What is "isomorphism"

Isomorphism \I`so*mor"phism\, n. [Cf. F. isomorphisme.]

1. (Crystallog.) A similarity of crystalline form between substances of similar composition, as between the sulphates of barium ( BaSO4) and strontium ( SrSO4). It is sometimes extended to include similarity of form between substances of unlike composition, which is more properly called hom[oe]omorphism.

2. (Math.) A one-to-one mapping of one set onto another set which preserves the relations between the elements of the domains of the sets.

from German Isomorphismus, 1828, coined by German chemist Eilhard Mitscherlich (1794-1863) from isomorph; see isomorphic. Related: Isomorph.

n. Similarity of form

n. (biology) similarity or identity of form or shape or structure [syn: isomorphy]

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that admits an inverse. Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.

For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if and only if it is bijective.

In topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. In mathematical analysis, where the morphisms are differentiable functions, isomorphisms are also called diffeomorphisms.

A canonical isomorphism is a canonical map that is an isomorphism. Two objects are said to be canonically isomorphic if there is a canonical isomorphism between them. For example, the canonical map from a finite-dimensional vector space V to its second dual space is a canonical isomorphism; on the other hand, V is isomorphic to its dual space but not canonically in general.

Isomorphisms are formalized using category theory. A morphism in a category is an isomorphism if it admits a two-sided inverse, meaning that there is another morphism in that category such that and , where 1 and 1 are the identity morphisms of X and Y, respectively.

In sociology, an isomorphism is a similarity of the processes or structure of one organization to those of another, be it the result of imitation or independent development under similar constraints. There are three main types of institutional isomorphism: normative, coercive and mimetic. The development that these three types of isomorphism promote can also create isomorphic paradoxes that hinder such development. Specifically, these isomorphic paradoxes are related to an organization's remit, resources, accountability, and professionalization.

The concept of institutional isomorphism was primarily developed by Paul DiMaggio and Walter Powell. The concept appears in their classical paper The iron cage revisited: institutional isomorphism and collective rationality in organizational fields from 1983.

Isomorphism in the context of globalization, is an idea of contemporary national societies that is addressed by the institutionalization of world models constructed and propagated through global cultural and associational processes. As it is emphasized by realist theories the heterogeneity of economic and political resource or local cultural origins by the micro-phenomenological theories, many ideas suggest that the trajectory of change in in political units is towards homogenization around the world. Such similarities so called isomorphic changes are found by researchers, explaining, despite of all possible configurations of local economic forces, power relationships, and forms of traditional culture it might consist of, an previously-isolated island society that made contact with the rest of the globe would quickly take on standardized forms and appear to be similar to a hundred other nation-states around the world. Isomorphic developments of same conclusion are reported from nay nation-states' features, that is, constitutional forms highlighting both state power and individual rights, mass schooling systems organized around a fairly standard curriculum, rationalized economic and demographic record keeping and data systems, antinatalist population control policies intended to enhance national development, formally equalized female status and rights, expanded human rights in general, expansive environmental policies, development-oriented economic policy, universalistic welfare systems, standard definitions of disease and health care, and even some basic demographic variables. These isomorphisms are difficultly accounted by theories reasoning from the differences among national economies and cultural traditions, however, they are sensible outcomes if nation-states are enactments of the world cultural order.

In biology, an isomorphism is a similarity of form or structure between organisms, generally between organisms with independent ancestries, e.g. after convergent evolution. Two organisms exhibiting isomorphism are referred to as isomorphs.

The separate evolution of camera eyes in vertebrates and cephalopods (and insects, and many more, in as many as fifty separate instances), is an example of isomorphism. So is the evolution of wings in insects, pterosaurs, birds, and bats.

Isomorphism may refer to:

• Isomorphism, in mathematics, logic, philosophy, and information theory, a mapping that preserves the structure of the mapped entities, in particular:
  • Graph isomorphism a mapping that preserves the edges and vertices of a graph
  • Group isomorphism a mapping that preserves the group structure
  • Ring isomorphism a mapping that preserves both the additive and multiplicative structure of a ring
  • Isomorphism theorems theorems that assert that some homomorphisms involving quotients and subobjects are isomorphisms
• Isomorphism (sociology), a similarity of the processes or structure of one organization to those of another
• Isomorphism (biology), a similarity of form or structure between organisms
• Isomorphism (crystallography), a similarity of crystal form
• Isomorphism (Gestalt psychology), a correspondence between a stimulus array and the brain state created by that stimulus
• Cybernetic isomorphism, a recursive property of viable systems, as defined by Anthony Stafford Beer

The term isomorphism literally means sameness (iso) of form (morphism). In Gestalt psychology, Isomorphism is the idea that perception and the underlying physiological representation are similar because of related Gestalt qualities. Isomorphism refers to a correspondence between a stimulus array and the brain state created by that stimulus, and is based on the idea that the objective brain processes underlying and correlated with particular phenomenological experiences functionally have the same form and structure as those subjective experiences.

Isomorphism can also be described as the similarity in the gestalt patterning of a stimulus and the activity in the brain while perceiving the stimulus. More generally, this concept is an expression of the materialist view that the properties of mind and consciousness are a direct consequence of the electrochemical interactions within the physical brain.

A commonly used example of isomorphism is the phi phenomenon, in which a row of lights flashing in sequence creates the illusion of motion. It is argued that the brain state created by this stimulus matches the brain state created by a patch of light moving from one location to another. The stimulus is perceived as motion because the subjective percept of spatial structure is correlated with electric fields in the brain whose spatial pattern mirrors the spatial structure in the perceived world.

In crystallography crystals are described as '''isomorphous ''' if they are closely similar in shape. Historically crystal shape was defined by measuring the angles between crystal faces with a goniometer. In modern usage isomorphous crystals belong to the same space group.

Double sulfates, such as Tutton's salt, with the generic formula MM(SO).6HO, where M is an alkali metal and M is a divalent ion of Mg, Mn, Fe, Co, Ni, Cu or Zn form a series of isomorphous compounds which were important in the nineteenth century in establishing the correct atomic weights of the transition elements. Alums, such as KAl(SO).12HO, are another series of isomorphous compounds, though there are three series of alums with similar external structures, but slightly different internal structures. Many spinels are also isomorphous.

In order to form isomorphous crystals two substances must have the same chemical formulation, they must contain atoms which have corresponding chemical properties and the sizes of corresponding atoms should be similar. These requirements ensure that the forces within and between molecules and ions are approximately similar and result in crystals that have the same internal structure. Even though the space group is the same, the unit cell dimensions will be slightly different because of the different sizes of the atoms involved.