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In geometry, **Euclidean space** encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and similar spaces of higher dimension. It is named after the Ancient Greek mathematician Euclid of Alexandria.^{ [1] } The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions.

**Geometry** is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

In mathematics, a *plane* is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point, a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.

**Three-dimensional space** is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension.

- Definition
- History
- Intuitive overview
- Technical definition
- Euclidean structure
- Distance
- Angle
- Rotations and reflections
- Euclidean group
- Non-Cartesian coordinates
- Geometric shapes
- Lines, planes, and other subspaces
- Line segments and triangles
- Polytopes and root systems
- Curves
- Balls, spheres, and hypersurfaces
- Topology
- Applications
- Alternatives and generalizations
- Curved spaces
- Indefinite quadratic form
- Other number fields
- Infinite dimensions
- See also
- Footnotes
- References
- External links

Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions were also used to define rational numbers as ratios of commensurable lengths. When algebra and mathematical analysis became developed enough, this relation reversed and now it is more common to define Euclidean spaces from vector spaces, which allows using Cartesian coordinates and the power of algebra and calculus. This means that points are specified with tuples of real numbers, called coordinate vectors, and geometric shapes are defined by equations and inequalities relating these coordinates. This approach also has the advantage of easily allowing the generalization of geometry to Euclidean spaces of more than three dimensions.

An **axiom** or **postulate** is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek *axíōma* (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'

In mathematics, a **theorem** is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally *deductive*, in contrast to the notion of a scientific law, which is *experimental*.

In mathematics, a **rational number** is any number that can be expressed as the quotient or fraction *p*/*q* of two integers, a numerator *p* and a non-zero denominator *q*. Since *q* may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "**the rationals**", the **field of rationals** or the **field of rational numbers** is usually denoted by a boldface **Q** ; it was thus denoted in 1895 by Giuseppe Peano after *quoziente*, Italian for "quotient".

From the modern viewpoint, there is essentially only one Euclidean space of each dimension. While Euclidean space is defined by a set of axioms, these axioms do not specify how the points are to be represented.^{ [2] } Euclidean space can, as one possible choice of representation, be modeled using Cartesian coordinates. In this case, the Euclidean space is then modeled by the real coordinate space (**R**^{n}) of the same dimension. In one dimension, this is the real line; in two dimensions, it is the Cartesian plane; and in higher dimensions it is a coordinate space with three or more real number coordinates. Mathematicians denote the n-dimensional Euclidean space by **E**^{n} if they wish to emphasize its Euclidean nature, but **R**^{n} is used as well since the latter is assumed to have the standard Euclidean structure, and these two structures are not always distinguished. Euclidean spaces have finite dimension.^{ [3] }

In mathematics, **real coordinate space** of n dimensions, written **R**^{n} is a coordinate space that allows several real variables to be treated as a single variable. With various numbers of dimensions, **R**^{n} is used in many areas of pure and applied mathematics, as well as in physics. With component-wise addition and scalar multiplication, it is the prototypical real vector space and is a frequently used representation of Euclidean *n*-space. Due to the latter fact, geometric metaphors are widely used for **R**^{n}, namely a plane for **R**^{2} and three-dimensional space for **R**^{3}.

In mathematics, the **real line**, or **real number line** is the line whose points are the real numbers. That is, the real line is the set **R** of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space, a metric space, a topological space, a measure space, or a linear continuum.

In mathematics, a **coordinate space** is a space in which an ordered list of coordinates, each from a set, collectively determine an element of the space – in short, a space with a coordinate system.

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 are too basic for being mathematically proved. 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.

**Greek mathematics** refers to mathematics texts and advances written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture and language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word "mathematics" itself derives from the Ancient Greek: μάθημα, romanized: *máthēma*Attic Greek: [má.tʰɛː.ma]Koine Greek: [ˈma.θi.ma], meaning "subject of instruction". The study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations.

The * Elements* is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC. It is a collection of definitions, postulates, propositions, and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines.

**Synthetic geometry** is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compass and straightedge, to draw conclusions and solve problems.

In 1637, René Descartes introduced Cartesian coordinates and showed that this allows reducing geometric problems to algebraic computations with numbers. This reduction of geometry to algebra was a major change of point of view, as, until then, the real numbers—that is, rational numbers and non-rational numbers together–were defined in terms of geometry, as lengths and distance.

**René Descartes** was a French philosopher, mathematician, and scientist. A native of the Kingdom of France, he spent about 20 years (1629–1649) of his life in the Dutch Republic after serving for a while in the Dutch States Army of Maurice of Nassau, Prince of Orange and the Stadtholder of the United Provinces. One of the most notable intellectual figures of the Dutch Golden Age, Descartes is also widely regarded as one of the founders of modern philosophy.

**Algebra** is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

In mathematics, a **real number** is a value of a continuous quantity that can represent a distance along a line. The adjective *real* in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.

Euclidean geometry was not applied in spaces of more than three dimensions until the 19th century. Ludwig Schläfli generalized Euclidean geometry to spaces of *n* dimensions 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 number of dimensions.^{ [4] }

**Ludwig Schläfli** was a Swiss mathematician, specialising in geometry and complex analysis who was one of the key figures in developing the notion of higher-dimensional spaces. The concept of multidimensionality has come to play a pivotal role in physics, and is a common element in science fiction.

In elementary geometry, a **polytope** is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions *n* as an *n*-dimensional polytope or ** n-polytope**. Flat sides mean that the sides of a (

In three-dimensional space, a **Platonic solid** is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex. Five solids meet these criteria:

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.

One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle. 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 about a fixed point in the plane, in which every point in the plane turns about 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 the notions of distance, angle, translation, and rotation for a mathematically described space. 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 define such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product.^{ [3] } The reason for working with arbitrary vector spaces instead of **R**^{n} is that it is often preferable to work in a *coordinate-free* manner (that is, without choosing a preferred basis). For then:

- the vectors in the vector space correspond to the points of the Euclidean plane,
- the addition operation in the vector space corresponds to translation, and
- the inner product implies notions of angle and distance, which can be used to define rotation.

Once the Euclidean plane has been described in this language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. (However, rotations are more subtle in high dimensions, and visualizing high-dimensional spaces remains difficult, even for experienced mathematicians.)

A Euclidean space is not technically a vector space but rather an affine space, on which a vector space acts by translations, or, conversely, a Euclidean vector is the difference (displacement) in an ordered pair of points, not a single point. Intuitively, the distinction says merely that there is no canonical choice of where the origin should go in the space, because it can be translated anywhere. When a certain point is chosen, it can be declared the origin and subsequent calculations may ignore the difference between a point and its coordinate vector, as said above. See point–vector distinction for details.

These are distances between points and the angles between lines or vectors, which satisfy certain conditions (see below), which makes a set of points a Euclidean space. The natural way to obtain these quantities is by introducing and using the standard inner product (also known as the dot product) on **R**^{n}.^{ [3] } The inner product of any two real n-vectors **x** and **y** is defined by

where x_{i} and y_{i} are ith coordinates of vectors **x** and **y** respectively. The result is always a real number.

The inner product of **x** with itself is always non-negative. This product allows us to define the "length" of a vector **x** through square root:

This length function satisfies the required properties of a norm and is called the **Euclidean norm** on **R**^{n}.

Finally, one can use the norm to define a metric (or distance function) on **R**^{n} by

This distance function is called the Euclidean metric. This formula expresses a special case of the Pythagorean theorem.

This distance function (which makes a metric space) is sufficient to define all Euclidean geometry, including the dot product. Thus, a real coordinate space together with this Euclidean structure is called **Euclidean space**. Its vectors form an inner product space (in fact a Hilbert space), and a normed vector space.

The metric space structure is the main reason behind the use of real numbers **R**, not some other ordered field, as the mathematical foundation of Euclidean (and many other) spaces. Euclidean space is a complete metric space, a property which is impossible to achieve operating over rational numbers, for example.

The (**non-reflex**) **angle**θ (0° ≤ *θ* ≤ 180°) between vectors **x** and **y** is then given by

where arccos is the arccosine function. It is useful only for *n* > 1,^{ [footnote 1] } and the case *n* = 2 is somewhat special. Namely, on an oriented Euclidean plane one can define an angle between two vectors as a number defined modulo 1 turn (usually denoted as either 2π or 360°), such that ∠**y** **x** = −∠**x** **y**. This oriented angle is equal either to the angle θ from the formula above or to −*θ*. If one non-zero vector is fixed (such as the first basis vector), then each non-zero vector is uniquely defined by its magnitude and angle.

The angle does not change if vectors **x** and **y** are multiplied by positive numbers.

Unlike the aforementioned situation with distance, the scale of angles is the same in pure mathematics, physics, and computing. It does not depend on the scale of distances; all distances may be multiplied by some fixed factor, and all angles will be preserved. Usually, the angle is considered a dimensionless quantity, but there are different units of measurement, such as radian (preferred in pure mathematics and theoretical physics) and degree (°) (preferred in most applications).

Symmetries of a Euclidean space are transformations which preserve the Euclidean metric (called * isometries *). Although aforementioned translations are most obvious of them, they have the same structure for any affine space and do not show a distinctive character of Euclidean geometry. Another family of symmetries leave one point fixed, which may be seen as the origin without loss of generality. All transformations, which preserves the origin and the Euclidean metric, are linear maps. Such transformations Q must, for any **x** and **y**, satisfy:

- (explain the notation),

Such transforms constitute a group called the * orthogonal group *O(*n*). Its elements Q are exactly solutions of a matrix equation

where Q^{T} is the transpose of Q and *I* is the identity matrix.

But a Euclidean space is orientable.^{ [footnote 2] } Each of these transformations either preserves or reverses orientation depending on whether its determinant is +1 or −1 respectively. Only transformations which preserve orientation, which form the *special orthogonal* group SO(*n*), are considered (proper) rotations. This group has, as a Lie group, the same dimension *n*(*n* − 1) /2 and is the identity component of O(*n*).

Group | Diffeo- morphic to | Isomorphic to |
---|---|---|

SO(1) | {1} | |

SO(2) | S^{1} | U(1) |

SO(3) | RP^{3} | SU(2) / {±1} |

SO(4) | (S^{3} × S^{3}) / {±1} | (SU(2) × SU(2)) / {±1} |

Note: elements of SU(2) are also known as versors. |

Groups SO(*n*) are well-studied for *n* ≤ 4. There are no non-trivial rotations in 0- and 1-spaces. Rotations of a Euclidean plane (*n* = 2) are parametrized by the angle (modulo 1 turn). Rotations of a 3-space are parametrized with axis and angle, whereas a rotation of a 4-space is a superposition of two 2-dimensional rotations around perpendicular planes.

Among linear transforms in O(*n*) which reverse the orientation are hyperplane reflections. This is the only possible case for *n* ≤ 2, but starting from three dimensions, such isometry in the general position is a rotoreflection.

The Euclidean group E(*n*), also referred to as the group of all isometries ISO(*n*), treats translations, rotations, and reflections in a uniform way, considering them as group actions in the context of group theory, and especially in Lie group theory. These group actions preserve the Euclidean structure.

As the group of all isometries, ISO(*n*), the Euclidean group is important because it makes Euclidean geometry a case of Klein geometry, a theoretical framework including many alternative geometries.

The structure of Euclidean spaces – distances, lines, vectors, angles (up to sign), and so on – is invariant under the transformations of their associated Euclidean group. For instance, translations form a commutative subgroup that acts freely and transitively on **E**^{n}, while the stabilizer of any point there is the aforementioned O(*n*).

Along with translations, rotations, reflections, as well as the identity transformation, Euclidean motions comprise also glide reflections (for *n* ≥ 2), screw operations and rotoreflections (for *n* ≥ 3), and even more complex combinations of primitive transformations for *n* ≥ 4.

The group structure determines which conditions a metric space needs to satisfy to be a Euclidean space:

- Firstly, a metric space must be translationally invariant with respect to some (finite-dimensional) real vector space. This means that the space itself is an affine space, that the space is
*flat*, not curved, and points do not have different properties, and so any point can be translated to any other point. - Secondly, the metric must correspond in the aforementioned way to some positive-defined quadratic form on this vector space, because point stabilizers have to be isomorphic to O(
*n*).

Cartesian coordinates are arguably the standard, but not the only possible option for a Euclidean space. Affine coordinates and barycentric coordinates are compatible with the affine structure of **E**^{n}, but make formulae for angles and distances more complicated.

Another approach, which goes in line with ideas of differential geometry and conformal geometry, is orthogonal coordinates, where coordinate hypersurfaces of different coordinates are orthogonal, although curved. Examples include the polar coordinate system on Euclidean plane, the second important plane coordinate system.

See below about expression of the Euclidean structure in curvilinear coordinates.

Polar coordi- nates: see Angle above |

The simplest (after points) objects in Euclidean space are flats, or Euclidean *subspaces* of lesser dimension. Points are 0-dimensional flats, 1-dimensional flats are called * (straight) lines *, and 2-dimensional flats are * planes *. (*n* − 1)-dimensional flats are called * hyperplanes *.

Any two distinct points lie on exactly one line. Any line and a point outside it lie on exactly one plane. More generally, the properties of flats and their incidence of Euclidean space are shared with affine geometry, whereas the affine geometry is devoid of distances and angles.

The sum of angles of a triangle is an important problem, which exerted a great influence on 19th-century mathematics. In a Euclidean space it invariably equals 180°, or a half-turn |

This is not only a line which a pair (*A*, *B*) of distinct points defines. Points on the line which lie between A and B, together with A and B themselves, constitute a line segment *A* *B*. Any line segment has the length, which equals to distance between A and B. If *A* = *B*, then the segment is degenerate and its length equals to 0, otherwise the length is positive.

A (non-degenerate) triangle is defined by three points not lying on the same line. Any triangle lies on one plane. The concept of triangle is not specific to Euclidean spaces, but Euclidean triangles have numerous special properties and define many derived objects.

A triangle can be thought of as a 3-gon on a plane, a special (and the first meaningful in Euclidean geometry) case of a polygon.

The Platonic solids are the five polyhedra that are most regular in a combinatoric sense, but also, their symmetry groups are embedded into O(3) | ||

Pair of dual tetrahedra | Cube and octahedron | Dodecahedron and icosahedron |

Polytope is a concept that generalizes polygons on a plane and polyhedra in 3-dimensional space (which are among the earliest studied geometrical objects). A simplex is a generalization of a line segment (1-simplex) and a triangle (2-simplex). A tetrahedron is a 3-simplex.

The concept of a polytope belongs to affine geometry, which is more general than Euclidean. But Euclidean geometry distinguish * regular polytopes *. For example, affine geometry does not see the difference between an equilateral triangle and a right triangle, but in Euclidean space the former is regular and the latter is not.

Root systems are special sets of Euclidean vectors. A root system is often identical to the set of vertices of a regular polytope.

The root system G _{2} | An orthogonal projection of the 2 _{31} polytope, whose vertices are elements of the E_{7} root system |

Since Euclidean space is a metric space, it is also a topological space with the natural topology induced by the metric. The metric topology on **E**^{n} is called the **Euclidean topology**, and it is identical to the standard topology on **R**^{n}. A set is open if and only if it contains an open ball around each of its points; in other words, open balls form a base of the topology. The topological dimension of the Euclidean n-space equals n, which implies that spaces of different dimension are not homeomorphic. A finer result is the invariance of domain, which proves that any subset of n-space, that is (with its subspace topology) homeomorphic to an open subset of n-space, is itself open.

Aside from countless uses in fundamental mathematics, a Euclidean model of the physical space can be used to solve many practical problems with sufficient precision. Two usual approaches are a fixed, or *stationary* reference frame (i.e. the description of a motion of objects as their positions that change continuously with time), and the use of Galilean space-time symmetry (such as in Newtonian mechanics). To both of them the modern Euclidean geometry provides a convenient formalism; for example, the space of Galilean velocities is itself a Euclidean space (see relative velocity for details).

Topographical maps and technical drawings are planar Euclidean. An idea behind them is the scale invariance of Euclidean geometry, that permits to represent large objects in a small sheet of paper, or a screen.

Although Euclidean spaces are no longer considered to be the only possible setting for a geometry, they act as prototypes for other geometric objects. Ideas and terminology from Euclidean geometry (both traditional and analytic) are pervasive in modern mathematics, where other geometric objects share many similarities with Euclidean spaces, share part of their structure, or embed Euclidean spaces.

A smooth manifold is a Hausdorff topological space that is locally diffeomorphic to Euclidean space. Diffeomorphism does not respect distance and angle, but if one additionally prescribes a smoothly varying inner product on the manifold's tangent spaces, then the result is what is called a Riemannian manifold. Put differently, a Riemannian manifold is a space constructed by deforming and patching together Euclidean spaces. Such a space enjoys notions of distance and angle, but they behave in a curved, non-Euclidean manner. The simplest Riemannian manifold, consisting of **R**^{n} with a constant inner product, is essentially identical to Euclidean n-space itself. Less trivial examples are n-sphere and hyperbolic spaces. Discovery of the latter in the 19th century was branded as the non-Euclidean geometry.

Also, the concept of a Riemannian manifold permits an expression of the Euclidean structure in any smooth coordinate system, via metric tensor. From this tensor one can compute the Riemann curvature tensor. Where the latter equals to zero, the metric structure is locally Euclidean (it means that at least some open set in the coordinate space is isometric to a piece of Euclidean space), no matter whether coordinates are affine or curvilinear.

If one replaces the inner product of a Euclidean space with an indefinite quadratic form, the result is a pseudo-Euclidean space. Smooth manifolds built from such spaces are called pseudo-Riemannian manifolds. Perhaps their most famous application is the theory of relativity, where flat spacetime is a pseudo-Euclidean space called Minkowski space, where rotations correspond to motions of hyperbolic spaces mentioned above. Further generalization to curved spacetimes form pseudo-Riemannian manifolds, such as in general relativity.

Another line of generalization is to consider other number fields than one of real numbers. Over complex numbers, a Hilbert space can be seen as a generalization of Euclidean dot product structure, although the definition of the inner product becomes a sesquilinear form for compatibility with metric structure.

- Function of several real variables, a coordinate presentation of a function on a Euclidean space
- Geometric algebra, an alternative algebraic formalism
- High-dimensional space
- Real coordinate space, a frequently used representation of Euclidean space
- Vector calculus, a standard algebraic formalism
- Vector space

- ↑ On the real line (
*n*= 1) any two non-zero vectors are either parallel or antiparallel depending on whether their signs match or oppose. There are no angles between 0 and 180°. - ↑ It is
**R**^{n}which is orient**ed**because of the ordering of elements of the standard basis. Although an orientation is not an attribute of the Euclidean structure, there are only two possible orientations, and any linear automorphism either keeps orientation or reverses (swaps the two).

A **Cartesian coordinate system** is a coordinate system that specifies each point uniquely in a plane by a set of numerical **coordinates**, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a *coordinate axis* or just *axis* of the system, and the point where they meet is its *origin*, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

In mathematics, the **Euclidean distance** or **Euclidean metric** is the "ordinary" straight-line distance between two points in Euclidean space. With this distance, Euclidean space becomes a metric space. The associated norm is called the **Euclidean norm.** Older literature refers to the metric as the **Pythagorean metric**. A generalized term for the Euclidean norm is the **L ^{2} norm** or L

In differential geometry, a (**smooth**) **Riemannian manifold** or (**smooth**) **Riemannian space**(*M*, *g*) is a real, smooth manifold *M* equipped with an inner product *g*_{p} on the tangent space *T*_{p}*M* at each point *p* that varies smoothly from point to point in the sense that if *X* and *Y* are differentiable vector fields on *M*, then *p* ↦ *g*_{p}(*X*|_{p}, *Y*|_{p}) is a smooth function. The family *g*_{p} of inner products is called a Riemannian metric. These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.

In mathematical physics, **Minkowski space** is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be an immediate consequence of the postulates of special relativity.

In mathematics, **affine geometry** is what remains of Euclidean geometry when not using the metric notions of distance and angle.

In mathematics, an **isometry** is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

In mathematics, **conformal geometry** is the study of the set of angle-preserving (conformal) transformations on a space.

In mathematics, an **affine space** is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

**Rotation** in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (*n* − 1)-dimensional flat of fixed points in a n-dimensional space. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude.

In the branch of mathematics called differential geometry, an **affine connection** is a geometric object on a smooth manifold which *connects* nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan and Hermann Weyl. The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space **R**^{n} by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an *n*-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension *n*. In this more precise terminology, a manifold is referred to as an ** n-manifold**.

In mathematics, a **differentiable manifold** is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

Affine geometry, broadly speaking, is the study of the geometrical properties of lines, planes, and their higher dimensional analogs, in which a notion of "parallel" is retained, but no metrical notions of distance or angle are. Affine spaces differ from linear spaces in that they do not have a distinguished choice of origin. So, in the words of Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps." Accordingly, a **complex affine space**, that is an affine space over the complex numbers, is like a complex vector space, but without a distinguished point to serve as the origin.

In mathematics, a **space** is a set with some added structure.

In mathematics and theoretical physics, a **pseudo-Euclidean space** is a finite-dimensional real *n*-space together with a non-degenerate quadratic form *q*. Such a quadratic form can, given a suitable choice of basis (*e*_{1}, ..., *e*_{n}), be applied to a vector *x* = *x*_{1}*e*_{1} + ... + *x*_{n}*e*_{n}, giving

In geometry, a **point reflection** or **inversion in a point** is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess **point symmetry**; if it is invariant under point reflection through its center, it is said to possess **central symmetry** or to be **centrally symmetric.**

**Two-dimensional space** is a geometric setting in which two values are required to determine the position of an element. In mathematics, it is commonly represented by the symbol ℝ^{2}. For a generalization of the concept, see dimension.

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- ↑ Ball, W.W. Rouse (1960) [1908].
*A Short Account of the History of Mathematics*(4th ed.). Dover Publications. pp. 50–62. ISBN 0-486-20630-0. - ↑ Gabi, Aalex. "What is the difference between Euclidean and Cartesian spaces?".
*Mathematics Stack Exchange*. Mathematics Stack Exchange. - 1 2 3 E.D. Solomentsev (7 February 2011). "Euclidean space".
*Encyclopedia of Mathematics*. Springer. Retrieved 1 May 2014. - ↑ Coxeter, H.S.M. (1973) [1948].
*Regular Polytopes*(3rd ed.). New York: Dover. pp. 141–144.Schläfli ... discovered them before 1853 -- a time when Cayley, Grassman and Möbius were the only other people who had ever conceived of the possibility of geometry in more than three dimensions.

- Hazewinkel, Michiel, ed. (2001) [1994], "Euclidean space",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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