Transformation (geometry) in the context of Real coordinate space

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X).

For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure.

In geometry, there exist various rotation formalisms to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational (or angular) kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

According to Euler's rotation theorem, the rotation of a rigid body (or three-dimensional coordinate system with a fixed origin) is described by a single rotation about some axis. Such a rotation may be uniquely described by a minimum of three real parameters. However, for various reasons, there are several ways to represent it. Many of these representations use more than the necessary minimum of three parameters, although each of them still has only three degrees of freedom.

In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (using the semigroup operation) is associated with the composite of the two corresponding transformations. The terminology conveys the idea that the elements of the semigroup are acting as transformations of the set. From an algebraic perspective, a semigroup action is a generalization of the notion of a group action in group theory. From the computer science point of view, semigroup actions are closely related to automata: the set models the state of the automaton and the action models transformations of that state in response to inputs.

An important special case is a monoid action or act, in which the semigroup is a monoid and the identity element of the monoid acts as the identity transformation of a set. From a category theoretic point of view, a monoid is a category with one object, and an act is a functor from that category to the category of sets. This immediately provides a generalization to monoid acts on objects in categories other than the category of sets.

Biological or process structuralism is a school of biological thought that objects to an exclusively Darwinian or adaptationist explanation of natural selection such as is described in the 20th century's modern synthesis. It proposes instead that evolution is guided differently, by physical forces which shape the development of an animal's body, and sometimes implies that these forces supersede selection altogether.

Structuralists have proposed different mechanisms that might have guided the formation of body plans. Before Darwin, Étienne Geoffroy Saint-Hilaire argued that animals shared homologous parts, and that if one was enlarged, the others would be reduced in compensation. After Darwin, D'Arcy Thompson hinted at vitalism and offered geometric explanations in his classic 1917 book On Growth and Form. Adolf Seilacher suggested mechanical inflation for "pneu" structures in Ediacaran biota fossils such as Dickinsonia. Günter P. Wagner argued for developmental bias, structural constraints on embryonic development. Stuart Kauffman favoured self-organisation, the idea that complex structure emerges holistically and spontaneously from the dynamic interaction of all parts of an organism. Michael Denton argued for laws of form by which Platonic universals or "Types" are self-organised. Stephen J. Gould and Richard Lewontin proposed biological "spandrels", features created as a byproduct of the adaptation of nearby structures. Gerd B. Müller and Stuart A. Newman argued that the appearance in the fossil record of most of the phyla in the Cambrian explosion was "pre-Mendelian" evolution caused by physical factors. Brian Goodwin, described by Wagner as part of "a fringe movement in evolutionary biology", denies that biological complexity can be reduced to natural selection, and argues that pattern formation is driven by morphogenetic fields.