Euclidean space

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Euclidean_space 

A point in three-dimensional Euclidean space can be located by three coordinates.

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.

Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his Elements, with the great innovation of proving all properties of the space as theorems, by starting from a few fundamental properties, called postulates, which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (parallel postulate).

After the introduction at the end of the 19th century of non-Euclidean geometries, the old postulates were re-formalized to define Euclidean spaces through axiomatic theory. Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article. In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space.

There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are isomorphic. Therefore, it is usually possible to work with a specific Euclidean space, denoted ${\displaystyle {\mathbf{E}}^n}$ or ${\displaystyle {\mathbb{E}}^n}$, which can be represented using Cartesian coordinates as the real n-space ${\displaystyle {\mathbb{R}}^n}$ equipped with the standard dot product.

Definition

History of the definition

Euclidean space was introduced by ancient Greeks as an abstraction of our physical space. Their great innovation, appearing in Euclid's Elements was to build and prove all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of the lack of more basic tools. These properties are called postulates, or axioms in modern language. This way of defining Euclidean space is still in use under the name of synthetic geometry.

In 1637, René Descartes introduced Cartesian coordinates, and showed that these allow reducing geometric problems to algebraic computations with numbers. This reduction of geometry to algebra was a major change in point of view, as, until then, the real numbers were defined in terms of lengths and distances.

Euclidean geometry was not applied in spaces of dimension more than three until the 19th century. Ludwig Schläfli generalized Euclidean geometry to spaces of dimension n, using both synthetic and algebraic methods, and discovered all of the regular polytopes (higher-dimensional analogues of the Platonic solids) that exist in Euclidean spaces of any dimension.

Despite the wide use of Descartes' approach, which was called analytic geometry, the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract vector spaces allowed their use in defining Euclidean spaces with a purely algebraic definition. This new definition has been shown to be equivalent to the classical definition in terms of geometric axioms. It is this algebraic definition that is now most often used for introducing Euclidean spaces.

Motivation of the modern definition

One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as motions) on the plane. One is translation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is rotation around a fixed point in the plane, in which all points in the plane turn around that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (usually considered as subsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by some sequence of translations, rotations and reflections (see below).

In order to make all of this mathematically precise, the theory must clearly define what is a Euclidean space, and the related notions of distance, angle, translation, and rotation. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of units of length and other physical dimensions: the distance in a "mathematical" space is a number, not something expressed in inches or metres.

The standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is as a set of points on which a real vector space acts – the space of translations which is equipped with an inner product. The action of translations makes the space an affine space, and this allows defining lines, planes, subspaces, dimension, and parallelism. The inner product allows defining distance and angles.

The set ${\displaystyle {\mathbb{R}}^n}$ of n-tuples of real numbers equipped with the dot product is a Euclidean space of dimension n. Conversely, the choice of a point called the origin and an orthonormal basis of the space of translations is equivalent with defining an isomorphism between a Euclidean space of dimension n and ${\displaystyle {\mathbb{R}}^n}$ viewed as a Euclidean space.

It follows that everything that can be said about a Euclidean space can also be said about ${\displaystyle {\mathbb{R}}^n.}$ Therefore, many authors, especially at elementary level, call ${\displaystyle {\mathbb{R}}^n}$ the standard Euclidean space of dimension n, or simply the Euclidean space of dimension n.

Origin-free illustration of the Euclidean plane

A reason for introducing such an abstract definition of Euclidean spaces, and for working with ${\displaystyle {\mathbb{E}}^n}$ instead of ${\displaystyle {\mathbb{R}}^n}$ is that it is often preferable to work in a coordinate-free and origin-free manner (that is, without choosing a preferred basis and a preferred origin). Another reason is that there is no standard origin nor any standard basis in the physical world.

Technical definition

A Euclidean vector space is a finite-dimensional inner product space over the real numbers.

A Euclidean space is an affine space over the reals such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called Euclidean affine spaces to distinguish them from Euclidean vector spaces.

If E is a Euclidean space, its associated vector space (Euclidean vector space) is often denoted ${\displaystyle \overrightarrow{E}.}$ The dimension of a Euclidean space is the dimension of its associated vector space.

The elements of E are called points, and are commonly denoted by capital letters. The elements of ${\displaystyle \overrightarrow{E}}$ are called Euclidean vectors or free vectors. They are also called translations, although, properly speaking, a translation is the geometric transformation resulting from the action of a Euclidean vector on the Euclidean space.

The action of a translation v on a point P provides a point that is denoted P + v. This action satisfies

$${\displaystyle P+(v+w)=(P+v)+w.}$$

Note: The second + in the left-hand side is a vector addition; each other + denotes an action of a vector on a point. This notation is not ambiguous, as, to distinguish between the two meanings of +, it suffices to look at the nature of its left argument.

The fact that the action is free and transitive means that, for every pair of points (P, Q), there is exactly one displacement vector v such that P + v = Q. This vector v is denoted Q − P or ${\displaystyle \overrightarrow{PQ}\phantom{\frac{}{}}.}$

As previously explained, some of the basic properties of Euclidean spaces result from the structure of affine space. They are described in § Affine structure and its subsections. The properties resulting from the inner product are explained in § Metric structure and its subsections.

Prototypical examples

For any vector space, the addition acts freely and transitively on the vector space itself. Thus a Euclidean vector space can be viewed as a Euclidean space that has itself as the associated vector space.

A typical case of Euclidean vector space is ${\displaystyle {\mathbb{R}}^n}$ viewed as a vector space equipped with the dot product as an inner product. The importance of this particular example of Euclidean space lies in the fact that every Euclidean space is isomorphic to it. More precisely, given a Euclidean space E of dimension n, the choice of a point, called an origin and an orthonormal basis of ${\displaystyle \overrightarrow{E}}$ defines an isomorphism of Euclidean spaces from E to ${\displaystyle {\mathbb{R}}^n.}$

As every Euclidean space of dimension n is isomorphic to it, the Euclidean space ${\displaystyle {\mathbb{R}}^n}$ is sometimes called the standard Euclidean space of dimension n.

Affine structure

Main article: Affine space

Some basic properties of Euclidean spaces depend only on the fact that a Euclidean space is an affine space. They are called affine properties and include the concepts of lines, subspaces, and parallelism, which are detailed in next subsections.

Subspaces

Main article: Flat (geometry)

Let E be a Euclidean space and ${\displaystyle \overrightarrow{E}}$ its associated vector space.

A flat, Euclidean subspace or affine subspace of E is a subset F of E such that

$${\displaystyle \overrightarrow{F}=\{\overrightarrow{PQ}|P\in F,Q\in F\}\phantom{\frac{}{}}}$$

as the associated vector space of F is a linear subspace (vector subspace) of ${\displaystyle \overrightarrow{E}.}$ A Euclidean subspace F is a Euclidean space with ${\displaystyle \overrightarrow{F}}$ as the associated vector space. This linear subspace ${\displaystyle \overrightarrow{F}}$ is also called the direction of F.

If P is a point of F then

$${\displaystyle F=\{P+v|v\in \overrightarrow{F}\}.}$$

Conversely, if P is a point of E and ${\displaystyle \overrightarrow{V}}$ is a linear subspace of ${\displaystyle \overrightarrow{E},}$ then

$${\displaystyle P+\overrightarrow{V}=\{P+v|v\in \overrightarrow{V}\}}$$

is a Euclidean subspace of direction ${\displaystyle \overrightarrow{V}}$. (The associated vector space of this subspace is ${\displaystyle \overrightarrow{V}}$.)

A Euclidean vector space ${\displaystyle \overrightarrow{E}}$ (that is, a Euclidean space that is equal to ${\displaystyle \overrightarrow{E}}$) has two sorts of subspaces: its Euclidean subspaces and its linear subspaces. Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector.

Lines and segments

Main article: Line (geometry)

In a Euclidean space, a line is a Euclidean subspace of dimension one. Since a vector space of dimension one is spanned by any nonzero vector, a line is a set of the form

$${\displaystyle \{P+\lambda \overrightarrow{PQ}|\lambda \in \mathbb{R}\},\phantom{\frac{}{}}}$$

where P and Q are two distinct points of the Euclidean space as a part of the line.

It follows that there is exactly one line that passes through (contains) two distinct points. This implies that two distinct lines intersect in at most one point.

A more symmetric representation of the line passing through P and Q is

$${\displaystyle \{O+(1-\lambda )\overrightarrow{OP}+\lambda \overrightarrow{OQ}|\lambda \in \mathbb{R}\},\phantom{\frac{}{}}}$$

where O is an arbitrary point (not necessary on the line).

In a Euclidean vector space, the zero vector is usually chosen for O; this allows simplifying the preceding formula into

$${\displaystyle \{(1-\lambda )P+\lambda Q|\lambda \in \mathbb{R}\}.}$$

A standard convention allows using this formula in every Euclidean space, see Affine space § Affine combinations and barycenter.

The line segment, or simply segment, joining the points P and Q is the subset of points such that 0 ≤ 𝜆 ≤ 1 in the preceding formulas. It is denoted PQ or QP; that is

$${\displaystyle PQ=QP=\{P+\lambda \overrightarrow{PQ}|0\le \lambda \le 1\}.\phantom{\frac{}{}}}$$

Parallelism

Main article: Parallel (geometry)

Two subspaces S and T of the same dimension in a Euclidean space are parallel if they have the same direction (i.e., the same associated vector space). Equivalently, they are parallel, if there is a translation vector v that maps one to the other:

$${\displaystyle T=S+v.}$$

Given a point P and a subspace S, there exists exactly one subspace that contains P and is parallel to S, which is ${\displaystyle P+\overrightarrow{S}.}$ In the case where S is a line (subspace of dimension one), this property is Playfair's axiom.

It follows that in a Euclidean plane, two lines either meet in one point or are parallel.

The concept of parallel subspaces has been extended to subspaces of different dimensions: two subspaces are parallel if the direction of one of them is contained in the direction to the other.

Metric structure

The vector space ${\displaystyle \overrightarrow{E}}$ associated to a Euclidean space E is an inner product space. This implies a symmetric bilinear form

$${\displaystyle \begin{array}{rl}\overrightarrow{E}\times \overrightarrow{E}& \to \mathbb{R}\\ (x,y)& \mapsto ⟨x,y⟩\end{array}}$$

that is positive definite (that is ${\displaystyle ⟨x,x⟩}$ is always positive for x ≠ 0).

The inner product of a Euclidean space is often called dot product and denoted x ⋅ y. This is specially the case when a Cartesian coordinate system has been chosen, as, in this case, the inner product of two vectors is the dot product of their coordinate vectors. For this reason, and for historical reasons, the dot notation is more commonly used than the bracket notation for the inner product of Euclidean spaces. This article will follow this usage; that is ${\displaystyle ⟨x,y⟩}$ will be denoted x ⋅ y in the remainder of this article.

The Euclidean norm of a vector x is

$${\displaystyle \Vert x\Vert =\sqrt{x\cdot x}.}$$

The inner product and the norm allows expressing and proving metric and topological properties of Euclidean geometry. The next subsection describe the most fundamental ones. In these subsections, E denotes an arbitrary Euclidean space, and ${\displaystyle \overrightarrow{E}}$ denotes its vector space of translations.

Distance and length

Main article: Euclidean distance

The distance (more precisely the Euclidean distance) between two points of a Euclidean space is the norm of the translation vector that maps one point to the other; that is

$${\displaystyle d(P,Q)=\Vert \overrightarrow{PQ}\Vert .\phantom{\frac{}{}}}$$

The length of a segment PQ is the distance d(P, Q) between its endpoints P and Q. It is often denoted ${\displaystyle |PQ|}$.

The distance is a metric, as it is positive definite, symmetric, and satisfies the triangle inequality

$${\displaystyle d(P,Q)\le d(P,R)+d(R,Q).}$$

Moreover, the equality is true if and only if a point R belongs to the segment PQ. This inequality means that the length of any edge of a triangle is smaller than the sum of the lengths of the other edges. This is the origin of the term triangle inequality.

With the Euclidean distance, every Euclidean space is a complete metric space.

Orthogonality

Main articles: Perpendicular and Orthogonality

Two nonzero vectors u and v of ${\displaystyle \overrightarrow{E}}$ (the associated vector space of a Euclidean space E) are perpendicular or orthogonal if their inner product is zero:

$${\displaystyle u\cdot v=0}$$

Two linear subspaces of ${\displaystyle \overrightarrow{E}}$ are orthogonal if every nonzero vector of the first one is perpendicular to every nonzero vector of the second one. This implies that the intersection of the linear subspaces is reduced to the zero vector.

Two lines, and more generally two Euclidean subspaces (A line can be considered as one Euclidean subspace.) are orthogonal if their directions (the associated vector spaces of the Euclidean subspaces) are orthogonal. Two orthogonal lines that intersect are said perpendicular.

Two segments AB and AC that share a common endpoint A are perpendicular or form a right angle if the vectors ${\displaystyle \overrightarrow{AB}\phantom{\frac{}{}}}$ and ${\displaystyle \overrightarrow{AC}\phantom{\frac{}{}}}$ are orthogonal.

If AB and AC form a right angle, one has

$${\displaystyle |BC{|}^2=|AB{|}^2+|AC{|}^2.}$$

This is the Pythagorean theorem. Its proof is easy in this context, as, expressing this in terms of the inner product, one has, using bilinearity and symmetry of the inner product:

$${\displaystyle \begin{array}{rl}|BC{|}^2& =\overrightarrow{BC}\cdot \overrightarrow{BC}\phantom{\frac{}{}}\\ & =(\overrightarrow{BA}+\overrightarrow{AC})\cdot (\overrightarrow{BA}+\overrightarrow{AC})\\ & =\overrightarrow{BA}\cdot \overrightarrow{BA}+\overrightarrow{AC}\cdot \overrightarrow{AC}-2\overrightarrow{AB}\cdot \overrightarrow{AC}\\ & =\overrightarrow{AB}\cdot \overrightarrow{AB}+\overrightarrow{AC}\cdot \overrightarrow{AC}\\ & =|AB{|}^2+|AC{|}^2.\end{array}}$$

Here, ${\displaystyle \overrightarrow{AB}\cdot \overrightarrow{AC}=0\phantom{\frac{}{}}}$ is used since these two vectors are orthogonal.

Angle

Main article: Angle

Positive and negative angles on the oriented plane

The (non-oriented) angle θ between two nonzero vectors x and y in ${\displaystyle \overrightarrow{E}}$ is

$${\displaystyle \theta =\arccos \left(\frac{x\cdot y}{|x||y|}\right)}$$

where arccos is the principal value of the arccosine function. By Cauchy–Schwarz inequality, the argument of the arccosine is in the interval [−1, 1]. Therefore θ is real, and 0 ≤ θ ≤ π (or 0 ≤ θ ≤ 180 if angles are measured in degrees).

Angles are not useful in a Euclidean line, as they can be only 0 or π.

In an oriented Euclidean plane, one can define the oriented angle of two vectors. The oriented angle of two vectors x and y is then the opposite of the oriented angle of y and x. In this case, the angle of two vectors can have any value modulo an integer multiple of 2π. In particular, a reflex angle π < θ < 2π equals the negative angle −π < θ − 2π < 0.

The angle of two vectors does not change if they are multiplied by positive numbers. More precisely, if x and y are two vectors, and λ and μ are real numbers, then

$${\displaystyle \mathrm{angle}(\lambda x,\mu y)=\{\begin{array}{l}\mathrm{angle}(x,y)\text{if }\lambda \text{ and }\mu \text{ have the same sign}\\ \pi -\mathrm{angle}(x,y)\text{otherwise}.\end{array}}$$

If A, B, and C are three points in a Euclidean space, the angle of the segments AB and AC is the angle of the vectors ${\displaystyle \overrightarrow{AB}\phantom{\frac{}{}}}$ and ${\displaystyle \overrightarrow{AC}.\phantom{\frac{}{}}}$ As the multiplication of vectors by positive numbers do not change the angle, the angle of two half-lines with initial point A can be defined: it is the angle of the segments AB and AC, where B and C are arbitrary points, one on each half-line. Although this is less used, one can define similarly the angle of segments or half-lines that do not share an initial point.

The angle of two lines is defined as follows. If θ is the angle of two segments, one on each line, the angle of any two other segments, one on each line, is either θ or π − θ. One of these angles is in the interval [0, π/2], and the other being in [π/2, π]. The non-oriented angle of the two lines is the one in the interval [0, π/2]. In an oriented Euclidean plane, the oriented angle of two lines belongs to the interval [−π/2, π/2].

Cartesian coordinates

See also: Cartesian coordinate system

Every Euclidean vector space has an orthonormal basis (in fact, infinitely many in dimension higher than one, and two in dimension one), that is a basis ${\displaystyle (e_1,\dots ,e_n)}$ of unit vectors (${\displaystyle \Vert e_i\Vert =1}$) that are pairwise orthogonal (${\displaystyle e_i\cdot e_j=0}$ for i ≠ j). More precisely, given any basis ${\displaystyle (b_1,\dots ,b_n),}$ the Gram–Schmidt process computes an orthonormal basis such that, for every i, the linear spans of ${\displaystyle (e_1,\dots ,e_i)}$ and ${\displaystyle (b_1,\dots ,b_i)}$ are equal.

Given a Euclidean space E, a Cartesian frame is a set of data consisting of an orthonormal basis of ${\displaystyle \overrightarrow{E},}$ and a point of E, called the origin and often denoted O. A Cartesian frame ${\displaystyle (O,e_1,\dots ,e_n)}$ allows defining Cartesian coordinates for both E and ${\displaystyle \overrightarrow{E}}$ in the following way.

The Cartesian coordinates of a vector v of ${\displaystyle \overrightarrow{E}}$ are the coefficients of v on the orthonormal basis ${\displaystyle e_1,\dots ,e_n.}$ For example, the Cartesian coordinates of a vector ${\displaystyle v}$ on an orthonormal basis ${\displaystyle (e_1,e_2,e_3)}$ (that may be named as ${\displaystyle (x,y,z)}$ as a convention) in a 3-dimensional Euclidean space is ${\displaystyle ({\alpha }_1,{\alpha }_2,{\alpha }_3)}$ if ${\displaystyle v={\alpha }_1{e}_1+{\alpha }_2{e}_2+{\alpha }_3{e}_3}$. As the basis is orthonormal, the i-th coefficient ${\displaystyle {\alpha }_i}$ is equal to the dot product ${\displaystyle v\cdot e_i.}$

The Cartesian coordinates of a point P of E are the Cartesian coordinates of the vector ${\displaystyle \overrightarrow{OP}.\phantom{\frac{}{}}}$

Other coordinates

3-dimensional skew coordinates

Main article: Coordinate system

As a Euclidean space is an affine space, one can consider an affine frame on it, which is the same as a Euclidean frame, except that the basis is not required to be orthonormal. This define affine coordinates, sometimes called skew coordinates for emphasizing that the basis vectors are not pairwise orthogonal.

An affine basis of a Euclidean space of dimension n is a set of n + 1 points that are not contained in a hyperplane. An affine basis define barycentric coordinates for every point.

Many other coordinates systems can be defined on a Euclidean space E of dimension n, in the following way. Let f be a homeomorphism (or, more often, a diffeomorphism) from a dense open subset of E to an open subset of ${\displaystyle {\mathbb{R}}^n.}$ The coordinates of a point x of E are the components of f(x). The polar coordinate system (dimension 2) and the spherical and cylindrical coordinate systems (dimension 3) are defined this way.

For points that are outside the domain of f, coordinates may sometimes be defined as the limit of coordinates of neighbour points, but these coordinates may be not uniquely defined, and may be not continuous in the neighborhood of the point. For example, for the spherical coordinate system, the longitude is not defined at the pole, and on the antimeridian, the longitude passes discontinuously from –180° to +180°.

This way of defining coordinates extends easily to other mathematical structures, and in particular to manifolds.

Isometries

An isometry between two metric spaces is a bijection preserving the distance, that is

$${\displaystyle d(f(x),f(y))=d(x,y).}$$

In the case of a Euclidean vector space, an isometry that maps the origin to the origin preserves the norm

$${\displaystyle \Vert f(x)\Vert =\Vert x\Vert ,}$$

since the norm of a vector is its distance from the zero vector. It preserves also the inner product

$${\displaystyle f(x)\cdot f(y)=x\cdot y,}$$

since

$${\displaystyle x\cdot y=\frac{1}{2}\left(\Vert x+y{\Vert }^2-\Vert x{\Vert }^2-\Vert y{\Vert }^2\right).}$$

An isometry of Euclidean vector spaces is a linear isomorphism.

An isometry ${\displaystyle f:E\to F}$ of Euclidean spaces defines an isometry ${\displaystyle \overrightarrow{f}:\overrightarrow{E}\to \overrightarrow{F}}$ of the associated Euclidean vector spaces. This implies that two isometric Euclidean spaces have the same dimension. Conversely, if E and F are Euclidean spaces, O ∈ E, O′ ∈ F, and ${\displaystyle \overrightarrow{f}:\overrightarrow{E}\to \overrightarrow{F}}$ is an isometry, then the map ${\displaystyle f:E\to F}$ defined by

$${\displaystyle f(P)=O^{\prime }+\overrightarrow{f}(\overrightarrow{OP})\phantom{\frac{}{}}}$$

is an isometry of Euclidean spaces.

It follows from the preceding results that an isometry of Euclidean spaces maps lines to lines, and, more generally Euclidean subspaces to Euclidean subspaces of the same dimension, and that the restriction of the isometry on these subspaces are isometries of these subspaces.

Isometry with prototypical examples

If E is a Euclidean space, its associated vector space ${\displaystyle \overrightarrow{E}}$ can be considered as a Euclidean space. Every point O ∈ E defines an isometry of Euclidean spaces

$${\displaystyle P\mapsto \overrightarrow{OP},\phantom{\frac{}{}}}$$

which maps O to the zero vector and has the identity as associated linear map. The inverse isometry is the map

$${\displaystyle v\mapsto O+v.}$$

A Euclidean frame ⁠${\displaystyle (O,e_1,\dots ,e_n)}$⁠ allows defining the map

$${\displaystyle \begin{array}{rl}E& \to {\mathbb{R}}^n\\ P& \mapsto (e_1\cdot \overrightarrow{OP},\dots ,e_n\cdot \overrightarrow{OP}),\phantom{\frac{}{}}\end{array}}$$

which is an isometry of Euclidean spaces. The inverse isometry is

$${\displaystyle \begin{array}{rl}{\mathbb{R}}^n& \to E\\ (x_1\dots ,x_n)& \mapsto \left(O+x_1{e}_1+\cdots +x_n{e}_n\right).\end{array}}$$

This means that, up to an isomorphism, there is exactly one Euclidean space of a given dimension.

This justifies that many authors talk of ${\displaystyle {\mathbb{R}}^n}$ as the Euclidean space of dimension n.

Euclidean group

Main articles: Euclidean group and Rigid transformation

An isometry from a Euclidean space onto itself is called Euclidean isometry, Euclidean transformation or rigid transformation. The rigid transformations of a Euclidean space form a group (under composition), called the Euclidean group and often denoted E(n) of ISO(n).

The simplest Euclidean transformations are translations

$${\displaystyle P\to P+v.}$$

They are in bijective correspondence with vectors. This is a reason for calling space of translations the vector space associated to a Euclidean space. The translations form a normal subgroup of the Euclidean group.

A Euclidean isometry f of a Euclidean space E defines a linear isometry ${\displaystyle \overrightarrow{f}}$ of the associated vector space (by linear isometry, it is meant an isometry that is also a linear map) in the following way: denoting by Q – P the vector ${\displaystyle \overrightarrow{PQ},\phantom{\frac{}{}}}$ if O is an arbitrary point of E, one has

$${\displaystyle \overrightarrow{f}(\overrightarrow{OP})=f(P)-f(O).\phantom{\frac{}{}}}$$

It is straightforward to prove that this is a linear map that does not depend from the choice of O.

The map ${\displaystyle f\to \overrightarrow{f}}$ is a group homomorphism from the Euclidean group onto the group of linear isometries, called the orthogonal group. The kernel of this homomorphism is the translation group, showing that it is a normal subgroup of the Euclidean group.

The isometries that fix a given point P form the stabilizer subgroup of the Euclidean group with respect to P. The restriction to this stabilizer of above group homomorphism is an isomorphism. So the isometries that fix a given point form a group isomorphic to the orthogonal group.

Let P be a point, f an isometry, and t the translation that maps P to f(P). The isometry ${\displaystyle g=t^{-1}\circ f}$ fixes P. So ${\displaystyle f=t\circ g,}$ and the Euclidean group is the semidirect product of the translation group and the orthogonal group.

The special orthogonal group is the normal subgroup of the orthogonal group that preserves handedness. It is a subgroup of index two of the orthogonal group. Its inverse image by the group homomorphism ${\displaystyle f\to \overrightarrow{f}}$ is a normal subgroup of index two of the Euclidean group, which is called the special Euclidean group or the displacement group. Its elements are called rigid motions or displacements.

Rigid motions include the identity, translations, rotations (the rigid motions that fix at least a point), and also screw motions.

Typical examples of rigid transformations that are not rigid motions are reflections, which are rigid transformations that fix a hyperplane and are not the identity. They are also the transformations consisting in changing the sign of one coordinate over some Euclidean frame.

As the special Euclidean group is a subgroup of index two of the Euclidean group, given a reflection r, every rigid transformation that is not a rigid motion is the product of r and a rigid motion. A glide reflection is an example of a rigid transformation that is not a rigid motion or a reflection.

All groups that have been considered in this section are Lie groups and algebraic groups.

Topology

Main article: Real n-space § Topological properties

The Euclidean distance makes a Euclidean space a metric space, and thus a topological space. This topology is called the Euclidean topology. In the case of ${\displaystyle {\mathbb{R}}^n,}$ this topology is also the product topology.

The open sets are the subsets that contains an open ball around each of their points. In other words, open balls form a base of the topology.

The topological dimension of a Euclidean space equals its dimension. This implies that Euclidean spaces of different dimensions are not homeomorphic. Moreover, the theorem of invariance of domain asserts that a subset of a Euclidean space is open (for the subspace topology) if and only if it is homeomorphic to an open subset of a Euclidean space of the same dimension.

Euclidean spaces are complete and locally compact. That is, a closed subset of a Euclidean space is compact if it is bounded (that is, contained in a ball). In particular, closed balls are compact.

Axiomatic definitions

The definition of Euclidean spaces that has been described in this article differs fundamentally of Euclid's one. In reality, Euclid did not define formally the space, because it was thought as a description of the physical world that exists independently of human mind. The need of a formal definition appeared only at the end of 19th century, with the introduction of non-Euclidean geometries.

Two different approaches have been used. Felix Klein suggested to define geometries through their symmetries. The presentation of Euclidean spaces given in this article, is essentially issued from his Erlangen program, with the emphasis given on the groups of translations and isometries.

On the other hand, David Hilbert proposed a set of axioms, inspired by Euclid's postulates. They belong to synthetic geometry, as they do not involve any definition of real numbers. Later G. D. Birkhoff and Alfred Tarski proposed simpler sets of axioms, which use real numbers (see Birkhoff's axioms and Tarski's axioms).

In Geometric Algebra, Emil Artin has proved that all these definitions of a Euclidean space are equivalent. It is rather easy to prove that all definitions of Euclidean spaces satisfy Hilbert's axioms, and that those involving real numbers (including the above given definition) are equivalent. The difficult part of Artin's proof is the following. In Hilbert's axioms, congruence is an equivalence relation on segments. One can thus define the length of a segment as its equivalence class. One must thus prove that this length satisfies properties that characterize nonnegative real numbers. Artin proved this with axioms equivalent to those of Hilbert.

Usage

Since the ancient Greeks, Euclidean space has been used for modeling shapes in the physical world. It is thus used in many sciences, such as physics, mechanics, and astronomy. It is also widely used in all technical areas that are concerned with shapes, figure, location and position, such as architecture, geodesy, topography, navigation, industrial design, or technical drawing.

Space of dimensions higher than three occurs in several modern theories of physics; see Higher dimension. They occur also in configuration spaces of physical systems.

Beside Euclidean geometry, Euclidean spaces are also widely used in other areas of mathematics. Tangent spaces of differentiable manifolds are Euclidean vector spaces. More generally, a manifold is a space that is locally approximated by Euclidean spaces. Most non-Euclidean geometries can be modeled by a manifold, and embedded in a Euclidean space of higher dimension. For example, an elliptic space can be modeled by an ellipsoid. It is common to represent in a Euclidean space mathematical objects that are a priori not of a geometrical nature. An example among many is the usual representation of graphs.

Other geometric spaces

Since the introduction, at the end of 19th century, of non-Euclidean geometries, many sorts of spaces have been considered, about which one can do geometric reasoning in the same way as with Euclidean spaces. In general, they share some properties with Euclidean spaces, but may also have properties that could appear as rather strange. Some of these spaces use Euclidean geometry for their definition, or can be modeled as subspaces of a Euclidean space of higher dimension. When such a space is defined by geometrical axioms, embedding the space in a Euclidean space is a standard way for proving consistency of its definition, or, more precisely for proving that its theory is consistent, if Euclidean geometry is consistent (which cannot be proved).

Affine space

Main article: Affine space

A Euclidean space is an affine space equipped with a metric. Affine spaces have many other uses in mathematics. In particular, as they are defined over any field, they allow doing geometry in other contexts.

As soon as non-linear questions are considered, it is generally useful to consider affine spaces over the complex numbers as an extension of Euclidean spaces. For example, a circle and a line have always two intersection points (possibly not distinct) in the complex affine space. Therefore, most of algebraic geometry is built in complex affine spaces and affine spaces over algebraically closed fields. The shapes that are studied in algebraic geometry in these affine spaces are therefore called affine algebraic varieties.

Affine spaces over the rational numbers and more generally over algebraic number fields provide a link between (algebraic) geometry and number theory. For example, the Fermat's Last Theorem can be stated "a Fermat curve of degree higher than two has no point in the affine plane over the rationals."

Geometry in affine spaces over a finite fields has also been widely studied. For example, elliptic curves over finite fields are widely used in cryptography.

Projective space

Main article: Projective space

Originally, projective spaces have been introduced by adding "points at infinity" to Euclidean spaces, and, more generally to affine spaces, in order to make true the assertion "two coplanar lines meet in exactly one point". Projective space share with Euclidean and affine spaces the property of being isotropic, that is, there is no property of the space that allows distinguishing between two points or two lines. Therefore, a more isotropic definition is commonly used, which consists as defining a projective space as the set of the vector lines in a vector space of dimension one more.

As for affine spaces, projective spaces are defined over any field, and are fundamental spaces of algebraic geometry.

Non-Euclidean geometries

Main article: Non-Euclidean geometry

Non-Euclidean geometry refers usually to geometrical spaces where the parallel postulate is false. They include elliptic geometry, where the sum of the angles of a triangle is more than 180°, and hyperbolic geometry, where this sum is less than 180°. Their introduction in the second half of 19th century, and the proof that their theory is consistent (if Euclidean geometry is not contradictory) is one of the paradoxes that are at the origin of the foundational crisis in mathematics of the beginning of 20th century, and motivated the systematization of axiomatic theories in mathematics.

Curved spaces

Main article: Curved space

Further information: Manifold and Riemannian manifold

A manifold is a space that in the neighborhood of each point resembles a Euclidean space. In technical terms, a manifold is a topological space, such that each point has a neighborhood that is homeomorphic to an open subset of a Euclidean space. Manifolds can be classified by increasing degree of this "resemblance" into topological manifolds, differentiable manifolds, smooth manifolds, and analytic manifolds. However, none of these types of "resemblance" respect distances and angles, even approximately.

Distances and angles can be defined on a smooth manifold by providing a smoothly varying Euclidean metric on the tangent spaces at the points of the manifold (these tangent spaces are thus Euclidean vector spaces). This results in a Riemannian manifold. Generally, straight lines do not exist in a Riemannian manifold, but their role is played by geodesics, which are the "shortest paths" between two points. This allows defining distances, which are measured along geodesics, and angles between geodesics, which are the angle of their tangents in the tangent space at their intersection. So, Riemannian manifolds behave locally like a Euclidean space that has been bent.

Euclidean spaces are trivially Riemannian manifolds. An example illustrating this well is the surface of a sphere. In this case, geodesics are arcs of great circle, which are called orthodromes in the context of navigation. More generally, the spaces of non-Euclidean geometries can be realized as Riemannian manifolds.

Pseudo-Euclidean space

An inner product of a real vector space is a positive definite bilinear form, and so characterized by a positive definite quadratic form. A pseudo-Euclidean space is an affine space with an associated real vector space equipped with a non-degenerate quadratic form (that may be indefinite).

A fundamental example of such a space is the Minkowski space, which is the space-time of Einstein's special relativity. It is a four-dimensional space, where the metric is defined by the quadratic form

$${\displaystyle x^2+y^2+z^2-t^2,}$$

where the last coordinate (t) is temporal, and the other three (x, y, z) are spatial.

To take gravity into account, general relativity uses a pseudo-Riemannian manifold that has Minkowski spaces as tangent spaces. The curvature of this manifold at a point is a function of the value of the gravitational field at this point.

Bisexual theory

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Bisexual_theory 

 

Bisexual theory is a field of critical theory, inspired by queer theory and bisexual politics, that foregrounds bisexuality as both a theoretical focus and as an epistemology. Bisexual theory emerged most prominently in the 1990s, in response to the burgeoning field of queer theory, and queer studies more broadly, frequently employing similar post-structuralist approaches but redressing queer theory's tendency towards bisexual erasure.

In their critique of the frequent elision of bisexuality in queer theory, Serena Anderlini-D'Onofrio and Jonathan Alexander write, "a queer theory that misses bisexuality's querying of normative sexualities is itself too mastered by the very normative and normalizing binaries it seeks to unsettle".

Scholars who have been discussed in relation to bisexual theory include Ibrahim Abdurrahman Farajajé, Steven Angelides, Elisabeth Däumer, Jo Eadie, Shiri Eisner, Marjorie Garber, Donald E. Hall, Clare Hemmings, Michael du Plessis, Maria Pramaggiore, Merl Storr, and Kenji Yoshino.

History

Bisexual theory emerged in the 1990s, inspired by and responding to the emergence of queer theory. Elisabeth Däumer's 1992 article, "Queer Ethics; or, the Challenge of Bisexuality to Lesbian Ethics", was the first major publication to theorise bisexuality in relation to queer and feminist theory.

In 1993, at the 11th National Bisexual Conference in the UK, a group of bisexual scholars formed Bi Academic Intervention. The same group published a volume of bisexual theory in 1997, entitled The Bisexual Imaginary: Representation, Identity and Desire. In 1995, Marjorie Garber released Vice Versa: Bisexuality and the Eroticism of Everyday Life, a monograph that aimed to reveal a 'bi-erotics' observable across disparate cultural locations, which however, drew some criticism due to its alleged ahistoricism. In 1996, Maria Pramaggiore and Donald E. Hall edited the collection RePresenting Bisexualities: Subjects and Cultures of Fluid Desire, which turned a bisexual theoretical lens to questions of representation. Chapters of bisexual theory also appeared in Activating Theory: Lesbian, Gay Bisexual Politics (1993) and Queer Studies: A Lesbian, Gay, Bisexual, and Transgender Anthology (1996).

The Journal of Bisexuality was first published in 2000 by the Taylor & Francis Group under the Routledge imprint, and its editors-in-chief have included Fritz Klein, Jonathan Alexander, Brian Zamboni, James D. Weinrich, and M. Paz Galupo.

In 2000, law scholar Kenji Yoshino published the influential article "The Epistemic Contract of Bisexual Erasure", which argues that "Straights and gays have an investment in stabilizing sexual orientation categories. The shared aspect of this investment is the security that all individuals draw from rigid social orderings." In 2001, Steven Angelides published A History of Bisexuality, in which he argues that bisexuality has operated historically as a structural other to sexual identity itself. In 2002, Clare Hemmings published Bisexual Spaces: A Geography of Sexuality and Gender in which she explores bisexuality's functions in geographical, political, theoretical, and cultural spaces.

In 2004, Jonathan Alexander and Karen Yescavage co-edited Bisexuality and Transgenderism: InterSEXions of the Others, which considers the intersections of bisexual and transgender identities.

Shiri Eisner's Bi: Notes for a Bisexual Revolution was released in 2013. This radical manifesto combines feminist, transgender, queer, and bisexual activism with theoretical work to establish a blueprint for bisexual revolution.

Epistemologies

One of the ways in which bisexual theorists have deployed bisexuality critically has been the formulation of bisexual epistemologies that ask how bisexuality generates or is given meaning.

Elisabeth Däumer suggests that bisexuality can be "an epistemological as well as ethical vantage point from which we can examine and deconstruct the bipolar framework of gender and sexuality."

Authors like Maria Pramaggiore and Jo Eadie repurposed the idea of bisexual people being "on the fence" in order to theorise an "epistemology of the fence":

a place of in-betweenness and indecision. Often precariously placed atop a structure that divides and demarcates, bisexual epistemologies have the capacity to reframe regimes and regions of desire by defaming and/or reframing in porous, nonexclusive ways... Bisexual epistemologies—ways of apprehending, organizing, and intervening in the world that refuse one-to-one correspondences between sex acts and identity, between erotic objects and sexualities, between identification and desire—acknowledge fluid desires and their continual construction and deconstruction of the desiring subject.

Clare Hemmings outlines three forms that bisexual epistemological approaches have tended to take:

The first locates bisexuality as outside conventional categories of sexuality and gender; the second locates it as critically inside those same categories; and the third focuses on the importance of bisexuality in the discursive formation of "other" identities.

Critiques

In his 1996 article, Jonathan Dollimore observes a trend he terms ‘wishful theory’ in bisexual theoretical work. Dollimore critiques bisexual theory's fabrication of eclectic theoretical narratives with little attention to how they relate to social reality, and its assumption of a subversive position that resists a consideration of how bisexual identity itself might be subverted. Dollimore contends that bisexual theory is "passing, if not closeted, as post-modern theory, safely fashioning itself as a suave doxa."

In her 1999 article, Merl Storr suggests that contemporary bisexual identity, community, organization, and politics are rooted in early postmodernity. By identifying this relation, Storr observes the postmodern themes of indeterminacy, instability, fragmentation, and flux that characterize bisexual theory and parses how these concepts might be reflected upon critically.

One of the problems Clare Hemmings identifies with bisexual epistemological approaches is that bisexuality becomes metaphorized to the point that it is unrecognizable to bisexual people, a critique that has also been made in transgender studies to the allegorization of trans feminine realities.

Genetic drift

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Genetic_drift

Genetic drift, also known as random genetic drift, allelic drift or the Wright effect, is the change in the frequency of an existing gene variant (allele) in a population due to random chance.

Genetic drift may cause gene variants to disappear completely and thereby reduce genetic variation. It can also cause initially rare alleles to become much more frequent and even fixed.

When few copies of an allele exist, the effect of genetic drift is more notable, and when many copies exist, the effect is less notable (due to the law of large numbers). In the middle of the 20th century, vigorous debates occurred over the relative importance of natural selection versus neutral processes, including genetic drift. Ronald Fisher, who explained natural selection using Mendelian genetics, held the view that genetic drift plays at most a minor role in evolution, and this remained the dominant view for several decades. In 1968, population geneticist Motoo Kimura rekindled the debate with his neutral theory of molecular evolution, which claims that most instances where a genetic change spreads across a population (although not necessarily changes in phenotypes) are caused by genetic drift acting on neutral mutations. In the 1990s, constructive neutral evolution was proposed which seeks to explain how complex systems emerge through neutral transitions.

Analogy with marbles in a jar

The process of genetic drift can be illustrated using 20 marbles in a jar to represent 20 organisms in a population. Consider this jar of marbles as the starting population. Half of the marbles in the jar are red and half are blue, with each colour corresponding to a different allele of one gene in the population. In each new generation, the organisms reproduce at random. To represent this reproduction, randomly select a marble from the original jar and deposit a new marble with the same colour into a new jar. This is the "offspring" of the original marble, meaning that the original marble remains in its jar. Repeat this process until 20 new marbles are in the second jar. The second jar will now contain 20 "offspring", or marbles of various colours. Unless the second jar contains exactly 10 red marbles and 10 blue marbles, a random shift has occurred in the allele frequencies.

If this process is repeated a number of times, the numbers of red and blue marbles picked each generation fluctuates. Sometimes, a jar has more red marbles than its "parent" jar and sometimes more blue. This fluctuation is analogous to genetic drift – a change in the population's allele frequency resulting from a random variation in the distribution of alleles from one generation to the next.

In any one generation, no marbles of a particular colour could be chosen, meaning they have no offspring. In this example, if no red marbles are selected, the jar representing the new generation contains only blue offspring. If this happens, the red allele has been lost permanently in the population, while the remaining blue allele has become fixed: all future generations are entirely blue. In small populations, fixation can occur in just a few generations.

In this simulation, each black dot on a marble signifies that it has been chosen for copying (reproduction) one time. Fixation in the blue "allele" occurs within five generations.

Probability and allele frequency

The mechanisms of genetic drift can be illustrated with a very simple example. Consider a very large colony of bacteria isolated in a drop of solution. The bacteria are genetically identical except for a single gene with two alleles labeled A and B, which are neutral alleles, meaning that they do not affect the bacteria's ability to survive and reproduce; all bacteria in this colony are equally likely to survive and reproduce. Suppose that half the bacteria have allele A and the other half have allele B. Thus, A and B each has an allele frequency of 1/2.

The drop of solution then shrinks until it has only enough food to sustain four bacteria. All other bacteria die without reproducing. Among the four that survive, 16 possible combinations for the A and B alleles exist:
(A-A-A-A), (B-A-A-A), (A-B-A-A), (B-B-A-A),
(A-A-B-A), (B-A-B-A), (A-B-B-A), (B-B-B-A),
(A-A-A-B), (B-A-A-B), (A-B-A-B), (B-B-A-B),
(A-A-B-B), (B-A-B-B), (A-B-B-B), (B-B-B-B).

Since all bacteria in the original solution are equally likely to survive when the solution shrinks, the four survivors are a random sample from the original colony. The probability that each of the four survivors has a given allele is 1/2, and so the probability that any particular allele combination occurs when the solution shrinks is

${\displaystyle \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}=\frac{1}{16}.}$

(The original population size is so large that the sampling effectively happens with replacement). In other words, each of the 16 possible allele combinations is equally likely to occur, with probability 1/16.

Counting the combinations with the same number of A and B gives the following table:

A	B	Combinations	Probability
4	0	1	1/16
3	1	4	4/16
2	2	6	6/16
1	3	4	4/16
0	4	1	1/16

As shown in the table, the total number of combinations that have the same number of A alleles as of B alleles is six, and the probability of this combination is 6/16. The total number of other combinations is ten, so the probability of unequal number of A and B alleles is 10/16. Thus, although the original colony began with an equal number of A and B alleles, quite possibly, the number of alleles in the remaining population of four members will not be equal. The situation of equal numbers is actually less likely than unequal numbers. In the latter case, genetic drift has occurred because the population's allele frequencies have changed due to random sampling. In this example, the population contracted to just four random survivors, a phenomenon known as a population bottleneck.

The probabilities for the number of copies of allele A (or B) that survive (given in the last column of the above table) can be calculated directly from the binomial distribution, where the "success" probability (probability of a given allele being present) is 1/2 (i.e., the probability that there are k copies of A (or B) alleles in the combination) is given by:

${\displaystyle (\genfrac{}{}{0ex}{}{n}{k}){\left(\frac{1}{2}\right)}^k{\left(1-\frac{1}{2}\right)}^{n-k}=(\genfrac{}{}{0ex}{}{n}{k}){\left(\frac{1}{2}\right)}^n}$

where n=4 is the number of surviving bacteria.

Mathematical models

Mathematical models of genetic drift can be designed using either branching processes or a diffusion equation describing changes in allele frequency in an idealised population.

Wright–Fisher model

Consider a gene with two alleles, A or B. In diploidy, populations consisting of N individuals have 2N copies of each gene. An individual can have two copies of the same allele or two different alleles. The frequency of one allele is assigned p and the other q. The Wright–Fisher model (named after Sewall Wright and Ronald Fisher) assumes that generations do not overlap (for example, annual plants have exactly one generation per year) and that each copy of the gene found in the new generation is drawn independently at random from all copies of the gene in the old generation. The formula to calculate the probability of obtaining k copies of an allele that had frequency p in the last generation is then

${\displaystyle \frac{(2N)!}{k!(2N-k)!}{p}^k{q}^{2N-k}}$

where the symbol "!" signifies the factorial function. This expression can also be formulated using the binomial coefficient,

${\displaystyle (\genfrac{}{}{0ex}{}{2N}{k})p^k{q}^{2N-k}}$

Moran model

The Moran model assumes overlapping generations. At each time step, one individual is chosen to reproduce and one individual is chosen to die. So in each timestep, the number of copies of a given allele can go up by one, go down by one, or can stay the same. This means that the transition matrix is tridiagonal, which means that mathematical solutions are easier for the Moran model than for the Wright–Fisher model. On the other hand, computer simulations are usually easier to perform using the Wright–Fisher model, because fewer time steps need to be calculated. In the Moran model, it takes N timesteps to get through one generation, where N is the effective population size. In the Wright–Fisher model, it takes just one.

In practice, the Moran and Wright–Fisher models give qualitatively similar results, but genetic drift runs twice as fast in the Moran model.

Other models of drift

If the variance in the number of offspring is much greater than that given by the binomial distribution assumed by the Wright–Fisher model, then given the same overall speed of genetic drift (the variance effective population size), genetic drift is a less powerful force compared to selection. Even for the same variance, if higher moments of the offspring number distribution exceed those of the binomial distribution then again the force of genetic drift is substantially weakened.

Random effects other than sampling error

Random changes in allele frequencies can also be caused by effects other than sampling error, for example random changes in selection pressure.

One important alternative source of stochasticity, perhaps more important than genetic drift, is genetic draft. Genetic draft is the effect on a locus by selection on linked loci. The mathematical properties of genetic draft are different from those of genetic drift. The direction of the random change in allele frequency is autocorrelated across generations.

Drift and fixation

The Hardy–Weinberg principle states that within sufficiently large populations, the allele frequencies remain constant from one generation to the next unless the equilibrium is disturbed by migration, genetic mutations, or selection.

However, in finite populations, no new alleles are gained from the random sampling of alleles passed to the next generation, but the sampling can cause an existing allele to disappear. Because random sampling can remove, but not replace, an allele, and because random declines or increases in allele frequency influence expected allele distributions for the next generation, genetic drift drives a population towards genetic uniformity over time. When an allele reaches a frequency of 1 (100%) it is said to be "fixed" in the population and when an allele reaches a frequency of 0 (0%) it is lost. Smaller populations achieve fixation faster, whereas in the limit of an infinite population, fixation is not achieved. Once an allele becomes fixed, genetic drift comes to a halt, and the allele frequency cannot change unless a new allele is introduced in the population via mutation or gene flow. Thus even while genetic drift is a random, directionless process, it acts to eliminate genetic variation over time.

Rate of allele frequency change due to drift

Ten simulations of random genetic drift of a single given allele with an initial frequency distribution 0.5 measured over the course of 50 generations, repeated in three reproductively synchronous populations of different sizes. In these simulations, alleles drift to loss or fixation (frequency of 0.0 or 1.0) only in the smallest population.

Assuming genetic drift is the only evolutionary force acting on an allele, after t generations in many replicated populations, starting with allele frequencies of p and q, the variance in allele frequency across those populations is

${\displaystyle V_t\approx pq\left(1-\exp \left(-\frac{t}{2N_e}\right)\right)}$

Time to fixation or loss

Assuming genetic drift is the only evolutionary force acting on an allele, at any given time the probability that an allele will eventually become fixed in the population is simply its frequency in the population at that time. For example, if the frequency p for allele A is 75% and the frequency q for allele B is 25%, then given unlimited time the probability A will ultimately become fixed in the population is 75% and the probability that B will become fixed is 25%.

The expected number of generations for fixation to occur is proportional to the population size, such that fixation is predicted to occur much more rapidly in smaller populations. Normally the effective population size, which is smaller than the total population, is used to determine these probabilities. The effective population (N_e) takes into account factors such as the level of inbreeding, the stage of the lifecycle in which the population is the smallest, and the fact that some neutral genes are genetically linked to others that are under selection. The effective population size may not be the same for every gene in the same population.

One forward-looking formula used for approximating the expected time before a neutral allele becomes fixed through genetic drift, according to the Wright–Fisher model, is

${\displaystyle {\overline{T}}_{\text{fixed}}=\frac{-4N_e(1-p)\ln (1-p)}{p}}$

where T is the number of generations, N_e is the effective population size, and p is the initial frequency for the given allele. The result is the number of generations expected to pass before fixation occurs for a given allele in a population with given size (N_e) and allele frequency (p).

The expected time for the neutral allele to be lost through genetic drift can be calculated as

${\displaystyle {\overline{T}}_{\text{lost}}=\frac{-4N_{e}p}{1-p}\ln p.}$

When a mutation appears only once in a population large enough for the initial frequency to be negligible, the formulas can be simplified to

${\displaystyle {\overline{T}}_{\text{fixed}}=4N_e}$

for average number of generations expected before fixation of a neutral mutation, and

${\displaystyle {\overline{T}}_{\text{lost}}=2\left(\frac{N_e}{N}\right)\ln (2N)}$

for the average number of generations expected before the loss of a neutral mutation in a population of actual size N.

Time to loss with both drift and mutation

The formulae above apply to an allele that is already present in a population, and which is subject to neither mutation nor natural selection. If an allele is lost by mutation much more often than it is gained by mutation, then mutation, as well as drift, may influence the time to loss. If the allele prone to mutational loss begins as fixed in the population, and is lost by mutation at rate m per replication, then the expected time in generations until its loss in a haploid population is given by

${\displaystyle {\overline{T}}_{\text{lost}}\approx \{\begin{array}{l}{\displaystyle \frac{1}{m}},\text{ if }m{N}_e\ll 1\\ {\displaystyle \frac{\ln (m{N}_e)+\gamma }{m}}\text{ if }m{N}_e\gg 1\end{array}}$

where ${\displaystyle \gamma }$ is Euler's constant. The first approximation represents the waiting time until the first mutant destined for loss, with loss then occurring relatively rapidly by genetic drift, taking time ⁠1/m⁠ ≫ N_e. The second approximation represents the time needed for deterministic loss by mutation accumulation. In both cases, the time to fixation is dominated by mutation via the term ⁠1/m⁠, and is less affected by the effective population size.

Versus natural selection

In natural populations, genetic drift and natural selection do not act in isolation; both phenomena are always at play, together with mutation and migration. Neutral evolution is the product of both mutation and drift, not of drift alone. Similarly, even when selection overwhelms genetic drift, it can act only on variation that mutation provides.

While natural selection has a direction, guiding evolution towards heritable adaptations to the current environment, genetic drift has no direction and is guided only by the mathematics of chance. As a result, drift acts upon the genotypic frequencies within a population without regard to their phenotypic effects. In contrast, selection favors the spread of alleles whose phenotypic effects increase survival and/or reproduction of their carriers, lowers the frequencies of alleles that cause unfavorable traits, and ignores those that are neutral.

The law of large numbers predicts that when the absolute number of copies of the allele is small (e.g., in small populations), the magnitude of drift on allele frequencies per generation is larger. The magnitude of drift is large enough to overwhelm selection at any allele frequency when the selection coefficient is less than 1 divided by the effective population size. Non-adaptive evolution resulting from the product of mutation and genetic drift is therefore considered to be a consequential mechanism of evolutionary change primarily within small, isolated populations. The mathematics of genetic drift depend on the effective population size, but it is not clear how this is related to the actual number of individuals in a population. Genetic linkage to other genes that are under selection can reduce the effective population size experienced by a neutral allele. With a higher recombination rate, linkage decreases and with it this local effect on effective population size. This effect is visible in molecular data as a correlation between local recombination rate and genetic diversity, and negative correlation between gene density and diversity at noncoding DNA regions. Stochasticity associated with linkage to other genes that are under selection is not the same as sampling error, and is sometimes known as genetic draft in order to distinguish it from genetic drift.

Low allele frequency makes alleles more vulnerable to being eliminated by random chance, even overriding the influence of natural selection. For example, while disadvantageous mutations are usually eliminated quickly within the population, new advantageous mutations are almost as vulnerable to loss through genetic drift as are neutral mutations. Not until the allele frequency for the advantageous mutation reaches a certain threshold will genetic drift have no effect.

Population bottleneck

Main article: Population bottleneck

Changes in a population's allele frequency following a population bottleneck: the rapid and radical decline in population size has reduced the population's genetic variation.

A population bottleneck is when a population contracts to a significantly smaller size over a short period of time due to some random environmental event. In a true population bottleneck, the odds for survival of any member of the population are purely random, and are not improved by any particular inherent genetic advantage. The bottleneck can result in radical changes in allele frequencies, completely independent of selection.

The impact of a population bottleneck can be sustained, even when the bottleneck is caused by a one-time event such as a natural catastrophe. An interesting example of a bottleneck causing unusual genetic distribution is the relatively high proportion of individuals with total rod cell color blindness (achromatopsia) on Pingelap atoll in Micronesia. After a bottleneck, inbreeding increases. This increases the damage done by recessive deleterious mutations, in a process known as inbreeding depression. The worst of these mutations are selected against, leading to the loss of other alleles that are genetically linked to them, in a process of background selection. For recessive harmful mutations, this selection can be enhanced as a consequence of the bottleneck, due to genetic purging. This leads to a further loss of genetic diversity. In addition, a sustained reduction in population size increases the likelihood of further allele fluctuations from drift in generations to come.

A population's genetic variation can be greatly reduced by a bottleneck, and even beneficial adaptations may be permanently eliminated. The loss of variation leaves the surviving population vulnerable to any new selection pressures such as disease, climatic change or shift in the available food source, because adapting in response to environmental changes requires sufficient genetic variation in the population for natural selection to take place.

There have been many known cases of population bottleneck in the recent past. Prior to the arrival of Europeans, North American prairies were habitat for millions of greater prairie chickens. In Illinois alone, their numbers plummeted from about 100 million birds in 1900 to about 50 birds in the 1990s. The declines in population resulted from hunting and habitat destruction, but a consequence has been a loss of most of the species' genetic diversity. DNA analysis comparing birds from the mid century to birds in the 1990s documents a steep decline in the genetic variation in just the latter few decades. Currently the greater prairie chicken is experiencing low reproductive success.

However, the genetic loss caused by bottleneck and genetic drift can increase fitness, as in Ehrlichia.

Over-hunting also caused a severe population bottleneck in the northern elephant seal in the 19th century. Their resulting decline in genetic variation can be deduced by comparing it to that of the southern elephant seal, which were not so aggressively hunted.

Founder effect

Main article: Founder effect

When very few members of a population migrate to form a separate new population, the founder effect occurs. For a period after the foundation, the small population experiences intensive drift. In the figure this results in fixation of the red allele.

The founder effect is a special case of a population bottleneck, occurring when a small group in a population splinters off from the original population and forms a new one. The random sample of alleles in the just formed new colony is expected to grossly misrepresent the original population in at least some respects. It is even possible that the number of alleles for some genes in the original population is larger than the number of gene copies in the founders, making complete representation impossible. When a newly formed colony is small, its founders can strongly affect the population's genetic make-up far into the future.

A well-documented example is found in the Amish migration to Pennsylvania in 1744. Two members of the new colony shared the recessive allele for Ellis–Van Creveld syndrome. Members of the colony and their descendants tend to be religious isolates and remain relatively insular. As a result of many generations of inbreeding, Ellis–Van Creveld syndrome is now much more prevalent among the Amish than in the general population.

The difference in gene frequencies between the original population and colony may also trigger the two groups to diverge significantly over the course of many generations. As the difference, or genetic distance, increases, the two separated populations may become distinct, both genetically and phenetically, although not only genetic drift but also natural selection, gene flow, and mutation contribute to this divergence. This potential for relatively rapid changes in the colony's gene frequency led most scientists to consider the founder effect (and by extension, genetic drift) a significant driving force in the evolution of new species. Sewall Wright was the first to attach this significance to random drift and small, newly isolated populations with his shifting balance theory of speciation. Following after Wright, Ernst Mayr created many persuasive models to show that the decline in genetic variation and small population size following the founder effect were critically important for new species to develop. However, there is much less support for this view today since the hypothesis has been tested repeatedly through experimental research and the results have been equivocal at best.

History

Further information: Modern synthesis (20th century)

The role of random chance in evolution was first outlined by Arend L. Hagedoorn and Anna Cornelia Hagedoorn-Vorstheuvel La Brand in 1921. They highlighted that random survival plays a key role in the loss of variation from populations. Fisher (1922) responded to this with the first, albeit marginally incorrect, mathematical treatment of the "Hagedoorn effect". Notably, he expected that many natural populations were too large (an N ~10,000) for the effects of drift to be substantial and thought drift would have an insignificant effect on the evolutionary process. The corrected mathematical treatment and term "genetic drift" was later coined by a founder of population genetics, Sewall Wright. His first use of the term "drift" was in 1929, though at the time he was using it in the sense of a directed process of change, or natural selection. Random drift by means of sampling error came to be known as the "Sewall–Wright effect", though he was never entirely comfortable to see his name given to it. Wright referred to all changes in allele frequency as either "steady drift" (e.g., selection) or "random drift" (e.g., sampling error). "Drift" came to be adopted as a technical term in the stochastic sense exclusively. Today it is usually defined still more narrowly, in terms of sampling error, although this narrow definition is not universal. Wright wrote that the "restriction of "random drift" or even "drift" to only one component, the effects of accidents of sampling, tends to lead to confusion". Sewall Wright considered the process of random genetic drift by means of sampling error equivalent to that by means of inbreeding, but later work has shown them to be distinct.

In the early days of the modern evolutionary synthesis, scientists were beginning to blend the new science of population genetics with Charles Darwin's theory of natural selection. Within this framework, Wright focused on the effects of inbreeding on small relatively isolated populations. He introduced the concept of an adaptive landscape in which phenomena such as cross breeding and genetic drift in small populations could push them away from adaptive peaks, which in turn allow natural selection to push them towards new adaptive peaks. Wright thought smaller populations were more suited for natural selection because "inbreeding was sufficiently intense to create new interaction systems through random drift but not intense enough to cause random nonadaptive fixation of genes".

Wright's views on the role of genetic drift in the evolutionary scheme were controversial almost from the very beginning. One of the most vociferous and influential critics was colleague Ronald Fisher. Fisher conceded genetic drift played some role in evolution, but an insignificant one. Fisher has been accused of misunderstanding Wright's views because in his criticisms Fisher seemed to argue Wright had rejected selection almost entirely. To Fisher, viewing the process of evolution as a long, steady, adaptive progression was the only way to explain the ever-increasing complexity from simpler forms. But the debates have continued between the "gradualists" and those who lean more toward the Wright model of evolution where selection and drift together play an important role.

In 1968, Motoo Kimura rekindled the debate with his neutral theory of molecular evolution, which claims that most of the genetic changes are caused by genetic drift acting on neutral mutations.

The role of genetic drift by means of sampling error in evolution has been criticized by John H. Gillespie and William B. Provine, who argue that selection on linked sites is a more important stochastic force.