/heuh mol"euh jee, hoh-/, n., pl. homologies.1. the state of being homologous; homologous relation or correspondence.2. Biol.a. a fundamental similarity based on common descent.b. a structural similarity of two segments of one animal based on a common developmental origin.3. Chem. the similarity of organic compounds of a series in which each member differs from its adjacent compounds by a fixed increment, as by CH2.4. Math. a classification of figures according to certain topological properties.
* * *Similarity of the structure, physiology, or development of different species of organisms based on their descent from a common evolutionary ancestor.Analogy, by contrast, is a functional similarity of structure that is based not on common evolutionary origins but on mere similarity of use. The forelimbs of such widely differing mammals as humans, bats, and deer are homologous; the form of construction and the number of bones in each are practically identical and represent adaptive modifications of the forelimb structure of their shared ancestor. The wings of birds and insects, on the other hand, are merely analogous; they are used for flight in both types of organisms but do not share a common ancestral origin.
* * *in biology, similarity of the structure, physiology, or development of different species of organisms based upon their descent from a common evolutionary ancestor. Homology is contrasted with analogy, which is a functional similarity of structure based not upon common evolutionary origins but upon mere similarity of use. Thus the forelimbs of such widely differing mammals as humans, bats, and deer are homologous; the form of construction and the number of bones in these varying limbs are practically identical, and represent adaptive modifications of the forelimb structure of their common early mammalian ancestors. Analogous structures, on the other hand, can be represented by the wings of birds and of insects; the structures are used for flight in both types of organisms, but they have no common ancestral origin at the beginning of their evolutionary development. A 19th-century British biologist, Sir Richard Owen (Owen, Sir Richard), was the first to define both homology and analogy in precise terms.When two or more organs or structures are basically similar to each other in construction but are modified to perform different functions, they are said to be serially homologous. An example of this is a bat's wing and a whale's flipper. Both originated in the forelimbs of early mammalian ancestors, but they have undergone different evolutionary modification to perform the radically different tasks of flying and swimming, respectively. Sometimes it is unclear whether similarities in structure in different organisms are analogous or homologous. An example of this is the wings of bats and birds. These structures are homologous in that they are in both cases modifications of the forelimb bone structure of early reptiles. But birds' wings differ from those of bats in the number of digits and in having feathers for flight while bats have none. And most importantly, the power of flight arose independently in these two different classes of vertebrates; in birds while they were evolving from early reptiles, and in bats after their mammalian ancestors had already completely differentiated from reptiles. Thus, the wings of bats and birds can be viewed as analogous rather than homologous upon a more rigorous scrutiny of their morphological differences and evolutionary origins.in mathematics, a basic notion of algebraic topology (topology). Intuitively, two curves in a plane or other two-dimensional surface are homologous if together they bound a region—thereby distinguishing between an inside and an outside. Similarly, two surfaces within a three-dimensional space are homologous if together they bound a three-dimensional region lying within the ambient space.There are many ways of making this intuitive notion precise. The first mathematical steps were taken in the 19th century by the German Bernhard Riemann (Riemann, Bernhard) and the Italian Enrico Betti (Betti, Enrico), with the introduction of “Betti numbers” in each dimension, referring to the number of independent (suitably defined) objects in that dimension that are not boundaries. Informally, Betti numbers refer to the number of times that an object can be “cut” before splitting into separate pieces; for example, a sphere has Betti number 0 since any cut will split it in two, while a cylinder has Betti number 1 since a cut along its longitudinal axis will merely result in a rectangle. A more extensive treatment of homology was carried out in n dimensions at the beginning of the 20th century by the French mathematician Henri Poincaré (Poincaré, Henri), leading to the notion of a homology group in each dimension, apparently first formulated about 1925 by the German mathematician Emmy Noether (Noether, Emmy). The two basic facts about homology groups for a surface or a higher-dimensional topological manifold are: (1) if the groups are defined by means of a triangulation, a cellular subdivision, or other artifact, the resulting groups do not depend on the particular choices made along the way; and (2) the homology groups are a topological invariant, so that if two surfaces or higher-dimensional spaces are homeomorphic (homeomorphism), then their homology groups in each dimension are isomorphic (see foundations of mathematics: Isomorphic structures (mathematics, foundations of) and mathematics: Algebraic topology (mathematics)).Homology plays a fundamental role in analysis; indeed, Riemann was led to it by questions involving integration on surfaces. The basic reason is because of Green's theorem (see George Green (Green, George)) and its generalizations, which express certain integrals over a domain in terms of integrals over the boundary. As a consequence, certain important integrals over curves will have the same value for any two curves that are homologous. This is in turn reflected in physics in the study of conservative vector spaces (vector space) and the existence of potentials.Robert Osserman
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