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We have discovered a new family of three-dimensional crystal sphere packings that are strictly jammed (i.e., mechanically stable) and yet possess an anomalously low density. This family constitutes an uncountably infinite number of crystal…
We give a combinatorial characterization of minimally rigid planar frameworks with orientation-preserving crystallographic symmetry, under the constraint of forced symmetry. The main theorems are proved by extending the methods of the first…
We present a scheme to explicitly construct and classify general topological states jointly protected by an onsite symmetry group and a spatial symmetry group. We show that all these symmetry protected topological states can be…
We give a new combinatorial proof for the number of convex polyominoes whose minimum enclosing rectangle has given dimensions. We also count the subclass of these polyominoes that contain the lower left corner of the enclosing rectangle…
Interaction between dipolar forces, such as permanent magnets, generally leads to the formation of one-dimensional chains and rings. We investigated whether it was possible to let dipoles self-assemble into three-dimensional structures by…
If all tiles in a tiling are congruent, the tiling is called monohedral. Tiling by convex polygons is called edge-to-edge if any two convex polygons are either disjoint or share one vertex or one entire edge in common. In this paper, we…
The kinetics of phase transition processes often governs the resulting material microstruc-ture. Using optical microscopy, we here investigate the formation and stabilization of a po-rous crystalline microstructure forming in low-salt…
Higher-order topological insulators are established as topological crystalline insulators protected by crystalline symmetries. One celebrated example is the second-order topological insulator in three dimensions that hosts chiral hinge…
Mathematical crystal chemistry views crystal structures as the optimal solutions of mathematical optimization problem formalizing inorganic structural chemistry. This paper introduces the minimum and maximum atomic radii depending on the…
Polymorphism contributes to the diversity of nature, so that even materials having identical chemical compositions exhibit variations in properties because of different lattice symmetries. Thus, if stacked together into multilayers,…
In this paper, we consider polyhedral realizations for crystal bases $B(\lambda)$ of irreducible integrable highest weight modules of a quantized enveloping algebra $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a classical affine Lie algebra…
Thin Lie algebras are Lie algebras L, graded over the positive integers, with all homogeneous components of dimension at most two, and satisfying a more stringent but natural narrowness condition modeled on an analogous one for pro-p…
The paper studies the minimum polynomial degrees of $p$-elements in cross-characteristic representations of simple groups of exceptional Lie type whose BN-pair rank is at most 2. Specifically, we prove that the degree in question equals the…
We study the problems of covering or partitioning a polygon $P$ (possibly with holes) using a minimum number of small pieces, where a small piece is a connected sub-polygon contained in an axis-aligned unit square. For covering, we seek to…
In this paper it is proved that there exist periodic monohedral tilings and finite seeds of colored tiles, which force non-periodic coloring of the whole plane
Novel ordered intermetallic compounds have stimulated much interest. Ru-Al alloys are a prominent class of high-temperature structural materials, but the experimentally reported crystal structure of the intermetallic Ru2Al5 phase remains…
We give a simple proof of T. Stehling's result, that in any normal tiling of the plane with convex polygons with number of sides not less than six, all tiles except the finite number are hexagons.
We study tilings of regions in the square lattice with L-shaped trominoes. Deciding the existence of a tiling with L-trominoes for an arbitrary region in general is NP-complete, nonetheless, we identify restrictions to the problem where it…
Colloidal crystal structures with complexity and diversity rivaling atomic and molecular crystals have been predicted and obtained for hard particles by entropy maximization. However, so far homochiral colloidal crystals, which are…
Reliable and robust methods of predicting the crystal structure of a compound, based only on its chemical composition, is crucial to the study of materials and their applications. Despite considerable ongoing research efforts, crystal…