Isotropy

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• Within mathematics
  Mathematics
  
  Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
  , 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 homogeneous space is a similar concept....
  s: Some manifold
  Manifold
  
  In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
  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 continuous function by symmetry in a transitivity way....
  . A manifold can be homogeneous without being isotropic. But if it is inhomogeneous, it is necessarily anisotropic.
• Isotropic quadratic form
  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 it evaluates to zero. Otherwise the quadratic form is anisotropic....
  : A quadratic form
  Quadratic form
  
  In mathematics, a quadratic form is a homogeneous polynomial of Degree_ two in a number of variables. For example,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 coordinates, but the isotropic chart is also often useful....
   on an Isotropic chart for Lorentzian manifolds.

• Cosmology: The Big Bang
  Big Bang
  
  The Big Bang is the physical cosmology model of the initial conditions and subsequent development of the universe supported by the most comprehensive and accurate explanations from current scientific method and observation....
   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 physical cosmology, the cosmological principle is an assumption, or working hypothesis, about the large scale structure of the cosmos, stating that:...
  . 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 the second largest known Large-scale structure of the cosmos in the Universe ....
   features, but small compared to so-called multi-verse scenarios.

• Cell biology: If the properties of the cell wall
  Cell (biology)
  
  The cell is the structural and functional unit of all known Life organisms. It is the smallest unit of an organism that is classified as living, and is often called the building bricks of life....
   are more or less the same everywhere, it is said to be isotropic. The interior of the cell is anisotropic due to intracellular organelles
  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 membrane....
  .

• Radio broadcasting: In radio
  Radio
  
  Radio is the transmission of signals, by modulation of electromagnetic radiation with frequency below those of visible light.Electromagnetic radiation radio propagation by means of oscillating electromagnetic fields that pass through the air and the vacuum of space....
  , an isotropic antenna is an idealized "radiating element
  Radiator
  
  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
  
  A reference is a relation between Object in which one object designates by linking to another object. Such relations as these may occur in a variety of domains, including logic, computer science, time, art and scholarship....
  ; 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 energy flux of an electromagnetic field. It is named after its inventor John Henry Poynting....
  ) 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 equationwhere ∇2 is the Laplace operator, k is the wavenumber, and A is the amplitude....
  . The gain of an arbitrary antenna is usually reported in decibel
  Decibel
  
  The decibel is a logarithmic units of measurement that expresses the magnitude of a physical quantity relative to a specified or implied reference level....
  s relative to an isotropic antenna, and is expressed as dBi or dB(i).

• Physiology: 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.

• Materials: In the study of mechanical
  Mechanics
  
  Mechanics is the branch of physics concerned with the behaviour of physical body when subjected to forces or Displacement , and the subsequent effect of the bodies on their environment....
   properties of materials, "isotropic" means having identical values of a property in all crystallographic directions.

• Optics: 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. Specifically, visible transmittance is this fraction for visible light....
   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....
   material under a polarizing microscope having the polarizers crossed: If the crystallites are larger than the resolution limit, they will be visible.

• 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
  Microfabrication
  
  Microfabrication or micromanufacturing are the terms to describe processes of fabrication of miniature structures, of micrometre sizes and smaller....
   of integrated circuits and MEMS devices.

• 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. All metals are isotropic.

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

• Electromagnetics: An isotropic medium is one such that the permittivity, e, and permeability, µ, of the medium are uniform in all directions of the medium, the most simple instance being free space.

See also

• 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 coordinates, but the isotropic chart is also often useful....
  
• Transverse isotropy
  Transverse isotropy
  
  A transversely isotropic material is symmetry 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 and engineering, and materials science, bi-isotropics are any of a class of materials having the special optical property that it can twist the polarization of light in either refraction or transmission....