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3D rotation group

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By defini

Icosahedral symmetry

In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the

Icosian calculus

The icosian calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856.In modern terms, he gave a group presentation of the icosahedral rota

Triptych

A triptych (/ˈtrɪptɪk/ TRIP-tik; from the Greek adjective τρίπτυχον "triptukhon" ("three-fold"), from tri, i.e., "three" and ptysso, i.e., "to fold" or ptyx, i.e., "fold") is a work of art (usually a

Plane of rotation

In geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotati

Representation theory of the Galilean group

In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin (normally explained in Wigner's classification of relativistic mechanics) in terms of the representation

6-j symbol

Wigner's 6-j symbols were introduced by Eugene Paul Wigner in 1940 and published in 1965. They are defined as a sum over products of four Wigner 3-j symbols, The summation is over all six mi allowed b

Gankyil

The Gankyil (Tibetan: དགའ་འཁྱིལ།, Lhasa IPA: [/kã˥ kʲʰiː˥/]) or "wheel of joy" (Sanskrit: ānanda-cakra) is a symbol and ritual tool used in Tibetan and East Asian Buddhism. It is composed of three (so

Pauli matrices

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (σ), they

Taijitu

In Chinese philosophy, a taijitu (simplified Chinese: 太极图; traditional Chinese: 太極圖; pinyin: tàijítú; Wade–Giles: t'ai⁴chi²t'u²) is a symbol or diagram (图; tú) representing Taiji (太极; tàijí; 'utmost e

Tetrahedral symmetry

A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all (not nec

Angular velocity

In physics, angular velocity or rotational velocity (ω or Ω), also known as angular frequency vector, is a pseudovector representation of how fast the angular position or orientation of an object chan

Ambigram

An ambigram is a calligraphic design that has several interpretations as written. The term was coined by Douglas Hofstadter in 1983. Most often, ambigrams appear as visually symmetrical words. When fl

Circle of a sphere

A circle of a sphere is a circle that lies on a sphere. Such a circle can be formed as the intersection of a sphere and a plane, or of two spheres. Circles of a sphere are the spherical geometry analo

Doublet state

In quantum mechanics, a doublet is a composite quantum state of a system with an effective spin of 1/2, such that there are two allowed values of the spin component, −1/2 and +1/2. Quantum systems wit

Rigid body dynamics

In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid (i.e. th

3-j symbol

In quantum mechanics, the Wigner 3-j symbols, also called 3-jm symbols, are an alternative to Clebsch–Gordan coefficients for the purpose of adding angular momenta. While the two approaches address ex

Clebsch–Gordan coefficients

In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstat

Triplet state

In quantum mechanics, a triplet is a quantum state of a system with a spin of quantum number s=1, such that there are three allowed values of the spin component, ms = −1, 0, and +1. Spin, in the conte

9-j symbol

In physics, Wigner's 9-j symbols were introduced by Eugene Paul Wigner in 1937. They are related to recoupling coefficients in quantum mechanics involving four angular momenta

Spin (physics)

Spin is a conserved quantity carried by elementary particles, and thus by composite particles (hadrons) and atomic nuclei. Spin is one of two types of angular momentum in quantum mechanics, the other

Borjgali

Borjgali (Georgian: ბორჯღალი; also Borjgala or Borjgalo) is a Georgian symbol of the Sun and eternity. The borjgali is often represented with seven rotating wings over the tree of life which can be us

Racah W-coefficient

Racah's W-coefficients were introduced by Giulio Racah in 1942. These coefficients have a purely mathematical definition. In physics they are used in calculations involving the quantum mechanical desc

Laplace–Runge–Lenz vector

In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star o

Slater integrals

In mathematics and mathematical physics, Slater integrals are certain integrals of products of three spherical harmonics. They occur naturally when applying an orthonormal basis of functions on the un

Armenian eternity sign

The Armenian eternity sign (Armenian: Յաւերժութեան Նշան, romanized: haverzhut’yan nshan) or Arevakhach (Արեւախաչ, "Sun Cross") is an ancient Armenian national symbol and a symbol of the national ident

Spin magnetic moment

In physics, mainly quantum mechanics and particle physics, a spin magnetic moment is the magnetic moment caused by the spin of elementary particles. For example, the electron is an elementary spin-1/2

Galilei-covariant tensor formulation

The Galilei-covariant tensor formulation is a method for treating non-relativistic physics using the extended Galilei group as the representation group of the theory. It is constructed in the light co

Three hares

The three hares (or three rabbits) is a circular motif or meme appearing in sacred sites from East Asia, the Middle East and to the churches of Devon, England (as the "Tinners' Rabbits"), and historic

Total angular momentum quantum number

In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentu

Spherical harmonics

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific

Hexafoil

The hexafoil is a design with six-fold dihedral symmetry composed from six vesica piscis lenses arranged radially around a central point, often shown enclosed in a circumference of another six lenses.

Wigner D-matrix

The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanic

Multiplet

In physics and particularly in particle physics, a multiplet is the state space for 'internal' degrees of freedom of a particle, that is, degrees of freedom associated to a particle itself, as opposed

Intersystem crossing

Intersystem crossing (ISC) is an isoenergetic radiationless process involving a transition between the two electronic states with different spin multiplicity.

Rotational symmetry

Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is t

Rigid body

In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains

Magnetic quantum number

In atomic physics, the magnetic quantum number (ml) is one of the four quantum numbers (the other three being the principal, azimuthal, and spin) which describe the unique quantum state of an electron

Octahedral symmetry

A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same

Angular momentum coupling

In quantum mechanics, the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. For instance, the orbit an

Rotational invariance

In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument.

Swastika

The swastika (卐 or 卍) is an ancient religious and cultural symbol, predominantly in various Eurasian, as well as some African and American cultures, now also widely recognized for its appropriation by

Axial symmetry

Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis. For example, a baseball bat without trademark or other design, or a

Magnetic braking (astronomy)

Magnetic braking is a theory explaining the loss of stellar angular momentum due to material getting captured by the stellar magnetic field and thrown out at great distance from the surface of the sta

Solid harmonics

In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions . There are two kinds: the regular solid harmonic

Spin quantum number

In atomic physics, the spin quantum number is a quantum number (designated ms) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particl

Triquetra

The triquetra (/traɪˈkwɛtrə/ treye-KWEH-truh; from the Latin adjective triquetrus "three-cornered") is a triangular figure composed of three interlaced arcs, or (equivalently) three overlapping vesica

Harmonic tensors

No description available.

Solids with icosahedral symmetry

Platonic solids - regular polyhedra (all faces of the same type) Archimedean solids - polyhedra with more than one polygon face type. Catalan solids - duals of the Archimedean solids.

Spin-weighted spherical harmonics

In special functions, a topic in mathematics, spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphe

Laplace expansion (potential)

In physics, the Laplace expansion of potentials that are directly proportional to the inverse of the distance, such as Newton's gravitational potential or Coulomb's electrostatic potential, expresses

Rotation operator (quantum mechanics)

This article concerns the rotation operator, as it appears in quantum mechanics.

Spinor spherical harmonics

In quantum mechanics, the spinor spherical harmonics (also known as spin spherical harmonics, spinor harmonics and Pauli spinors) are special functions defined over the sphere. The spinor spherical ha

Zonal spherical harmonics

In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spheric

Hexagram

A hexagram (Greek) or sexagram (Latin) is a six-pointed geometric star figure with the Schläfli symbol {6/2}, 2{3}, or {{3}}. Since there are no true regular continuous hexagrams, the term is instead

Angular momentum operator

In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of ato

Radial function

In mathematics, a radial function is a function defined on a Euclidean space Rn whose value at each point depends only on the distance between that point and the origin. For example, a radial function

Triskelion

A triskelion or triskeles is an ancient motif consisting of a triple spiral exhibiting rotational symmetry.The spiral design can be based on interlocking Archimedean spirals, or represent three bent h

Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body ar

Azimuthal quantum number

The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second o

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