## Description

The term **isomorphic** refers to quantities which have *identical or similar form, shape, or structure*. In mathematics, an isomorphism defines a *one-to-one correspondence between two mathematical sets*. In ergodynamics, isomorphic quantities are defined by equations of identical form. If isomorphic quantities are not expressed in identical units, then these quantities are expressed in different formats which can be converted to identical untis. Example: electric force [V=J/C] and chemical force [Jol=J/mol] are ismorphic forces; the electrical format [J/C] can be converted to the chemical format [J/mol] by the Faraday constant. Units not only give meaning to the numerical value of a quantity, but units provide also an abbreviated common language to communicate and compare isomorphic quantities. In irreversible thermodynamics, isomorphic forces are referred to as *generalized* forces.

**Reference:** MitoPedia: Ergodynamics, Gnaiger 2020 BEC MitoPathways

## Some quotes

- ".. the word 'isomorphism' was defined as an information-preserving transformation. .. The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where "corresponding" means that the two parts play similar roles in their respective structures. .. The perception of an isomorphism between two known structures is a significant advance in knowledge — and I claim that it is such perceptions of isomorphism which create meanings in the minds of people." — Hofstadter DR (1979) Gödel, Escher, Bach: An eternal golden braid. A metaphorical fugue on minds and machines in the spirit of Lewis Carroll. Harvester Press:499 pp. - »Bioblast link«

- "Relationships between apparently different subjects are as creatively important in mathematics as they are in any discipline. The relationship hints at some underlying truth which enriches both subjects. .. Barry Mazur: 'Mathematicians studying elliptic equations might not be well versed in things modular, and conversely. Then along comes the Taniyama-Shimura conjecture which is the grand surmise that there's a bridge between these two completely different worlds. Mathematicians love to build bridges.' The value of mathematical bridges is enormous. They enable communities of mathematicians who have been living on separate islands to exchange ideas and explore each other's creations. Mathematics consists of islands of knowledge in a sea of ignorance. For example, there is the island occupied by geometers who study shape and form, and then there is the island of probability where mathemtaticians discuss risk and chance. There are dozens of such islands, each one whith its own unique language, incomprehensible to the inhabitants of other islands. The language of geometry is quite different to the language of probability, and the slang of calculus is meaningless to those who speak only statistics." — Singh Simon (1997) Fermat's last theorem. Fourth Estate, London 340 pp. - »Bioblast link«

- "The symbolic language of logistics is allegedly an ideal mode of representation that makes all content explicit; it stands in isomorphic relation to the objects it describes. .. .. the adequacy of a scientific theory is characterized in terms of a relation of isomorphism between theory and model. .. It is often assumed that the best way to think of representation in mathematics is in terms of isomorphism between structures, .." — Grosholz Emily R (2007) Representation and productive ambiguity in mathematics and the sciences. Oxford Univ Press 312 pp. - »Bioblast link«

- "The isomorphic method uses mathematical expressions complementary to a focus on the terms and units of physicochemical quantities." — Gnaiger E (2020) Mitochondrial pathways and respiratory control. An introduction to OXPHOS analysis. 5th ed. Bioenerg Commun 2020.2: 112 pp. - »Bioblast link«

**MitoPedia concepts:**
Ergodynamics