Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describe situations where properties vary systematically, dependent on direction. Isotropic radiation has the same intensity regardless of the direction of

measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...

, and an isotropic field exerts the same action regardless of how the test particle is oriented.

Mathematics

Within

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, ''isotropy'' has a few different meanings: ; Isotropic manifolds: A

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

is isotropic if the geometry on the manifold is the same regardless of direction. A similar concept is homogeneity. ;

Isotropic quadratic form In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector sp ...

: A

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...

''q'' is said to be isotropic if there is a non-zero vector ''v'' such that ; such a ''v'' is an

isotropic vector In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real bilinear forms, definite quadratic forms a ...

or null vector. In complex geometry, a line through the origin in the direction of an isotropic vector is an isotropic line. ; Isotropic coordinates: Isotropic coordinates are coordinates on an isotropic chart for

Lorentzian manifolds Lorentzian may refer to * Cauchy distribution, also known as the Lorentz distribution, Lorentzian function, or Cauchy–Lorentz distribution * Lorentz transformation * Lorentzian manifold See also

*Lorentz (disambiguation) *Lorenz (disambiguati ...

. ;

Isotropy group In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...

:An isotropy group is the group of isomorphisms from any

object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...

to itself in a groupoid. An isotropy representation is a representation of an isotropy group. ; Isotropic position: A

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

over a vector space is in isotropic position if its

covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...

is the

identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...

. ; Isotropic vector field: The vector field generated by a point source is said to be ''isotropic'' if, for any spherical neighborhood centered at the point source, the magnitude of the vector determined by any point on the sphere is invariant under a change in direction. For an example, starlight appears to be isotropic.

Physics

; Quantum mechanics or particle physics: When a spinless particle (or even an unpolarized particle with spin) decays, the resulting decay distribution ''must'' be isotropic in the rest frame of the decaying particle regardless of the detailed physics of the decay. This follows from rotational invariance of the Hamiltonian, which in turn is guaranteed for a spherically symmetric potential. :

Kinetic theory of gases Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to its motion Art and enter ...

is also an example of isotropy. It is assumed that the molecules move in random directions and as a consequence, there is an equal probability of a molecule moving in any direction. Thus when there are many molecules in the gas, with high probability there will be very similar numbers moving in one direction as any other, demonstrating approximate isotropy. ;

Fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...

: Fluid flow is isotropic if there is no directional preference (e.g. in fully developed 3D turbulence). An example of anisotropy is in flows with a background density as gravity works in only one direction. The apparent surface separating two differing isotropic fluids would be referred to as an isotrope. ; Thermal expansion: A solid is said to be isotropic if the expansion of solid is equal in all directions when thermal energy is provided to the solid. ; Electromagnetics: An isotropic medium is one such that the permittivity, ε, and permeability, μ, of the medium are uniform in all directions of the medium, the simplest instance being free space. ; Optics: Optical isotropy means having the same optical properties in all directions. The individual reflectance or transmittance of the domains is averaged for micro-heterogeneous samples if the macroscopic reflectance or transmittance is to be calculated. This can be verified simply by investigating, e.g., a polycrystalline material under a polarizing microscope having the polarizers crossed: If the crystallites are larger than the resolution limit, they will be visible.

; Cosmology:The

Big Bang The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...

theory of the evolution of the observable universe assumes that space is isotropic. It also assumes that space is homogeneous. These two assumptions together are known as the

cosmological principle In modern physical cosmology, the cosmological principle is the notion that the spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throu ...

. As of 2006, the observations suggest that, on distance scales much larger than galaxies, galaxy clusters are "Great" features, but small compared to so-called

multiverse The multiverse is a hypothetical group of multiple universes. Together, these universes comprise everything that exists: the entirety of space, time, matter, energy, information, and the physical laws and constants that describe them. The di ...

scenarios. Here homogeneous means that the universe is the same everywhere (no preferred location) and isotropic implies that there is no preferred direction.

Materials science

In the study of mechanical properties of materials, "isotropic" means having identical values of a property in all directions. This definition is also used in geology and

mineralogy Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical (including optical) properties of minerals and mineralized artifacts. Specific studies within mineralogy include the proces ...

. Glass and metals are examples of isotropic materials. Common anisotropic materials include wood, because its material properties are different parallel and perpendicular to the grain, and layered rocks such as

slate Slate is a fine-grained, foliated, homogeneous metamorphic rock derived from an original shale-type sedimentary rock composed of clay or volcanic ash through low-grade regional metamorphism. It is the finest grained foliated metamorphic rock. ...

. Isotropic materials are useful since they are easier to shape, and their behavior is easier to predict. Anisotropic materials can be tailored to the forces an object is expected to experience. For example, the fibers in carbon fiber materials and

rebar Rebar (short for reinforcing bar), known when massed as reinforcing steel or reinforcement steel, is a steel bar used as a Tension (physics), tension device in reinforced concrete and reinforced masonry structures to strengthen and aid the concr ...

s in

reinforced concrete Reinforced concrete (RC), also called reinforced cement concrete (RCC) and ferroconcrete, is a composite material in which concrete's relatively low tensile strength and ductility are compensated for by the inclusion of reinforcement having hig ...

are oriented to withstand tension.

Microfabrication

In industrial processes, such as etching steps, isotropic means that the process proceeds at the same rate, regardless of direction. Simple chemical reaction and removal of a substrate by an acid, a solvent or a reactive gas is often very close to isotropic. Conversely, anisotropic means that the attack rate of the substrate is higher in a certain direction. Anisotropic etch processes, where vertical etch-rate is high, but lateral etch-rate is very small are essential processes in microfabrication of integrated circuits and MEMS devices.

Antenna (radio)

An isotropic antenna is an idealized " radiating element" used as a reference; an

antenna Antenna ( antennas or antennae) may refer to: Science and engineering * Antenna (radio), also known as an aerial, a transducer designed to transmit or receive electromagnetic (e.g., TV or radio) waves * Antennae Galaxies, the name of two collid ...

that broadcasts power equally (calculated by the

Poynting vector In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or '' power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt ...

) in all directions. The

gain Gain or GAIN may refer to: Science and technology * Gain (electronics), an electronics and signal processing term * Antenna gain * Gain (laser), the amplification involved in laser emission * Gain (projection screens) * Information gain in de ...

of an arbitrary antenna is usually reported in

decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...

s relative to an isotropic antenna, and is expressed as

dBi The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a p ...

or dB(i). In cells (a.k.a.

muscle fibers A muscle cell is also known as a myocyte when referring to either a cardiac muscle cell (cardiomyocyte), or a smooth muscle cell as these are both small cells. A skeletal muscle cell is long and threadlike with many nuclei and is called a muscl ...

), the term "

isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...

" refers to the light bands (

I bands I, or i, is the ninth letter and the third vowel letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''i'' (pronounced ), plural ...

) that contribute to the striated pattern of the cells. ;

Pharmacology Pharmacology is a branch of medicine, biology and pharmaceutical sciences concerned with drug or medication action, where a drug may be defined as any artificial, natural, or endogenous (from within the body) molecule which exerts a biochemica ...

: While it is well established that the skin provides an ideal site for the administration of local and systemic drugs, it presents a formidable barrier to the permeation of most substances. Most recently,

isotropic formulations Isotropic formulations are Chemical stability, thermodynamically stable microemulsions possessing lyotropic liquid crystal properties. They inhabit a state of matter and physical behaviour somewhere between conventional Liquid, liquids and that of s ...

have been used extensively in dermatology for drug delivery.

Computer science

; Imaging:We say a volume such as a

computed tomography A computed tomography scan (CT scan; formerly called computed axial tomography scan or CAT scan) is a medical imaging technique used to obtain detailed internal images of the body. The personnel that perform CT scans are called radiographers ...

has isotropic voxel spacing when the space between any two adjacent voxels is the same along each axis ''x, y, z''. E.g., voxel spacing is isotropic if the center of voxel ''(i, j, k)'' is 1.38 mm from that of ''(i+1, j, k)'', 1.38 mm from that of ''(i, j+1, k)'' and 1.38 mm from that of ''(i, j, k+1)'' for all indices ''i, j, k''.

Other sciences

; Economics and geography: An isotropic region is a region that has the same properties everywhere. Such a region is a construction needed in many types of models.

References

{{Reflist Orientation (geometry)