Category: Rotational symmetry

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
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
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
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
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 (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
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
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.
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
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
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
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