Marjorie Senechal on B N Delone
In Boris Nikolaevich Delone I found my real teacher. I never met him (though I later learned that I could have). But on the basis of his papers and through my friendship with Ravil V Galiulin (1940-2010), Nikolai P Dolbilin, and Mikhail I Shtogrin, I became and remain his disciple, trying always to emulate his clear and simple approach to crystallographic problems and his informal, lucid writing style. Delone's work has been the starting point for all of mine since then. I would like to describe here, briefly, what Delone had done that captivated me so thoroughly. In 1934, together with the crystallographer Nikolai N Padurov and the mathematician Aleksandr Danilovic Alexandrov, Delone published Mathematical Foundations of the Structural Analysis of Crystals (in Russian), in which they developed a new approach to crystal symmetry. Evgraf Stepanovich Fedorov (1853-1919) and other great mathematically-inclined crystallographers of the past had asked - and answered - such questions as: given a repeating pattern, what is its symmetry, or crystallographic, group? How many different crystallographic groups are there? Delone and his colleagues didn't take the pattern as given. They began with more general point sets, sets which could model the distribution of atoms in gases. These sets, which they called (r, R) systems (today they are called Delone sets), satisfy two axioms: first, no two points can be closer than the fixed distance r, and second, every sphere of radius R has at least one point of the set in or on it. From these simple hardcore and homogeneity axioms they drew a surprising amount of information about the geometry of the distribution of these points in space. Next, step-by-step, they constrained the way each point is surrounded by the others, until the point set became crystalline, or regular. Not only is this approach simple and elegant, it has turned out to be useful. Quasicrystals show us that sharp diffraction patterns are not the sole province of lattice structures. Evidently "order/disorder" is a spectrum, not a dichotomy. Delone sets are a tool for exploring that spectrum.
JOC/EFR November 2017
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