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In biology, symmetry refers to the balanced proportions and arrangement of parts in organisms, which can influence their development, behavior, and evolutionary adaptations. There are several types of symmetry observed in living organisms: 1. **Radial Symmetry**: Organisms exhibit radial symmetry when their body can be divided into multiple identical sections around a central axis. Examples include starfish and jellyfish.

Dichromatic symmetry is a concept that arises in the context of color theory and visual perception, particularly related to how we perceive and represent colors in a symmetrical manner. It often relates to the ways certain color combinations can be perceived as symmetrical or harmonious even when they are not identical. In art and design, dichromatic symmetry may refer to the use of two distinct colors that create a balanced and visually appealing composition.

Dihedral symmetry in three dimensions refers to the symmetry of three-dimensional objects that can be described by dihedral groups, which are related to the symmetries of polygons. Specifically, dihedral symmetry arises in the context of a polygon that has a certain number of sides, with a focus on its rotational and reflectional symmetries.

The term "Einstein Group" doesn't refer to a widely recognized concept in academia or other fields as of my last update in October 2023. However, it could relate to several different contexts depending on what you're referencing: 1. **Scientific Community**: It might refer to a group of physicists or researchers who focus on topics related to Einstein's theories, especially in the realms of relativity or quantum mechanics.

Elitzur's theorem is a result in quantum mechanics that deals with the relationship between measurement and quantum states. Specifically, it addresses the concept of "quantum erasure," which refers to the idea that certain measurements can potentially make it possible to restore information about a quantum system that was previously lost or obscured by other measurements. The most famous context in which Elitzur's theorem is discussed involves the double-slit experiment, a fundamental demonstration of quantum behavior.

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 **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.

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).

Explicit symmetry breaking refers to a situation in physics where a system that has a certain symmetry is made to lose that symmetry due to the introduction of some external influence or perturbation. This is different from spontaneous symmetry breaking, where the symmetry is broken by the dynamics of the system itself, without any external influence. In explicit symmetry breaking, the parameters of the system (like masses, coupling constants, or external fields) are adjusted in such a way that they actively favor one state over another.

Family symmetries refer to a concept in theoretical physics, particularly in the context of particle physics and the Standard Model. They involve the idea that certain symmetries can exist among different families or generations of particles. In the Standard Model, matter particles are classified into three generations, each containing particles such as quarks and leptons. Family symmetry suggests that these generations could be connected through some symmetry that goes beyond the conventional gauge symmetries that govern particle interactions.

Fock–Lorentz symmetry is a specific type of symmetry that arises in the context of relativistic quantum mechanics and quantum field theory. It relates to how physical systems behave under Lorentz transformations, which are mathematically expressed as the transformations that relate the coordinates of events in one inertial frame to those in another moving at a constant velocity relative to the first.

Hesse's principle of transfer is a concept in the philosophy of mathematics, particularly in the context of mathematical logic and set theory. It is named after the mathematician Heinrich Hesse. The principle addresses the nature of mathematical objects and the relationships between them, specifically how properties or structures can be "transferred" from one context to another.

The Higgs mechanism is a process in particle physics that explains how certain fundamental particles acquire mass through their interaction with the Higgs field. The Higgs field is a scalar field that permeates all of space, and it is associated with the Higgs boson, a fundamental particle confirmed by experiments at the Large Hadron Collider in 2012.

The Higgs sector refers to the part of the Standard Model of particle physics that describes the Higgs boson and the associated mechanisms that give mass to elementary particles. It plays a crucial role in explaining how particles acquire mass through the Higgs mechanism, which involves spontaneous symmetry breaking. Here's a breakdown of the key components of the Higgs sector: 1. **Higgs Field**: The Higgs sector is based on a scalar field known as the Higgs field, which permeates the universe.

Isometry is a concept in mathematics and geometry that refers to a transformation that preserves distances between points. In other words, an isometric transformation or mapping maintains the original size and shape of geometric figures, meaning the distances between any two points remain unchanged after the transformation. There are several types of isometric transformations, which include: 1. **Translations**: Moving every point of a figure the same distance in a specified direction.

A **Lie group** is a mathematical structure that combines concepts from algebra and geometry. It is defined as a group that is also a smooth manifold, which means it has a structure that allows for differentiation and smoothness.

Lie point symmetry is a concept from the field of differential equations and mathematical physics, named after the mathematician Sophus Lie. It specifically refers to symmetries of differential equations that can be expressed in terms of point transformations of the independent and dependent variables. In simpler terms, if a differential equation remains invariant under a transformation that is generated by a continuous group of transformations, then it possesses a Lie point symmetry.