On symmetries of higher-order elastic constants

This commentary summarizes methods and results of symbolic computations in the recent article of Telyatnik [Acta Cryst. (2024), A80, 394–404]. Brief background on second- and higher-order elastic constants, symmetry constraints, and classic works preceding the contributions of Telyatnik is included for context.

A milestone for the solution to the lattice sphere covering problem in dimension n = 6

The complete classification of (primitive, generic) parallelohedra in a given dimension is a challenging computational task and is currently known only up to dimension n = 5. The paper by Dutour Sikirić & van Woerden [Acta Cryst. (2025), A81, https://doi.org/10.1107/S2053273324010143] presents a milestone for this task.

Complete classification of six-dimensional iso-edge domains

We enumerate the 55083357 iso-edge subdivisions of six-dimensional translational lattices. We report on the use of the method of canonical forms that allows us to apply hashing techniques used in modern databases.

The general equation of δ direct methods and the novel SMAR algorithm residuals using the absolute value of ρ and the zero conversion of negative ripples

The general equation of the δ direct methods is established and used to define one of the two residuals of the SMAR phasing algorithm. These residuals utilize the absolute value of ρ and the zero conversion of slightly negative Fourier ripples (≥50% of the cell volume) and lead to overdetermination for atomic resolution diffraction data. Due to its architecture, the SMAR algorithm is particularly well suited for Deep Learning. Alternatively, when solved for ρ, the general equation provides a simple derivation of the already known δ_M tangent formula, the core of the δ_M recycling algorithm.

Small-angle rigid-unit modes requiring linear strain compensation

The linear-algebraic RUM-search method is extended to include modes that require linear compensation from lattice strains.

A computationally efficient quasi-harmonic study of ice polymorphs using the FFLUX force field

The next-generation machine-learning force field FFLUX is applied to ice polymorphs Ih, II and XV. Under the quasi-harmonic approximation, Gibbs free energies are calculated using FFLUX at a significantly reduced computational cost compared with the commonly used density functional theory methods. However, the parametrized non-bonded potentials negatively affect the accuracy of the model, leading to large errors in the free energies calculated.

Periodic graphs with coincident edges: folding-ladder and related graphs

We explore a special class of periodic graphs, ladder graphs, whose edges can coincide when embedded vertices are moved. Many of these exhibit additional non-crystallographic graph symmetries.

Simulation of reciprocal-space mapping using a new analytical solution of kinematical X-ray diffraction in a crystal

New analytical solutions describing the diffraction of a restricted X-ray beam from a thin crystal are obtained in the kinematical approximation. X-ray reciprocal-space mapping for a silicon crystal was simulated using analytical solutions, as well as calculated using numerical methods based on 2D recurrence relations and the Takagi–Taupin equations.

Growth functions of periodic space tessellations

For any periodic tilings of the plane by congruent polygons, growth functions are derived that count the numbers of vertices, edges and faces as the coverage expands. The functions are computed as polynomial formulas in a Python program and analyzed graphically using orphic diagrams.

Visual Group Theory: A Computer-Oriented Geometric Introduction