Related papers: New Views of Crystal Symmetry
This paper shows how beginning with Justus Grassmann's work, Hermann Grassmann was influenced in his mathematical thinking by crystallography. H. Grassmann's Ausdehnungslehre in turn had a decisive influence on W.K. Clifford in the genesis…
The notion of a geometric crystal was introduced by A.Berenstein and D.Kazhdan, motivated by the needs of representation theory of p-adic groups. It was shown by A.Braverman, A.Berenstein, and D.Kazhdan that some particular geometric…
"Symmetry" was one of the most important methodological themes in 20th-century physics and is probably going to play no lesser role in physics of the 21st century. As used today, there are a variety of interpretations of this term, which…
In crystallography, a structure is typically represented by the arrangement of atoms in the direct space. Furthermore, space group symmetry and Wyckoff site notations are applied to characterize crystal structures with only a few variables.…
Most of crystalline materials exhibit a hysteresis on their deformation curve when mechanically loaded in alternating directions. This Bauschinger effect is the signature of mechanisms existing at the atomic scale and controlling the…
The equilibrium shape of crystals is a fundamental property of both aesthetic appeal and practical import. It is also a visible macro-manifestation of the underlying atomic-scale forces and chemical makeup, most conspicuous in…
In this paper we completely settle the Almgren problem in $\mathbb R^3$ under some generic conditions on the potential and tension functions. The problem, among other things, appears in classical thermodynamics when one is to understand if…
Crystal skeletons were introduced by Maas-Gari\'epy in 2023 by contracting quasi-crystal components in a crystal graph. On the representation theoretic level, crystal skeletons model the expansion of Schur functions into Gessel's…
Crystal structures can be viewed as assemblies of space-filling polyhedra, which play a critical role in determining material properties such as ionic conductivity and dielectric constant. However, most conventional crystal structure…
Let $G$ be a connected reductive group over $\CC$ and let $G^{\vee}$ be the Langlands dual group. Crystals for $G^{\vee}$ were introduced by Kashiwara as certain ``combinatorial skeletons'' of finite-dimensional representations of…
Herein, fundamentals of topology and symmetry breaking are used to understand crystallization and geometrical frustration in topologically close-packed structures. This frames solidification from a new perspective that is unique from…
Geometrical constructions, such as the tangent construction on the molar free energy for determining whether a particular composition of a solution, is stable, are related to similar tangent constructions on the orientation-dependent…
Crystal structure design is important for the discovery of new highly functional materials because crystal structure strongly influences material properties. Crystal structures are composed of space-filling polyhedra, which affect material…
This contribution discusses the geometry of $k$D crystal cells given by $(k+1)$ points in a projective space $\R^{n+1}$. We show how the concepts of barycentric and fractional (crystallographic) coordinates, reciprocal vectors and dual…
Generating novel crystalline materials has the potential to lead to advancements in fields such as electronics, energy storage, and catalysis. The defining characteristic of crystals is their symmetry, which plays a central role in…
The present work revisits the classical Wulff problem restricted to crystalline integrands, a class of surface energies that gives rise to finitely faceted crystals. The general proof of the Wulff theorem was given by J.E. Taylor (1978) by…
We measure the local contact forces at both the top and bottom boundaries of three-dimensional face-centered-cubic and hexagonal-close-packed granular crystals in response to an external force applied to a small area at the top surface.…
This paper was motivated by the articles "Same or different - that is the question" in CrystEngComm (July 2020) and "Change to the definition of a crystal" in the IUCr newsletter (June 2021). Experimental approaches to crystal comparisons…
By Tits' deformation argument, a generic Iwahori--Hecke algebra $H$ associated to a finite Coxeter group $W$ is abstractly isomorphic to the group algebra of $W$. Lusztig has shown how one can construct an explicit isomorphism, provided…
Symmetries are so involved in crystallographic detail that it is indispensable when crystal cell or lattice is mentioned. For nanocrystals, however, many are 'systemic symmetries', meaning that the whole structures of nanocrystal particles…