Isotropy is uniformity in all orientations; it is derived from the Greek isos (ἴσος, "equal") and tropos (τρόπος, "way"). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, 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, and an isotropic field exerts the same action regardless of how the test particle is oriented.

Mathematics

Within mathematics, isotropy has a few different meanings:

Isotropic manifolds

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

Isotropic quadratic form

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

Isotropy group

An isotropy group is the group of isomorphisms from any object to itself in a groupoid.^{[dubious – discuss][1]} An isotropy representation is a representation of an isotropy group.

Isotropic position

A probability distribution over a vector space is in isotropic position if its covariance matrix is the identity.

Isotropic vector

A vector in Euclidean 3-space is said to be isotropic if ${\displaystyle x^{2}+y^{2}+z^{2}=0}$. To better understand this equation, le

Within mathematics, isotropy has a few different meanings:

Isotropic manifolds

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

Isotropic quadratic form

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

Isotropy group

An isotropy group is the group of isomorphisms from any object to itself in a groupoid.^{[dubious – discuss]}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. Glass and metals are examples of isotropic materials.^[3] Common anisotropic materials include wood, because its material properties are different parallel and perpendicular to the grain, and layered rocks such as slate.

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 rebars in reinforced concrete 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 that broadcasts power equally (calculated by the Poynting vector) in all directions. The gain of an arbitrary antenna is usually reported in decibels relative to an isotropic antenna, and is expressed as dBi or dB(i).

Biology

Cell biology

If the properties of the cell wall are more or less the same everywhere, it is said to be isotropic. The interior of the cell is anisotropic due to intracellular organelles.

Physiology

In skeletal muscle cells (a.k.a. muscle fibers), the term "isotropic" refers to the light bands (I bands) that contribute to the striated pattern of the cells.

Pharmacology

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.^[4] Most recently, isotropic formulations have been used extensively in dermatology for drug delivery.^[5]

Computer science

Imaging

We say a volume such as a computed tomography 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.^[6]

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.

See also

• Rotational invariance
• Isotropic bands
• Isotropic coordinates
• Transverse isotropy
• Anisotropic
• Bi isotropic
• Symmetry

References

1. ^ A groupoid ${\displaystyle {\mathcal {G}}}$ is a category where all morphisms are isomorphisms, i.e., invertible. If ${\displaystyle G\in {\mathcal {G}}}$ is any object, then ${\displaystyle {\mathcal {G}}(G,G)}$ denotes its isotropy group: the group of isomorphisms from ${\displaystyle G}$ to ${\displaystyle G}$.
2. ^ ^{a b} "WMAP Big Bang Theory". Map.gsfc.nasa.gov. Retrieved 2014-03-06.
3. ^ "Anisotropy and Isotropy". Archived from the original on 2012-05-31. Retrieved 2012-05-26.
4. ^ Landman L. “The Epidermal Permeability Barrier.” Anatomy and Embryology (Berl) 1

   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 rebars in reinforced concrete are oriented to withstand tension.

   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)

   Cell