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Periodic space crystals are well established and widely used in physical sciences. Time crystals have been increasingly explored more recently, where time is disconnected from space. Periodic relativistic spacetime crystals on the other hand need to account for the mixing of space and time in special relativity through Lorentz transformation, and have been listed only in 2-dimensions. This work shows that there exists a transformation between the conventional Minkowski spacetime (MS) and what is referred to here as renormalized blended spacetime (RBS); they are shown to be equivalent descriptions of relativistic physics in flat spacetime. There are two elements to this reformulation of MS, namely, blending and renormalization. When observers in two inertial frames adopt each other’s clocks as their own, while retaining their original space coordinates; the observers become blended. This process reformulates the Lorentz boosts into Euclidean rotations while retaining the original spacetime hyperbola describing worldlines of constant spacetime length from the origin. By renormalizing the blended coordinates with an appropriate factor that is a function of the relative velocities between the various frames, the hyperbola is transformed into a Euclidean circle. With these two steps, one obtains the RBS coordinates complete with new light lines, but now with a Euclidean construction. One can now enumerate the RBS point and space groups in various dimensions with their mapping to the well-known space crystal groups. The RBS point group for flat isotropic RBS spacetime is identified to be that of cylinders in various dimensions: mm2 which is that of a rectangle in 2D, (∞⁄m)m which is that of a cylinder in 3D, and that of hypercylinder in 4D. An antisymmetry operation is introduced that can swap between space-like and time-like directions, leading to color spacetime groups. The formalism reveals RBS symmetries that are not readily apparent in the conventional MS formulation. Mathematica® script is provided for plotting the MS and RBS geometries discussed in the work. 

Symmetry is fundamental to understanding our physical world. An antisymmetry operation switches between two different states of a trait, such as two time states, position states, charge states, spin states, or chemical species. This review covers the fundamental concepts of antisymmetry and focuses on four antisymmetries, namely, spatial inversion in point groups, time reversal, distortion reversal, and wedge reversion. The distinction between classical and quantum mechanical descriptions of time reversal is presented. Applications of these antisymmetries—in crystallography, diffraction, determining the form of property tensors, classifying distortion pathways in transition state theory, finding minimum energy pathways, diffusion, magnetic structures and properties, ferroelectric and multiferroic switching, classifying physical properties in arbitrary dimensions, and antisymmetry-protected topological phenomena—are described. 

It is shown that there are 41 types of multivectors representing physical quantities in non-relativistic physics in arbitrary dimensions within the formalism of Clifford algebra. The classification is based on the action of three symmetry operations on a general multivector: spatial inversion, 1 , time-reversal, 1′, and a third that is introduced here, namely wedge reversion, 1 † . It is shown that the traits of `axiality' and `chirality' are not good bases for extending the classification of multivectors into arbitrary dimensions, and that introducing 1 † would allow for such a classification. Since physical properties are typically expressed as tensors, and tensors can be expressed as multivectors, this classification also indirectly classifies tensors. Examples of these multivector types from non-relativistic physics are presented. 

Subchalcogenides are uncommon, and their chemical bonding results from an interplay between metal–metal and metal–chalcogenide interactions. Herein, we present Ir 6 In 32 S 21 , a novel semiconducting subchalcogenide compound that crystallizes in a new structure type in the polar P 31 m space group, with unit cell parameters a = 13.9378(12) Å, c = 8.2316(8) Å, α = β = 90°, γ = 120°. The compound has a large band gap of 1.48(2) eV, and photoemission and Kelvin probe measurements corroborate this semiconducting behavior with a valence band maximum (VBM) of −4.95(5) eV, conduction band minimum of −3.47(5) eV, and a photoresponse shift of the Fermi level by ∼0.2 eV in the presence of white light. X-ray absorption spectroscopy shows absorption edges for In and Ir do not indicate clear oxidation states, suggesting that the numerous coordination environments of Ir 6 In 32 S 21 make such assignments ambiguous. Electronic structure calculations confirm the semiconducting character with a nearly direct band gap, and electron localization function (ELF) analysis suggests that the origin of the gap is the result of electron transfer from the In atoms to the S 3p and Ir 5d orbitals. DFT calculations indicate that the average hole effective masses near the VBM (1.19 m e ) are substantially smaller than the average electron masses near the CBM (2.51 m e ), an unusual feature for most semiconductors. The crystal and electronic structure of Ir 6 In 32 S 21 , along with spectroscopic data, suggest that it is neither a true intermetallic nor a classical semiconductor, but somewhere in between those two extremes. 

The nudged elastic band (NEB) method is a commonly used approach for the calculation of minimum energy pathways of kinetic processes. However, the final paths obtained rely heavily on the nature of the initially chosen path. This often necessitates running multiple calculations with differing starting points in order to obtain accurate results. Recently, it has been shown that the NEB algorithm can only conserve or raise the distortion symmetry exhibited by an initial pathway. Using this knowledge, symmetry- adapted perturbations can be generated and used as a tool to systematically lower the initial path symmetry, enabling the exploration of other low-energy pathways that may exist. Here, the group and representation theory details behind this process are presented and implemented in a standalone piece of software (DiSPy). The method is then demonstrated by applying it to the calculation of ferroelectric switching pathways in LiNbO3. Previously reported pathways are more easily obtained, with new paths also being found which involve a higher degree of atomic coordination.