physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

and

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''

anisotropy Anisotropy () is the structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit ve ...

''. ''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 to ...

, and an isotropic field exerts the same action regardless of how the test

particle In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...

is oriented.

Mathematics

Within

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, ''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

on the manifold is the same regardless of direction. A similar concept is

homogeneity Homogeneity and heterogeneity are concepts relating to the Uniformity (chemistry), uniformity of a Chemical substance, substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, ...

. ;

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 it is a definite quadratic form. More explicitly, 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 t ...

''q'' is said to be isotropic if there is a non-zero vector ''v'' such that ; such a ''v'' is an isotropic vector 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 lineshape (spectroscopy) * Lorentz transformation * Lorentzian manifold In mathematical phys ...

. ; Isotropy group:An isotropy group is the group of

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

s 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 a ...

to itself in a

groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...

. An isotropy representation is a representation of an isotropy group. ; Isotropic position: A

probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

over a

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

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. ; Isotropic vector field: The

vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...

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 Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...

: When a spinless particle (or even an unpolarized particle with spin) decays, the resulting decay distribution ''must'' be isotropic in the

rest frame In special relativity, the rest frame of a particle is the frame of reference (a coordinate system attached to physical markers) in which the particle is at rest. The rest frame of compound objects (such as a fluid, or a solid made of many vibrati ...

of the decaying particle - regardless of the detailed physics of the decay. This follows from

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. Mathematics Functions For example, the function : f(x,y) = ...

of the

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

, which in turn is guaranteed for a spherically symmetric potential. ; Gases: The

kinetic theory of gases The kinetic theory of gases is a simple classical model of the thermodynamic behavior of gases. Its introduction allowed many principal concepts of thermodynamics to be established. It treats a gas as composed of numerous particles, too small ...

also exemplifies 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, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...

: 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 Thermal expansion is the tendency of matter to increase in length, area, or volume, changing its size and density, in response to an increase in temperature (usually excluding phase transitions). Substances usually contract with decreasing temp ...

: 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 In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter (epsilon), is a measure of the electric polarizability of a dielectric material. A material with high permittivity polarizes more ...

, ε, and permeability, μ, of the medium are uniform in all directions of the medium, the simplest instance being free space. ;

Optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...

: Optical isotropy means having the same optical properties in all directions. The individual

reflectance The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic ...

transmittance Electromagnetic radiation can be affected in several ways by the medium in which it propagates. It can be Scattering, scattered, Absorption (electromagnetic radiation), absorbed, and Fresnel equations, reflected and refracted at discontinui ...

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, for example, a

polycrystalline A crystallite is a small or even microscopic crystal which forms, for example, during the cooling of many materials. Crystallites are also referred to as grains. Bacillite is a type of crystallite. It is rodlike with parallel longulites. S ...

material under a polarizing microscope having the polarizers crossed: If the crystallites are larger than the resolution limit, they will be visible.

;

Cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...

: The

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

, which underpins much of modern cosmology (including the

Big Bang The Big Bang is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models based on the Big Bang concept explain a broad range of phenomena, including th ...

theory of the evolution of the observable universe), assumes that the universe is both isotropic and homogeneous, meaning that the universe has no preferred location (is the same everywhere) and has no preferred direction. Observations made in 2006 suggest that, on distance-scales much larger than galaxies, galaxy clusters are "Great" features, but small compared to so-called

multiverse The multiverse is the hypothetical set of all universes. Together, these universes are presumed to comprise everything that exists: the entirety of space, time, matter, energy, information, and the physical laws and constants that describ ...

scenarios.

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 Geology (). is a branch of natural science concerned with the Earth and other astronomical objects, the rocks of which they are composed, and the processes by which they change over time. Modern geology significantly overlaps all other Earth ...

and

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

. Glass and metals are examples of isotropic materials. Common anisotropic materials include

wood Wood is a structural tissue/material found as xylem in the stems and roots of trees and other woody plants. It is an organic materiala natural composite of cellulosic fibers that are strong in tension and embedded in a matrix of lignin t ...

(because its material properties are different parallel to 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 ro ...

. 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 Carbon fiber-reinforced polymers (American English), carbon-fibre-reinforced polymers ( Commonwealth English), carbon-fiber-reinforced plastics, carbon-fiber reinforced-thermoplastic (CFRP, CRP, CFRTP), also known as carbon fiber, carbon comp ...

materials and

rebar Rebar (short for reinforcement bar or reinforcing bar), known when massed as reinforcing steel or steel reinforcement, is a tension device added to concrete to form ''reinforced concrete'' and reinforced masonry structures to strengthen and aid ...

s in

reinforced concrete Reinforced concrete, also called ferroconcrete or ferro-concrete, is a composite material in which concrete's relatively low tensile strength and ductility are compensated for by the inclusion of reinforcement having higher tensile strength or ...

are oriented to withstand tension.

Microfabrication

In industrial processes, such as

etching Etching is traditionally the process of using strong acid or mordant to cut into the unprotected parts of a metal surface to create a design in intaglio (incised) in the metal. In modern manufacturing, other chemicals may be used on other type ...

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 Microfabrication is the process of fabricating miniature structures of micrometre scales and smaller. Historically, the earliest microfabrication processes were used for integrated circuit fabrication, also known as "semiconductor manufacturing" ...

integrated circuits An integrated circuit (IC), also known as a microchip or simply chip, is a set of electronic circuits, consisting of various electronic components (such as transistors, resistors, and capacitors) and their interconnections. These components a ...

and

MEMS MEMS (micro-electromechanical systems) is the technology of microscopic devices incorporating both electronic and moving parts. MEMS are made up of components between 1 and 100 micrometres in size (i.e., 0.001 to 0.1 mm), and MEMS devices ...

devices.

Antenna (radio)

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

reference A reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. It is called a ''nam ...

; an antenna 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 wat ...

) in all directions. The gain 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, root-power, and field quantities, power or root-power quantity on a logarithmic scale. Two signals whos ...

s relative to an isotropic antenna, and is expressed as dBi or dB(i). In cells (a.k.a.

muscle fibers Skeletal muscle (commonly referred to as muscle) is one of the three types of vertebrate muscle tissue, the others being cardiac muscle and smooth muscle. They are part of the somatic nervous system, voluntary muscular system and typically are a ...

), the term "isotropic" refers to the light bands ( I bands) that contribute to the striated pattern of the cells.

Pharmacology Pharmacology is the science of drugs and medications, including a substance's origin, composition, pharmacokinetics, pharmacodynamics, therapeutic use, and toxicology. More specifically, it is the study of the interactions that occur betwee ...

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. Recently, isotropic formulations have been used extensively in dermatology for drug delivery.

Computer science

;

Imaging Imaging is the representation or reproduction of an object's form; especially a visual representation (i.e., the formation of an image). Imaging technology is the application of materials and methods to create, preserve, or duplicate images. ...

:A volume such as a

computed tomography A computed tomography scan (CT scan), formerly called computed axial tomography scan (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 or ...

is said to have isotropic

voxel In computing, a voxel is a representation of a value on a three-dimensional regular grid, akin to the two-dimensional pixel. Voxels are frequently used in the Data visualization, visualization and analysis of medical imaging, medical and scient ...

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 Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...

and

geography Geography (from Ancient Greek ; combining 'Earth' and 'write', literally 'Earth writing') is the study of the lands, features, inhabitants, and phenomena of Earth. Geography is an all-encompassing discipline that seeks an understanding o ...

: 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) Symmetry