# Isotropy

The term isotropy (from the Greek isos, ἴσος, “equal” and tropos, τρόπος, “way”) has several definitions depending on the field of interest. The opposite of isotropy is anisotropy.

In Mathematics we have the following definitions:

• Isotropic varieties (geometry): a variety is an isotropic if the geometry 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 exists a non-zero vector v such that q(v) = 0; therefore such v is either an isotropic vector or a 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 map for Lorentzian varieties.
• Isotropy group: an isotropy group is the group of isomorphisms from any object to itself in a groupid. An isotropy representation is a representation of an isotropy group.
• Isotropic position: a probability distribution on a vector space is in an isotropic position if its covariance matrix is the identity.
• Isotropic vector field: a vector field generated by a point source is said to be isotropic if, for any spherical quadrant centered on the point source, the magnitude of the vector determined by any point on the sphere is invariant with respect to a change in direction. For example, starlight appears to be isotropic.

In Physics we have the following definitions:

• In 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 system of the decaying particle (independently). This follows from the rotational invariance of the Hamiltonian, which in turn is guaranteed for a spherically symmetric potential. In special relativity, the rest system of a particle is the coordinate system (reference system) in which the particle is at rest. The kinetic theory of gases is an example of isotropy. Molecules are assumed to move in random directions and, therefore, there is an equal probability that a molecule will move 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: the flow of a fluid is isotropic if there is no directional preference (e.g. in case of fully developed 3D turbulence). An example of anisotropy, on the other hand, occurs in fluid flows that have a background density, since gravity acts in only one direction. The apparent surface area separating two different isotropic fluids would be referred to as isotropic.
• Thermal expansion: a solid is said to be isotropic if its expansion following an administration of heat, is equal in all directions.
• Electromagnetism: an isotropic medium is such that magnetic permectivity and magnetic permeability are uniform in all directions.
• Optics: optical isotropy means having the same optical properties in all directions. Reflectance or transmittance is averaged for microheterogeneous samples if macroscopic reflectance or transmittance is to be calculated. This can be verified by simply examining, for example, a polycrystalline material under a polarizing microscope with cross-polarizers: if the crystallites are larger than the resolution limit, they will be visible. Otherwise, for micro-homogeneous samples (domains smaller than the resolution limit), it is the dielectric tensor that is reduced to a scalar.
• Cosmology: the Big Bang 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. As of 2006, observations suggest that, on distance scales much larger than galaxies, galaxy clusters are “large” features, but small compared to so-called multiverse scenarios. Here homogeneous means that the universe is the same everywhere (no preferred location) and isotropic implies that there is no preferred direction.

In Biology we have the following definitions:

• Cell Biology: if the properties of the cell wall are more or less the same everywhere, it is said to be isotropic. The inside of the cell is anisotropic because of the intracellular organelles.
• Physiology: in skeletal muscle cells (i.e., muscle fibers), the term “isotropic” refers to the sarcomeres that contribute to the striated pattern of the cells.
• Pharmacology: although it is well known that the skin is an ideal site for local and systemic drug delivery, it presents a formidable barrier to permeation of most substances. More recently, isotropic formulations (thermodynamically stable microemulsions that possess lyotropic liquid crystal properties) have been widely used in dermatology for drug delivery.