Diffraction pattern of a crystal of a blue-green algae

Falling in Love with Mathematics and Crystallography

An op-ed article by Manil Suri (Professor of Mathematics, University of Maryland) in the New York Times on Monday, September 16
(http://www.nytimes.com/2013/09/16/opinion/how-to-fall-in-love-with-math.html) eloquently expresses the dismay of the author when he faces the shallow perception that the general public have for mathematics. The common expression “Do the math” typically implying the arithmetic of operating with numbers, as if balancing a checkbook or adding the family expenses would be the ultimate objective of mathematics.

He argues and illustrates that indeed ‘mathematics’ is infinitely more than just that. Through the ages, the realm of ‘mathematics’ has expanded the human mind and spirit from the Pythagorean’s perception of the world, to the use of calculus to accurately describe motion and change, and on to the non-Euclidian geometries to define and explain the space we live in. He argues that the richness of mathematics is comparable to the fullness of Art and Music with their similar unremitting searches although all in different domains.

The article immediately brought to my mind the thought: Isn’t that also true about other areas of science? What does Biochemistry suggest? Take your vitamins? In the worst possible scenario, an esoteric branch of science might just elicit silence or even an inconsequential response. How about ‘crystallography’, followed by ‘healing power of crystals’ or ‘crystals can predict the future’?

Unfortunately, many of the ill responses to mathematics discussed by the author of the article apply to many of the broader branches of science and quite often to the narrower specialties. And yet, in very different ways, all expand our knowledge of who we are and what surrounds us. Knowledge that later translates into conveniences or comforts that make our lives more enjoyable.

Going back to crystallography. Crystallography is an astonishing branch of science. Crystals have been studied from antiquity and fascinated human beings since they were discovered (see previous posting). Their external forms probably inspired the geometry of the Platonic solids and the mathematical description of its internal symmetry is related to ‘group theory’ and to artistic tessellations of two and three-dimensional spaces. Artists such as M.C. Escher (1898-1972) were inspired by their internal structure.

Fig.1. Diffraction pattern of a crystal of a blue-green algae protein C-Phycocyanin.

Fig.1. Diffraction pattern of a crystal of a blue-green algae protein named C-Phycocyanin.
A prismatic crystal of the protein is exposed to the X-rays oriented so that the hexagonal axis is along the direction of the X-rays. The symmetry of the diffraction pattern is a reflection of the symmetry of the crystal. A three dimensional data set of photographs like this permits the mapping of the atoms forming the protein in three-dimensions. Copyright C.A-Z.

Its true internal symmetry, however, was veiled from the human eyes and could only be revealed by X-rays. Those ‘naughty Roentgen rays’ were discovered by Wilhelm Conrad Roentgen (1845-1923) in 1895 and unsealed new fields in the history of science. Its mysterious nature and character unfathomable to Roentgen’s experimental probing: were they waves or a stream of particles? Eventually, they only yielded their innermost secret to the crystals themselves and the crystals in turn, confessed that they were arrays of atoms. Now the two were inseparable: crystals and X-rays.

The pattern of spots (i.e., ‘diffraction pattern’, Fig.1) resulting from the interaction of the X-rays with the atomic structure of matter inside the crystals has permitted the unveiling of the atomic universes that surround us, ever since the Braggs (William Henry and William Lawrence, father and son) revealed the structure of the commons salt (Sodium Chloride) in 1914.

Since then, the mathematical expansions of functions proposed by J.B. Joseph Fourier (1768-1830) in his 1822 memoir entitled the ‘The analytic theory of Heat’, have been the cornerstone of the mathematical underpinning of crystallography, permitting the discovery of the ever more complex atomic structures within the world that we inhabit. From the most common minerals to the double-helical structure of DNA that relates us to the ladder of organisms living and extinct. Isn’t this mind-blowing? Crystallography, is a division of science that combines mathematics, physics, instrumentation and chemistry to provide us with an unparalleled view of who we are and they world that we inhabit.

Without having to master the mathematics behind crystallography, any curious human mind can understand and appreciate all these wonders. Not even the ‘The Math Gene’ is necessary. I am not certain that the assertion by Stanford mathematician Keith Devlin that human beings are wired for mathematics is correct. Certainly, not all human beings are ‘wired’ or at least some are much more than others. However, I am certain that all human beings share the ‘curiosity gene’ at the most fundamental and ancestral level; you only have to look at the human child. Otherwise, we would not have arrived to where we are now. As scientists, each one of us in our own arcane specialties, need to learn and practice the avocation of transmitting this sense of wonder to our fellow humans so that our specialties are, if not fully understood, at least appreciated in the context of other human activities. This will make all of us richer and even more human.