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We consider hard-disc mixtures with disc sizes within ratio $\sqrt{2}-1$, that is, the small disc exactly fits in the hole between four large discs. For each prescribed stoichiometry of large and small discs, the densest packings are…
Motivated by modern applications like image processing and wireless sensor networks, we consider a variation of the famous Kepler Conjecture. Given any infinite set of unit balls covering the whole space, we want to know the optimal (lim…
It is well-known that every isosceles tetrahedron (disphenoid) admits infinitely many simple closed geodesics on its surface. They can be naturally enumerated by pairs of co-prime integers $n > m > 1$ with two additional cases $(1,0)$ and…
Delone sets are discrete point sets $X$ in $\mathbb{R}^d$ characterized by parameters $(r,R)$, where (usually) $2r$ is the smallest inter-point distance of $X$, and $R$ is the radius of a largest ``empty ball" that can be inserted into the…
The exploration of the densest sphere packings is a fundamental problem in mathematics and a wide variety of sciences including materials science. We present our exhaustive computational exploration of the densest ternary sphere packings…
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…
In view of solving problems of geometric realizability of polyhedra with given geometric constraints, we describe the space of geometric realizations of a simply-connected triangulated euclidean polyhedron in $\mathbb{R}^3$ up to similarity…
The global minima of clusters bound by a Dzugutov potential form non-compact polytetrahedral clusters mainly composed of interpenetrating and face-sharing 13-atom icosahedra. As the size increases, these icosahedral units first form linear…
We revisit the densest binary sphere packings (DBSP) under the periodic boundary conditions and present an updated phase diagram, including newly found 12 putative densest structures over the $x - \alpha$ plane, where $x$ is the relative…
We stress the importance of stochastic geometry as a branch of mathematical statistics particularly suited to model and investigate nontrivial spatial patterns. One of its key concepts, Voronoi tessellations, represents a versatile and…
Deformation in the lithosphere-asthenosphere system can be accommodated by faulting and plastic flow. However, incorporating structural data in models of distributed deformation still represents a challenge. Here, I present solutions for…
The phase diagram of model anisotropic particles with four attractive patches in a tetrahedral arrangement has been computed at two different values for the range of the potential, with the aim of investigating the conditions under which a…
Minimal energy shapes of closed, elastic shells with twelve pentagonal disclinations introduced in otherwise hexagonally coordinated crystalline lattice are studied. The geometry and the total energy of shells are studied as a function of…
Every surface that is intrinsically polyhedral can be represented by a portalgon: a collection of polygons in the Euclidean plane with some pairs of equally long edges abstractly identified. While this representation is arguably simpler…
A disordered (glassy) state has been searched in solid 3He deformed in the course of experiment employing precise measurements of pressure. The analysis of the temperature dependence of the crystal pressure measured at a constant volume…
There has long been a discrepancy between the size distributions of Ar$_n^+$ clusters measured by different groups regarding whether or not magic numbers appear at sizes corresponding to the closure of icosahedral (sub-)shells. We show that…
Magic numbers in finite particle systems correspond to specific system sizes that allow configurations with low free energy, often exhibiting closed surface shells to maximize the number of nearest neighbors. Since their discovery in atomic…
Quasicrystals are fascinating structures, characterized by strong positional order but lacking the periodicity of a crystal. In colloidal systems, quasicrystals are typically predicted for particles with complex or highly specific…
It is well known that the point group of the root lattice D_6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H_3, its roots and weights are determined in terms of those of D_6. Platonic and…
The mechanical properties of a granular sample depend frequently on the way the packing was prepared. However, is not well understood which properties of the packing store this information. Here we present an X-ray tomography study of three…