 x Isotropy Encyclopedia
Isotropy is uniformity in all orientations; it is derived from the Greek iso (equal) and tropos (direction). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy
Anisotropy
Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. It can be defined as a difference, when measured along different axes, in a material's physical or mechanical properties An example of anisotropy is the light...

. Anisotropy is also used to describe situations where properties vary systematically, dependent on direction. Isotropic radiation
Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. It radiates uniformly in all directions from a point source sometimes called an isotropic radiator. The radiation may...

has the same intensity regardless of the direction of measurement
Measurement
Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...

, and an isotropic field exerts the same action regardless of how the test particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

is oriented.

## Mathematics

Within mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, isotropy has a few different meanings:
Isotropic manifold
Isotropic manifold
In mathematics, an isotropic manifold is a manifold in which the geometry doesn't depend on directions. A simple example is the surface of a sphere....

s: Some manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s are isotropic, meaning that the geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

on the manifold is the same regardless of direction. A similar concept is homogeneity
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...

. A manifold can be homogeneous without being isotropic. But if it is inhomogeneous, it is necessarily anisotropic.

In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which it evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be...

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

q is said to be isotropic if there is a non-zero vector v such that q(v)=0.

Isotropic coordinates
Isotropic coordinates
In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres. There are several different types of coordinate chart which are adapted to this family of nested spheres; the best known is the Schwarzschild chart, but the isotropic chart is also often...

on an Isotropic chart for Lorentzian manifolds.

## Physics

Quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

or Particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

: 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
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...

of the Hamiltonian, which in turn is guaranteed for a spherically symmetric potential.

Kinetic theory
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

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, there will be an equal number moving in one direction as any other hence demonstrating isotropy.

Thermal expansion
Thermal expansion
Thermal expansion is the tendency of matter to change in volume in response to a change in temperature.When a substance is heated, its particles begin moving more and thus usually maintain a greater average separation. Materials which contract with increasing temperature are rare; this effect is...

: 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
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

: 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 most simple instance being free space.

Optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

: Optical isotropy means having the same optical properties in all directions. The individual reflectance or transmittance
Transmittance
In optics and spectroscopy, transmittance is the fraction of incident light at a specified wavelength that passes through a sample. A related term is absorptance, or absorption factor, which is the fraction of radiation absorbed by a sample at a specified wavelength...

of the domains is averaged if the macroscopic reflectance or transmittance is to be calculated. This can be verified simply by investigating, e.g., a polycrystalline
Polycrystalline
Polycrystalline materials are solids that are composed of many crystallites of varying size and orientation. The variation in direction can be random or directed, possibly due to growth and processing conditions. Fiber texture is an example of the latter.Almost all common metals, and many ceramics...

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

Cosmology
Cosmology
Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...

: The Big Bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...

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
Cosmological Principle
In modern physical cosmology, the cosmological principle is the working assumption that observers on Earth do not occupy an unusual or privileged location within the universe as a whole, judged as observers of the physical phenomena produced by uniform and universal laws of physics...

. As of 2006, the observations suggest that, on distance scales much larger than galaxies, galaxy clusters are "Great"
Great Wall (astronomy)
The Great Wall , sometimes specifically referred to as the CfA2 Great Wall, is one of the largest known super-structures in the Universe...

features, but small compared to so-called multi-verse scenarios.

### Materials science

Materials science
Materials science
Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. This scientific field investigates the relationship between the structure of materials at atomic or molecular scales and their macroscopic properties. It incorporates...

: 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/mineralogy.

Microfabrication
Microfabrication
Microfabrication is the term that describes processes of fabrication of miniature structures, of micrometre sizes and smaller. Historically the earliest microfabrication processes were used for integrated circuit fabrication, also known as "semiconductor manufacturing" or "semiconductor device...

: 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
Microfabrication
Microfabrication is the term that describes processes of fabrication of miniature structures, of micrometre sizes and smaller. Historically the earliest microfabrication processes were used for integrated circuit fabrication, also known as "semiconductor manufacturing" or "semiconductor device...

of integrated circuits and MEMS
Microelectromechanical systems
Microelectromechanical systems is the technology of very small mechanical devices driven by electricity; it merges at the nano-scale into nanoelectromechanical systems and nanotechnology...

devices.

An isotropic antenna is an idealized "radiating element
Radiators are heat exchangers used to transfer thermal energy from one medium to another for the purpose of cooling and heating. The majority of radiators are constructed to function in automobiles, buildings, and electronics...

" used as a reference
Reference
Reference is derived from Middle English referren, from Middle French rèférer, from Latin referre, "to carry back", formed from the prefix re- and ferre, "to bear"...

; an antenna that broadcasts power equally (calculated by the Poynting vector
Poynting vector
In physics, the Poynting vector can be thought of as representing the directional energy flux density of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside and Nikolay Umov independently co-invented the Poynting vector...

) in all directions. In practice, an isotropic antenna cannot exist, as equal radiation in all directions would be a violation of the Helmholtz wave equation
Helmholtz equation
The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation\nabla^2 A + k^2 A = 0where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...

. The gain of an arbitrary antenna is usually reported in decibel
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

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

## Biology

Cell biology
Cell biology
Cell biology is a scientific discipline that studies cells – their physiological properties, their structure, the organelles they contain, interactions with their environment, their life cycle, division and death. This is done both on a microscopic and molecular level...

: If the properties of the cell wall
Cell (biology)
The cell is the basic structural and functional unit of all known living organisms. It is the smallest unit of life that is classified as a living thing, and is often called the building block of life. The Alberts text discusses how the "cellular building blocks" move to shape developing embryos....

are more or less the same everywhere, it is said to be isotropic. The interior of the cell is anisotropic due to intracellular organelle
Organelle
In cell biology, an organelle is a specialized subunit within a cell that has a specific function, and is usually separately enclosed within its own lipid bilayer....

s.

Physiology
Physiology
Physiology is the science of the function of living systems. This includes how organisms, organ systems, organs, cells, and bio-molecules carry out the chemical or physical functions that exist in a living system. The highest honor awarded in physiology is the Nobel Prize in Physiology or...

: In skeletal muscle cells (a.k.a. muscle fibers), the term "isotropic
Isotropic bands
In physiology, isotropic bands are skeletal muscle cells that form the light bands that contribute to the striated pattern of the cells.Isotropic bands indicate the behavior of polarized light as it passes through I bands....

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

## Other sciences

Economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

and Geography
Geography
Geography is the science that studies the lands, features, inhabitants, and phenomena of Earth. A literal translation would be "to describe or write about the Earth". The first person to use the word "geography" was Eratosthenes...

: An isotropic region is a region which has the same properties everywhere. Such a region is a construction needed in many types of models.

• Rotational invariance
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...

• Isotropic bands
Isotropic bands
In physiology, isotropic bands are skeletal muscle cells that form the light bands that contribute to the striated pattern of the cells.Isotropic bands indicate the behavior of polarized light as it passes through I bands....

• Isotropic coordinates
Isotropic coordinates
In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres. There are several different types of coordinate chart which are adapted to this family of nested spheres; the best known is the Schwarzschild chart, but the isotropic chart is also often...

• Transverse isotropy
Transverse isotropy
A transversely isotropic material is one with physical properties which are symmetric about an axis that is normal to a plane of isotropy. This transverse plane has infinite planes of symmetry and thus, within this plane, the material properties are same in all directions...

• Anisotropic
• Bi isotropic
Bi isotropic
In physics, engineering and materials science, bi-isotropic materials have the special optical property that they can twist the polarization of light in either refraction or transmission. This does not mean all materials with twist effect fall in the bi-isotropic class... 