that an abstract study of tiles and building blocks can lead to a keen insight about the fundamental constituents of matter and a classification of all possible arrangements of them. It is a classic example of physicist Eugene Wigner referred to as the "unreasonable effectiveness of mathematics in the natural sciences." The story does not end there. With the development of quantum mechanics, group theory and symmetry principles have been used to predict the electronic, magnetic, elastic and other physical properties of solids. Emulating this triumph, physicists have successfully used symmetry principles to explain the fundamental constituents of nuclei and elementary particles, as well as the forces through which they interact.

As a young student first reading Weyl's book, crystallography seemed like the "ideal" of what one should be aiming for in science: elegant mathematics that provides a complete understanding of all physical possibilities. Ironically, many years later, I played a role in showing that my "ideal" was seriously flawed. In 1984, Dan Shechtman, Ilan Blech, Denis Gratias and John Cahn reported the discovery of a puzzling manmade alloy of aluminumand manganese with icosahedral symmetry. Icosahedral symmetry, with its six five-fold symmetry axes, is the most famous forbidden crystal symmetry. As luck would have it, Dov Levine (Technion) and I had been developing a hypothetical idea of a new form of solid that we dubbed quasicrystals, short for quasiperiodic crystals. (A quasiperiodic atomic arrangement means the atomic positions can be described by a sum of oscillatory functions whose frequencies have an irrational ratio.) We were inspired by a two-dimensional tiling invented by Sir Roger Penrose known as the Penrose tiling, comprised of two tiles arranged in a five-fold symmetric pattern. We showed that quasicrystals could exist in three dimensions and were not subject to the rules of crystallography. In fact, they could have any of the symmetries forbidden to crystals. Furthermore, we showed that the diffraction patterns predicted for icosahedral quasicrystals matched the Shechtman et al. observations. Since 1984, quasicrystals with other forbidden symmetries have been synthesized in the laboratory. The 2011 Nobel Prize in Chemistry was awarded to Dan Shechtman for his experimental breakthrough that changed our thinking about possible forms of matter. More recently, colleagues and I have found evidence that quasicrystals may have been among the first minerals to have formed in the solar system.

The crystallography I first encountered in Weyl's book, thought to be complete and immutable, turned out to be woefully incomplete, missing literally an uncountable number of possible symmetries for matter. Perhaps there is a lesson to be learned: While elegance and simplicity are often useful criteria for judging theories, they can sometimes mislead us into thinking we are right, when we are actually infinitely wrong.