Children: Symmetry
A list of space groups refers to a classification of the symmetrical arrangements in three-dimensional space that describe how atoms are organized in crystals. These groups are essential in the field of crystallography and solid-state physics because they provide a systematic way to categorize and understand the symmetry properties of crystalline materials. Space groups combine the concepts of point groups and translation operations.
Lorentz covariance is a fundamental principle in the theory of relativity that describes how the laws of physics remain invariant under Lorentz transformations, which relate the coordinates of events as observed in different inertial reference frames moving at constant velocities relative to each other. In more detail, Lorentz transformations include combinations of rotations and boosts (changes in velocity) that preserve the spacetime interval between events.
Misorientation generally refers to a condition in which two objects, such as materials, crystals, or cells, are oriented in a way that does not align with each other. This term is commonly used in various fields, including materials science, crystallography, and biology. In the context of crystallography, misorientation describes the angular difference between the crystallographic directions or planes of two adjacent grains or crystals.
Modular invariance is a concept that arises in various fields of theoretical physics, particularly in string theory, conformal field theory (CFT), and statistical mechanics. It refers to the property of a system or mathematical formulation that remains invariant (unchanged) under transformations related to modular arithmetic or modular transformations.
Molecular symmetry refers to the spatial arrangement of atoms in a molecule and how that arrangement can exhibit symmetrical properties. It is a key concept in chemistry that helps in understanding the physical and chemical properties of molecules, including their reactivity, polarity, and interaction with light (such as in spectroscopy).
The Murnaghan–Nakayama rule is a tool used in representation theory, specifically in the context of symmetric functions and the study of representations of the symmetric group. This rule provides a method for calculating the characters of the symmetric group when restricted to certain subgroups, particularly the Young subgroups.
A **non-Euclidean crystallographic group** refers to a symmetry group that arises in the study of lattices and patterns in geometries that are not based on Euclidean space. Crystallographic groups describe how a pattern can be repeated in space while maintaining certain symmetries, including rotations, translations, and reflections. In Euclidean geometry, the classifications of crystallographic groups are based on the 17 two-dimensional plane groups and the 230 three-dimensional space groups.
A one-dimensional symmetry group refers to a group of symmetries that act on a one-dimensional space, such as a line or an interval. In mathematical terms, this involves transformations that preserve certain properties of the space, specifically geometric or algebraic structures. ### Characteristics of One-Dimensional Symmetry Groups: 1. **Transformations**: The transformations in one-dimensional symmetry groups typically include translations, reflections, and rotations (though rotations in one dimension behave similarly to a reflection).
A **P-compact group** (or **p-compact group**) is a type of topological group that plays a significant role in algebraic topology and group theory. These groups generalize the notion of compact groups, which are topological groups that are compact as topological spaces, but allow for more general structures.
The Poincaré group is a fundamental algebraic structure in the field of theoretical physics, particularly in the context of special relativity and quantum field theory. It describes the symmetries of spacetime in four dimensions and serves as the group of isometries for Minkowski spacetime. The group includes the following transformations: 1. **Translations**: These are shifts in space and time.
In the context of crystallography and group theory, a **polar point group** refers to a specific category of symmetry groups associated with three-dimensional objects, where there is a distinguished direction or axis. This type of symmetry group is associated with systems that have a unique spatial orientation, allowing for distinctions between positive and negative versions of various properties, such as polarization or chirality. Polar point groups typically possess a non-centrosymmetric arrangement, meaning they lack a center of symmetry.
Polychromatic symmetry refers to the concept of symmetry that involves multiple colors or hues. In a broader context, it can be understood in various fields, including art, mathematics, and physics, where multiple dimensions or variations are considered. In art and design, polychromatic symmetry can be observed in patterns and compositions that exhibit symmetrical properties while using a diverse color palette. This contrasts with traditional symmetry, which often emphasizes uniformity in color as well as shape.
A regular polytope is a multi-dimensional geometric figure that is highly symmetrical, with identical shapes and arrangements in its structure. In general, a regular polytope can be defined as a convex polytope that is both uniform (its faces are the same type of regular polygon) and vertex-transitive (the structure looks the same from any vertex).
Rotational symmetry is a property of a shape or object that indicates it can be rotated around a central point by a certain angle and still look the same as it did before the rotation. In other words, if you were to rotate the object about its central point, it would match its original configuration at certain intervals of rotation.
Scale invariance is a property of certain systems or equations where the system's characteristics or behavior do not change under a rescaling of lengths, times, or other dimensions. In other words, if you magnify or reduce the size of the system (or the parameters involved), the system remains statistically or qualitatively the same. This concept is prevalent in various fields, including physics, economics, and biology.
Schoenflies notation is a system used in chemistry and molecular biology to describe the symmetry of molecules and molecular structures, particularly in the context of point groups in three-dimensional space. It provides a way to classify the symmetry of a molecule based on its geometric arrangements and symmetries. In Schoenflies notation, point groups are denoted by symbols that often consist of letters and numbers.
A screw axis is a concept in the field of crystallography and molecular symmetry that describes a particular type of symmetry operation. It refers to a combination of a rotation and a translation along the same axis. The screw axis is commonly denoted using a notation that combines a number (indicating the degree of rotation) and a fraction (indicating the translational component).
Soft Supersymmetry (SUSY) breaking refers to a set of mechanisms in particle physics that allow supersymmetric partners of known particles to have different masses without eliminating the essential symmetry properties of supersymmetry itself. In a supersymmetric theory, every known particle has a corresponding partner, or superpartner, with differing spin properties. However, these superpartners are not observed in experiments, which suggests that supersymmetry must be broken.
A space group is a mathematical classification used in crystallography that describes the symmetries of a crystal structure. Specifically, it combines the symmetries of both the lattice (the periodic arrangement of points in space) and the motif (the group of atoms associated with each lattice point). In other words, a space group encapsulates how a crystal can be transformed into itself through operations like translations, rotations, reflections, and inversions.
Spontaneous symmetry breaking is a phenomenon that occurs in various fields of physics, particularly in condensed matter physics, particle physics, and cosmology. It describes a situation in which a system that is symmetric under some transformation settles into an asymmetrical state. Despite the underlying laws or equations being symmetric, the actual observed state of the system does not exhibit this symmetry.