# Transformation

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## recombination

**recombination,**process of “shuffling” of genes by which new combinations can be generated. In recombination through sexual reproduction, the offspring's complete set of genes differs from that of either parent, being rather a combination of genes from both parents. In recombination by crossing over, alleles of genes are exchanged between homologous chromosomes during meiosis. This exchange results in the generation of new combinations of alleles on segments of chromosomes, counteracting the tendency of linked genes, i.e., genes on the same chromosome, to be always transmitted as a group. Various mechanisms for introducing new genetic material have been discovered in bacteria. These mechanisms have been used extensively to study gene structure and function. In transformation, a fragment of free deoxyribonucleic acid (DNA) is inserted in a recipient bacterium (see nucleic acid). The free DNA fragment comes from the chromosome of a bacterial cell that has been lysed, or dissolved. In transduction, genetic material is transferred from one bacterium to another by a carrier virus. When a virus enters a bacterium, its DNA can be inserted into the bacterial chromosome, reproducing along with the host chromosome in cell division. Subsequently, sometimes many bacterial generations later, the viral genetic material may detach from the bacterial chromosome, taking some of the bacterial chromosomal material along with it. The bacterium then lyses, the viral particle enters a new bacterium, and the viral particle, together with the bacterial genes it carries, is inserted into the chromosome of the new host. In conjugation, which occurs between bacteria of the same species and also between some bacteria of different species, either an entire chromosome or a part of one is transferred from a bacterium of a donor-mating strain to a bacterium of a recipient-mating strain. Donor strains, denoted male, contain a sex-factor particle composed of a nucleic acid; recipient, or female, strains lack the particle (see episome). Conjugation has been used to construct genetic maps, i.e., the ordering of genes along a chromosome. Evidence from transformation experiments was used to support the idea that DNA was the genetic material.

## Transformation (bacteria)

The addition of deoxyribonucleic acid (DNA) to living cells, thereby changing their genetic composition and properties. The recipient bacteria are usually closely related to the donor strain. The process may occur in natural conditions, for example, in a host animal infected with two parasitic strains, and indeed it might play a part in the rapid evolution of pathogenic bacteria. There are several species of bacteria in which transformation has been achieved in the laboratory.

That bacterial transformation is true genetic transmission on a small scale, rather than controlled mutation, is demonstrated by the following characteristics: (1) A specific trait is introduced, coming always from donors bearing the trait. (2) The trait is transferred by determinant, genelike material far less complex than whole cells or nuclei, and this material, DNA, is known to be present in gene-carrying chromosomes. (3) The trait is inherited by the progeny of the changed bacteria. (4) The progeny produce, when they grow, increased amounts of DNA carrying the specific property. (5) The traits are transferred as units exactly in the patterns in which they appear or in which they are induced by mutation. (6) The DNA transmits the full potentialities of the donor strain, whether these are in an expressed or in a latent state. (7) The traits are often attributable to the presence of a specific gene-determined enzyme protein. (8) Certain groups of determinants may occur “linked” within DNA molecules, just as genes may be linked, and if so, heat denaturation, radiation, or enzyme action will inactivate or separate them just to the extent that they can damage or break apart the DNA molecules. (9) Linked determinants, while transforming a new cell, may become exchanged (recombined) between themselves and their unmarked or unselective alternate forms in such a way that they bring about genetic variation, and in a pattern indicating the existence of larger organized genetic units. *See* Bacterial genetics, Gene

Through the application of a number of procedures prior to adding the DNA, transformation was extended first to many different bacterial species and then to eukaryotic cells. Today almost any cell type can be transformed. In some cases, tissues can be injected directly with naked DNA and transformed. However, unlike with bacteria, the naked DNA adds almost anywhere in the genome rather than recombining with its indigenous homolog. However, with special highly selective procedures, homologous recombination can be obtained. By treating embryonic stem cells and adding them to embryos that then go to term, specific and nonspecific transgenic animals can be obtained (for example, mice). *See* Genetic engineering

When the source of the DNA is some entity capable of independent replication, such as a virus or plasmid, the phenomenon is called transfection. If foreign DNA is then inserted into these entities, the result is recombinant DNA that can lead to transduction. *See* Molecular biology, Transduction (bacteria)

## Transformation

## Transformation

(pop culture)The vampire traditionally could transform itself into various **animals,** particularly a **bat,** a wolf, or a dog. It could also transform into a **dust**-like cloud or a **mist.** This attribute was often referred to as shape-shifting, and vampires figures often graded into shape-shifters, a particular kind of demon entity in European mythologies. It was also an attribute often tied to **witchcraft.** In the novel by **Bram Stoker, Dracula**‘s first recorded transformations were observed by **Jonathan Harker** but occurred in such a manner that neither Harker (nor the reader) realized what was happening. During the course of the novel, Dracula transformed into a wolf (to leave the ship that had wrecked at **Whitby,)** a bat (throughout the novel), and as mist (in order to enter **Mina Murray**‘s bedroom). In one of his encounters with the three vampire **brides** in **Castle Dracula,** Harker noted that they appeared to him first as a swirl of dust in the moonlight. Twenty-five years prior to Dracula, the transformation of the vampire into an animal had also been an integral part of **Sheridan Le Fanu**‘s story “Carmilla.” In that story, **Carmilla** transformed into a cat on several occasions.

The ability of vampires to transform into animals was part of the folklore of the majority of countries that include a vampire figure. The Japanese, for example, had a well-known tale of a vampire who assumed the form of the wife of Prince Nabeshima and then transformed into a cat to hunt her **victims.** In various countries in eastern Europe, the **Slavic** vampire could transform into a wide variety of animals, and on rare occasions, even some plants and farm implements. The ancient Roman vampire often appeared as a bird, a crow, or screech owl. Their transformation into wolves tied vampires to **werewolves,** the exact relationship being a matter of scholarly disagreement.

In *Dracula,***(1897) Abraham Van Helsing** also proposed the idea that vampire transformations were facilitated at certain hours. Dracula could change into various forms during the evening at will. However, during the daylight hours he could change only at noon, or exactly at sunrise and sunset. Most likely, Stoker borrowed this idea from Emily Gerard, who reported that Romanians believed that specific times of the day had special significance. Among these were the “exact hour of noon,” a precarious time because an evil spirit, Pripolniza, was active. The idea of special powers tied to certain times of the day was dropped by subsequent writers using the vampire theme.

More recent novels and movies have disagreed on the issue of the vampire’s power of transformation. The many remakes of *Dracula* have been fairly consistent with his ability to transform into an animal, at least a bat. In 1992’s ** Bram Stoker’s Dracula,** his ability to transform was a major sub-theme of the plot. However, the influential novels by

**Chelsea Quinn Yarbro**and

**Anne Rice**have denied the vampire’s ability to transform, as have the more recent novels of

**P. N. Elrod**and

**Elaine Bergstrom.**Yarbro, even more than Rice, stripped her vampires of most of their supernatural abilities, although they were left with great

**strength**and a long life.

The trend initiated by Rice and Yarbro carried over into many books and motion pictures that have stepped back from Dracula and created other vampire characters who exist in a contemporary setting. Thus, transformation has not been an element in the life of the vampires in such movies as *Vamp* (1986), ** Near Dark** (1987),

*The Lost Boys*(1987),

*Innocent Blood*(1992),

**(2005),**

*Bloodrayne**30 Days of Night*(2007), or

*Let the Right One In*(2008). Movies such as

*Van Helsing*(2004), the popular

**movie trilogy, and the several**

*Underworld**Twilight*(2009) movies featured werewolves who transformed from human to animal, while the vampires more or less stayed in human form.

The vampires in **television** series such as *Forever Knight, Buffy the Vampire Slayer,**Moonlight, Blood Ties,* or *Being Human* did not shift into animal forms. The series *True Blood* has included shapeshifters as an integral part of the story, but they are not vampires.

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Transformation

a fundamental concept of mathematics that arises in the study of correspondences between classes, particularly classes of geometric objects or classes of functions. For example, geometrical investigations often require that the dimensions of a figure be changed in the same ratio, that the radii of circles be increased by the same amount, or in general that with figures of some class there be associated other figures obtainable from the initial figures by certain rules. In the solution of differential equations by operational methods, the given functions are replaced by other, transformed, functions. Such correspondences are called transformations. More precisely, a transformation is a correspondence that associates with each element *x* of a set *X* a definite element *y* of another set *Y.* Logically, the concept of transformation coincides with those of function, mapping, and operator. The term “transformation” is used most often in geometry and functional analysis, and the correspondence between *ξ* and *y* = f(x) is usually considered to be one-to-one.

** Geometric transformations.** In geometry, point transformations are encountered most frequently. Such transformations associate with each point of a geometric object— that is, a curve, surface, or space—another point of that object. In other words, a point transformation is a mapping of a geometric object into itself. A point transformation carries every figure, which may be regarded as a set of points, into a new figure. The new figure is called the image of the original figure, and the original figure is called the preimage of the new figure. If a point transformation is one-to-one, the inverse transformation can be defined. A point transformation is called an identity transformation if under it the image of every point coincides with the preimage. If we limit ourselves for definiteness to point transformations of the plane, then such transformations can be specified analytically by equations of the form

*xʹ = f(x, y) yʹ* = *θ(x, y)*

where *x* and *y* are the coordinates of the preimage and *xʹ* and *yʹ* are the coordinates of the image in the same coordinate system.

Many important classes of point transformations form groups. For any two transformations in such a class, the class also contains the product of these transformations, where the product is defined as the result of the successive application of the transformations. In addition, for each transformation in the class, the class also contains the inverse transformation. The most important examples of groups of point transformations follow.

(1) The group of rotations of the plane about the origin, where the coordinates of the image are given by

*xʹ = x* cos α *— y* sin α

*yʹ =x* sin α *+ y* cos α

where α is the angle of rotation.

(2) The group of translations, where all points are displaced by the same vector *ai + bj:*

*xʹ = x + a yʹ = y + b*

(3) The group of motions consisting of transformations that preserve the distances between points and the orientations of the plane:

*xʹ = x* cos α — *y* sin α + a_{1}

*yʹ = x* sin α *+ y* cos α + b_{1}

(4) The group of motions and reflections consisting of transformations that preserve the distances between points in the plane. The set of motions and reflections that transform a figure into itself is called the symmetry group of the figure. This group determines the symmetry properties of the figure. For example, the symmetry group of a regular tetrahedron consists of the 4! = 24 transformations that permute its vertices.

(5) The group of similarity transformations is generated by motions, reflections, and homothety.

(6) The group of affine transformations consisting of one-to-one mappings of the plane onto itself such that straight lines are transformed into straight lines:

If *c _{1} = c_{2}*, the transformation is called a central affinity. If

*D*= 1, the transformation is called equiaffine; equiaffine transformations preserve the areas of figures.

(7) The group of projective transformations consisting of one-to-one transformations of the projective plane— the plane augmented by a “line at infinity”— that transform lines into lines:

From these equations it is evident that the line *ax + by + c* = 0 is transformed into the line at infinity.

(8) The group of circle-preserving transformations that is generated by motions, reflections, similarity transformations, and inversions. If the points of the plane are represented by complex numbers, the transformations of this group can be written in the form

or

where *w = xʹ+ iyʹ, z = x + iy*, and z = x *— iy.* Thus, these transformations coincide with the linear fractional transformations. The transformations of this group are characterized by the property carrying the set of lines and circles in the plane into itself. They are also characterized by the property of conformality.

Groups (1) through (7) are linear groups, since they transform straight lines into straight lines. Moreover, groups (1) and (2) are subgroups of group (3), and each subsequent group— that is, groups (4), (5), (6), and (7)— contains its predecessor. Groups (1) through (6) can be characterized as the set of projective transformations that leave unchanged some figure in the projective plane. For example, affine transformations preserve the line at infinity. Group (8) is a nonlinear group since the transformations in it can carry straight lines into circles. The transformations of groups (1) through (8) are birational transformations—that is, transformations in which *xʹ* and *yʹ* can be rationally expressed in terms of *x* and *y*, and vice versa.

In addition to point transformations, which establish a correspondence between points, geometry also uses transformations that establish correspondences between figures other than points. For example, in some geometric problems all circles are replaced by circles whose radii are increased by a certain factor. A transformation of the set of circles into itself is thereby defined.

Also considered are transformations that change the nature of the elements, that is, carry, for example, points into lines or lines into points. Thus, it is possible to associate with every point of *M(x, y)* the line *u _{x}ʹ* + v

_{y}ʹ = 1, where

*u*and

*v*are some functions of

*x*and

*y.*If

*u*and v are linear fractional functions of

*x*and y:

then we obtain the most general projective transformation of points of the plane into lines of the plane. If, moreover, *b _{1} = a_{2}, c_{1} = — a, and c_{2}= — b*, then we obtain a polarity relative to some conic. In particular, when

*u = x*and v =

*y*, we obtain the polarity relative to the circle

*x*=1. In this case, to each point in the xy-plane there corresponds a straight line in the

^{2}+ y^{2}*xʹ yʹ*-plane, and to a curve Γ in the xy-plane there corresponds the family of lines that are tangent to some curve Γ ‘or pass through the same point. A correspondence is thereby established between curves in the jy-plane, which are regarded as the sets of their points, and curves in the x’y’-plane, which are regarded as the envelopes of their tangents. More general transformations are given by the formula

*F(x, y, xʹ, yʹ) =*0. If

*x*and

*y*are specified, this formula determines a certain curve in the

*xʹ yʹ-*plane. A correspondence is thereby established between the points of one plane and a two-parameter set of curves of the other plane. This correspondence may be extended to a correspondence between the curves of one plane, which are regarded as the sets of their points, and curves of the other plane, which are regarded as the envelopes of the corresponding family of curves. Under this transformation, curves that touch in one plane are transformed into curves that touch in the other plane. Such transformations are therefore called contact transformations.

Transformations of multidimensional spaces, in particular, of three-dimensional spaces, are defined analogously to plane transformations. Each of the plane-transformation groups outlined above has a three-dimensional analogue, which is obtained from it by increasing the number of variables involved in the transformation. For example, the group of orthogonal transformations corresponds to group (1), and the group of nonsingular linear transformations corresponds to the group of central affinities. An example of a transformation group of four-dimensional space is the Lorentz group, which plays an important role in the theory of relativity. Transformations of multidimensional spaces are used in analysis to calculate multiple integrals, since they allow a given region of integration to be reduced to a simpler region.

For both plane-transformation groups and groups of transformations of multidimensional spaces, it is possible to define the concept of proximity of transformations. This concept permits the formation of continuous transformation groups.

For each transformation group, there exist properties of figures that are preserved under the transformations of the group. Such properties are said to be invariants with respect to the given transformation group. Thus, under the transformations of the group of motions the distance between two points is an invariant, under affine transformations the parallelism of lines and the ratio of the areas of two figures are invariants, and under projective transformations the cross ratio *AB/AD:CB/CD* of points *A, B, C*, and *D* on a line is an invariant. In accordance with the Erlangen program, to each transformation group there corresponds a branch of geometry that studies the properties that remain invariant under the transformations of that group. A distinction is accordingly made between, for example, the metric, affine, and projective properties of figures. In general, the larger the group, the more fundamental the connection between invariants and figures. The most general properties of figures are properties invariant under all topological transformations, that is, under all bicontinuous one-to-one transformations. These properties include dimension, connectedness, and orientability.

Transformations play an especially important role in the establishment of new theorems and the generalization of existing ones. If the statement of a theorem that has been proved for a figure *F* uses only those properties of *F* that are invariant under some transformation group, then the theorem is valid for all figures obtainable from *F* by means of transformations of that group. Such figures are said to be homologous, or equivalent, to *F* with respect to that group. This property of transformations is especially important if of several equivalent figures there is one that has the simplest properties in certain respects. Thus, a number of theorems of projective geometry were first established for circles and then carried over to any nondegenerated conic, since all nondegenerate conies are equivalent to the circle under the group of projective transformations. In the solution of geometric construction problems, transformations are often used to reduce the initial configuration to a form that simplifies the solution of the problem.

** Transformations of functions.** The theory of transformation groups is also very important for the theory of analytic functions, wherein classes of automorphic functions are studied. Such functions do not change under the transformations that form a certain group.

The concept of transformation plays an important role in functional analysis, where transformations of one set of functions into another are considered. Examples of such transformations are Fourier transforms and Laplace transforms. Under these transformations, with every function f there is associated another function Φ according to a specific rule. For example, the Fourier transform has the form

It, like the Laplace transform, belongs to the class of integral transforms defined by equations of the form

In a number of cases, transformations permit operations on functions to be replaced by simpler operations on the images of the functions— for example, differentiation may be replaced by multiplication by the independent variable. As a result, the solution of equations is facilitated.

Many equations can be written in the form f = *Af*, where f is the unknown function and *A* is a symbol for a transformation. In this case, the problem of solving the equation may be interpreted as that of finding a function that is invariant under the transformation. This viewpoint, which is called the fixed-point principle, permits the existence and uniqueness of solutions to be established in a number of cases. The fixed-point principle involves the use of fixed-point theorems such as the contraction mapping theorem of Banach.

### REFERENCE

Efimov, N. V.*Vysshaia geometriia*, 5th ed. Moscow, 1971.

Klein, F.

*Vysshaia geometriia.*Moscow-Leningrad, 1939. (Translated from German.)

Klein, F.

*Elementarnaia matematika s tochki zreniia vysshei: Lektsii*…, 2nd ed., vol. 2. Moscow-Leningrad, 1934. (Translated from German.)

Hadamard, J.

*Elementarnaia geometriia*, 4th ed., part 1. Moscow, 1957. (Translated from French.)

## Transformation

in genetics, the introduction of genetic information into a cell by means of isolated deoxyribonucleic acid (DNA). Transformation leads to the appearance in the transformed cell (transformant) and its offspring of new traits that are characteristic of the source of the DNA.

Transformation was discovered in 1928 by the British geneticist F. Griffith, who observed the heritable reestablishment of the synthesis of capsule-type polysaccharide in pneumococci when mice were infected with a mixture of encapsulated bacteria killed by overheating and cells lacking capsules. The bodies of mice that died during these experiments proved to be unique detectors, since the acquired capsule-type polysaccharide informed the cells that lacked capsules how to engender the infectious process that proved lethal.

Later experiments established that transformation also occurred when an extract made from destroyed encapsulated bacteria was added to the pneumococci that lacked capsules. In 1944 the Canadian geneticist O. Avery and his American co-workers established that DNA molecules were the factor causing transformation. This was the first proof that DNA acted as a carrier of hereditary information.

Transformation has also been discovered and studied in bacteria other than pneumococci. Experiments have been made with a number of easily discernible genetic characteristics, for example, resistance to the action of cellular toxins, or a need for certain growth factors. The results of these experiments, as well as the use of DNA with a radioisotope tag, have made it possible to assign a quantitative value to transformation.

Transformation in bacteria is regarded as a complex process that includes the following stages: fixation of DNA molecules by the recipient cell; penetration of DNA into the cell; inclusion of fragments of the transforming DNA in the chromosome of the host cell; and formation of pure transformed variants. The fixation of DNA occurs on a limited number of receptors located on the cell surface. The DNA bonded to the receptors remains sensitive to the enzyme deoxyribonuclease added to the medium. This leads to the decomposition of the DNA by the enzyme. Less than one minute after fixation, part of the DNA penetrates the cell.

Bacterial cells of the same strain differ greatly in their permeability to DNA. Cells of a bacterial population that accept foreign DNA are called competent. The number of competent cells in a population is insignificant and depends on the genetic characteristics of the bacteria and the stage of growth of the bacterial culture. The development of competence is associated with the synthesis of a specific protein that permits DNA to penetrate into the cell.

The average size of the DNA fragments that penetrate a cell is 5 × 10^{6} atomic mass units. Since a number of such fragments may enter a competent cell simultaneously, the total magnitude of DNA absorbed may approximately equal the size of a chromosome of the host cell. After the penetration of double-stranded DNA into a cell, one strand decomposes into mono- and oligo-nucleotides, and the other becomes structured into the chromosome of the host cell by means of ruptures and recombinations of the cell. The ensuing replication of this hybrid structure leads to the splintering off of pure transformant clones, whose offspring inherit the characteristic encoded by the DNA that penetrated the cell.

Transformation has made it possible to perform genetic analysis of bacteria in which such forms of genetic exchange as conjugation and transduction have not been described. Transformation is also a convenient method for investigating the influences of physical or chemical changes in DNA structure on the biological activity of DNA. The development of a method of transformation in *Escherichia coli* has made it possible to use not only fragments of a bacterial chromosome but the DNA of bacterial plasmids and phages in transformation. This method is widely used to introduce hybrid DNA into a cell during experiments in gene engineering.

There have been reports of successful experimental reproduction of transformation in cells of higher organisms. However, the process of transformation has not been sufficiently studied in these cases.

### REFERENCES

Hayes, W.*Genetika bakterii i bakteriofagov*. Moscow, 1965. (Translated from English.)

Prozorov, A. A.

*Geneticheskaia transformatsüa u mikroorganizmov*. Moscow, 1966.

Braun, W.

*Genetika bakterii*. Moscow, 1968. (Translated from English.)

Bresler, S. E.

*Molekuliarnaia biologiia*. Leningrad, 1973.

Stent, G.

*Molekuliarnaia genetika*. Moscow, 1974. Chapter 7. (Translated from English.)

A. L. TABACHNIK

## Transformation

a theatrical device. In the theater, the variety stage, and the circus, it is the ability of a performer to transform his appearance in a short time with the aid of makeup, wigs, costumes, and masks. The methods of transformation are widely used in the vaudeville theater. The greatest Soviet master of transformation is A. I. Raikin.

## transformation

[‚tranz·fər′mā·shən]## Transformation

Actaeon*Metamorphoses*]

*Dictionary*, 26]

*Dictionary*, 55]

*A Midsummer Night’s Dream*]

*Breast, The*

*The Breast*in Weiss, 55]

*Dictionary*, 180]

*Classical*, 28]

*Dictionary*]

*Pygmalion*; Am. Musical:

*My Fair Lady*in

*On Stage*, 373]

*Alice’s Adventures in Wonderland*]

*Orpheus in Hades*, Westerman, 271–272]

*Steppenwolf*]

*Dr. Jekyll and Mr. Hyde*in Magill I, 214]

*Jurgen*in Magill I, 464]

*The Return*in Magill II, 896]

*Metamorphoses*or

*The Golden*Ass in Magill I, 309]

*Dictionary*, 570]

*Dictionary*, 596]

*Odyssey*]

*Dictionary*, 664]

*Orlando*, Magill I, 698–700]

*Petrouchka*in Thompson, 1657]

*Dictionary*, 698]

*Pinocchio*]

*The Picture of Dorian Gray*]

*Cinderella*]

*Through the Looking-Glass*]

*Rhinoceros*

*The Metamorphosis*in Benét, 663]

*Dictionary*, 876]

*The Ring of the Nibelung*]

*Lady into Fox*, Magill I, 486]

*The Land of Oz*]

*Dictionary*, 901]

*Dictionary*, 1097]

*Dictionary*, 1148]

## transformation

**1.**

*Maths*

**a.**a change in position or direction of the reference axes in a coordinate system without an alteration in their relative angle

**b.**an equivalent change in an expression or equation resulting from the substitution of one set of variables by another

**2.**

*Physics*a change in an atomic nucleus to a different nuclide as the result of the emission of either an alpha-particle or a beta-particle

**3.**an apparently miraculous change in the appearance of a stage set

**4.**(in South Africa) a national strategy aimed at attaining national unity, promoting reconciliation through negotiated settlement and non-racism

## transformation

**foldoc.org**)

## geometry calculations

In 3D graphics rendering, the computation of the base properties for each point (vertex) of the triangles forming the objects in the 3D world. See graphics pipeline.**The Computer Language Company Inc**. All Rights reserved. THIS DEFINITION IS FOR PERSONAL USE ONLY. All other reproduction is strictly prohibited without permission from the publisher.