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Euclidean space

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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. [1]

191 relations: Abelian group, Abstraction, Addition, Affine geometry, Affine space, Algebra, Analytic geometry, Antiparallel (mathematics), Axiom, Axis–angle representation, Ball (mathematics), Base (topology), Calculus, Canonical form, Cartesian coordinate system, Cartesian product, Circle, Circle group, Classical mechanics, Complete metric space, Complex number, Conformal geometry, Congruence (geometry), Continuous function, Coordinate space, Coordinate system, Coordinate vector, Covering space, Cube, Curvature, Curved space, Curvilinear coordinates, Degeneracy (mathematics), Degree (angle), Diffeomorphism, Differentiable manifold, Differential geometry, Dimension, Dimensional analysis, Dimensionless quantity, Displacement (vector), Dodecahedron, Dot product, E7 (mathematics), Equation, Equilateral triangle, Euclid, Euclidean distance, Euclidean geometry, Euclidean vector, ..., Field (mathematics), Flat (geometry), Frame of reference, Function composition, Function of several real variables, G2 (mathematics), Galilean invariance, General position, General relativity, Geometric algebra, Geometric shape, Geometry, Glide reflection, Greek mathematics, Group (mathematics), Group action, Group theory, Hausdorff space, Hilbert space, Homeomorphism, Hyperbolic space, Hyperplane, Icosahedron, Identity component, Identity function, Identity matrix, If and only if, Improper rotation, Incidence (geometry), Inequality (mathematics), Inner product space, Invariance of domain, Invariant (mathematics), Inverse trigonometric functions, Isometry, Klein geometry, Law of cosines, Lebesgue covering dimension, Length, Lie group, Line (geometry), Line segment, Linear equation, Linear map, Mathematical analysis, Mathematical structure, Matrix (mathematics), Metric (mathematics), Metric space, Metric tensor, Minkowski space, Modular arithmetic, Motion (geometry), Motion (physics), N-sphere, Natural topology, Non-Euclidean geometry, Norm (mathematics), Normed vector space, Number, Octahedron, Open set, Ordered field, Orientation (vector space), Origin (mathematics), Orthogonal coordinates, Orthogonal group, Orthogonality, Physics, Pi, Plane (geometry), Platonic solid, Point (geometry), Polar coordinate system, Polygon, Polyhedron, Position (vector), Pseudo-Euclidean space, Pseudo-Riemannian manifold, Pythagorean theorem, Quadratic form, Radian, Rational number, Real coordinate space, Real line, Real number, Real projective space, Reflection (mathematics), Regular polytope, Relative velocity, Riemann curvature tensor, Riemannian manifold, Right triangle, Rotation (mathematics), Rotation group SO(3), Rotations in 4-dimensional Euclidean space, Scalar multiplication, Scale invariance, Screw axis, Sesquilinear form, Set (mathematics), Sign (mathematics), Simplex, Skew coordinates, Space, Spacetime, Special unitary group, Square (algebra), Square root, Standard basis, Subgroup, Subset, Subspace topology, Subtraction, Sum of angles of a triangle, Tangent space, Technical drawing, Tetrahedron, Theorem, Theory of relativity, Three-dimensional space, Time, Topographic map, Topological space, Total order, Translation (geometry), Transpose, Triangle, Triangle inequality, Trivial group, Turn (geometry), Two-dimensional space, Unit of length, Up to, Vector calculus, Vector space, Velocity, Versor, 2 31 polytope, 3, 3-sphere. Expand index (141 more) »

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

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Abstraction

Abstraction in its main sense is a conceptual process where general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods.

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Addition

Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.

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Affine geometry

In mathematics, affine geometry is what remains of Euclidean geometry when not using (mathematicians often say "when forgetting") the metric notions of distance and angle.

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Affine space

In mathematics, an affine space is a geometric structure that generalizes 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.

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Algebra

Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.

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Analytic geometry

In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system.

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Antiparallel (mathematics)

In geometry, anti-parallel lines can be defined with respect to either lines or angles.

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Axiom

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.

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Axis–angle representation

In mathematics, the axis–angle representation of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector indicating the direction of an axis of rotation, and an angle describing the magnitude of the rotation about the axis.

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Ball (mathematics)

In mathematics, a ball is the space bounded by a sphere.

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Base (topology)

In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.We are using a convention that the union of empty collection of sets is the empty set.

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Calculus

Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

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Canonical form

In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression.

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Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

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Cartesian product

In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.

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Circle

A circle is a simple closed shape.

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Circle group

In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers.

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Classical mechanics

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.

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Complete metric space

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).

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Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

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Conformal geometry

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

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Congruence (geometry)

In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.

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Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

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Coordinate space

In mathematics, a coordinate space is a space in which an ordered list of coordinates, each from a set (not necessarily the same set), collectively determine an element (or point) of the space – in short, a space with a coordinate system.

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Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

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Coordinate vector

In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis.

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Covering space

In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.

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Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

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Curvature

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.

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Curved space

Curved space often refers to a spatial geometry which is not "flat" where a flat space is described by Euclidean geometry.

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Curvilinear coordinates

In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved.

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Degeneracy (mathematics)

In mathematics, a degenerate case is a limiting case in which an element of a class of objects is qualitatively different from the rest of the class and hence belongs to another, usually simpler, class.

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Degree (angle)

A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle, defined so that a full rotation is 360 degrees.

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Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.

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Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

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Differential geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.

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Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

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Dimensional analysis

In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed.

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Dimensionless quantity

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned.

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Displacement (vector)

A displacement is a vector whose length is the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a straight line from the initial position to the final position of the point.

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Dodecahedron

In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces.

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Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

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E7 (mathematics)

In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7.

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Equation

In mathematics, an equation is a statement of an equality containing one or more variables.

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Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal.

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Euclid

Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".

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Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.

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Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

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Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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Flat (geometry)

In geometry, a flat is a subset of n-dimensional space that is congruent to a Euclidean space of lower dimension.

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Frame of reference

In physics, a frame of reference (or reference frame) consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements.

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Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

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Function of several real variables

In mathematical analysis, and applications in geometry, applied mathematics, engineering, natural sciences, and economics, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables.

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G2 (mathematics)

In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras \mathfrak_2, as well as some algebraic groups.

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Galilean invariance

Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames.

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General position

In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects.

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General relativity

General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.

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Geometric algebra

The geometric algebra (GA) of a vector space is an algebra over a field, noted for its multiplication operation called the geometric product on a space of elements called multivectors, which is a superset of both the scalars F and the vector space V. Mathematically, a geometric algebra may be defined as the Clifford algebra of a vector space with a quadratic form.

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Geometric shape

A geometric shape is the geometric information which remains when location, scale, orientation and reflection are removed from the description of a geometric object.

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Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

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Glide reflection

In 2-dimensional geometry, a glide reflection (or transflection) is a type of opposite isometry of the Euclidean plane: the composition of a reflection in a line and a translation along that line.

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Greek mathematics

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.

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Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

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Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

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Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

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Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.

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Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

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Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

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Hyperbolic space

In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.

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Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.

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Icosahedron

In geometry, an icosahedron is a polyhedron with 20 faces.

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Identity component

In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group.

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Identity function

Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.

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Identity matrix

In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.

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If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

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Improper rotation

In geometry, an improper rotation,.

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Incidence (geometry)

In geometry, an incidence relation is a binary relation between different types of objects that captures the idea being expressed when phrases such as "a point lies on a line" or "a line is contained in a plane" are used.

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Inequality (mathematics)

In mathematics, an inequality is a relation that holds between two values when they are different (see also: equality).

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Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

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Invariance of domain

Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn.

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Invariant (mathematics)

In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects.

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Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains).

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Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

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Klein geometry

In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program.

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Law of cosines

In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles.

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Lebesgue covering dimension

In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way.

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Length

In geometric measurements, length is the most extended dimension of an object.

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Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

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Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

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Line segment

In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.

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Linear equation

In mathematics, a linear equation is an equation that may be put in the form where x_1, \ldots, x_n are the variables or unknowns, and c, a_1, \ldots, a_n are coefficients, which are often real numbers, but may be parameters, or even any expression that does not contain the unknowns.

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Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

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Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

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Mathematical structure

In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.

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Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

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Metric (mathematics)

In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.

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Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

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Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.

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Minkowski space

In mathematical physics, Minkowski space (or Minkowski spacetime) is a combining 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.

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Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).

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Motion (geometry)

In geometry, a motion is an isometry of a metric space.

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Motion (physics)

In physics, motion is a change in position of an object over time.

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N-sphere

In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension.

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Natural topology

In any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question.

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Non-Euclidean geometry

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.

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Norm (mathematics)

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.

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Normed vector space

In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined.

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Number

A number is a mathematical object used to count, measure and also label.

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Octahedron

In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.

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Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

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Ordered field

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations.

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Orientation (vector space)

In mathematics, orientation is a geometric notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed.

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Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space.

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Orthogonal coordinates

In mathematics, orthogonal coordinates are defined as a set of d coordinates q.

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Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

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Orthogonality

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

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Physics

Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

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Pi

The number is a mathematical constant.

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Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.

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Platonic solid

In three-dimensional space, a Platonic solid is a regular, convex polyhedron.

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Point (geometry)

In modern mathematics, a point refers usually to an element of some set called a space.

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Polar coordinate system

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

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Polygon

In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit.

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Polyhedron

In geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices.

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Position (vector)

In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point P in space in relation to an arbitrary reference origin O. Usually denoted x, r, or s, it corresponds to the straight-line from O to P. The term "position vector" is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus.

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Pseudo-Euclidean space

In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional ''n''-space together with a non-degenerate quadratic form.

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Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold in which the metric tensor need not be positive-definite, but need only be a non-degenerate bilinear form, which is a weaker condition.

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Pythagorean theorem

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.

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Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.

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Radian

The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics.

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Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

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Real coordinate space

In mathematics, real coordinate space of dimensions, written R (also written with blackboard bold) is a coordinate space that allows several (''n'') real variables to be treated as a single variable.

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Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Real projective space

In mathematics, real projective space, or RPn or \mathbb_n(\mathbb), is the topological space of lines passing through the origin 0 in Rn+1.

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Reflection (mathematics)

In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.

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Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.

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Relative velocity

The relative velocity \vec_ (also \vec_ or \vec_) is the velocity of an object or observer B in the rest frame of another object or observer A.

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Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds.

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Riemannian manifold

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_pM 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 \mapsto g_p(X(p),Y(p)) is a smooth function.

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Right triangle

A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle).

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Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry.

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Rotation group SO(3)

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.

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Rotations in 4-dimensional Euclidean space

In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4).

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Scalar multiplication

In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra).

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Scale invariance

In physics, mathematics, statistics, and economics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, thus represent a universality.

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Screw axis

A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs.

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Sesquilinear form

In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space.

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Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

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Sign (mathematics)

In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative.

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Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

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Skew coordinates

A system of skew coordinates is a curvilinear coordinate system where the coordinate surfaces are not orthogonal, in contrast to orthogonal coordinates.

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Space

Space is the boundless three-dimensional extent in which objects and events have relative position and direction.

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Spacetime

In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.

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Special unitary group

In mathematics, the special unitary group of degree, denoted, is the Lie group of unitary matrices with determinant 1.

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Square (algebra)

In mathematics, a square is the result of multiplying a number by itself.

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Square root

In mathematics, a square root of a number a is a number y such that; in other words, a number y whose square (the result of multiplying the number by itself, or) is a. For example, 4 and −4 are square roots of 16 because.

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Standard basis

In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.

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Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

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Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

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Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).

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Subtraction

Subtraction is an arithmetic operation that represents the operation of removing objects from a collection.

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Sum of angles of a triangle

In several geometries, a triangle has three vertices and three sides, where three angles of a triangle are formed at each vertex by a pair of adjacent sides.

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Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.

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Technical drawing

Technical drawing, drafting or drawing, is the act and discipline of composing drawings that visually communicate how something functions or is constructed.

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Tetrahedron

In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.

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Theorem

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.

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Theory of relativity

The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity.

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Three-dimensional space

Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point).

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Time

Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future.

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Topographic map

In modern mapping, a topographic map is a type of map characterized by large-scale detail and quantitative representation of relief, usually using contour lines, but historically using a variety of methods.

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Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

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Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.

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Translation (geometry)

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.

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Transpose

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At).

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Triangle

A triangle is a polygon with three edges and three vertices.

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Triangle inequality

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

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Trivial group

In mathematics, a trivial group is a group consisting of a single element.

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Turn (geometry)

A turn is a unit of plane angle measurement equal to 2pi radians, 360 degrees or 400 gradians.

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Two-dimensional space

Two-dimensional space or bi-dimensional space is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point).

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Unit of length

A unit of length refers to any discrete, pre-established length or distance having a constant magnitude which is used as a reference or convention to express linear dimension.

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Up to

In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.

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Vector calculus

Vector calculus, or vector analysis, is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3.

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Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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Velocity

The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time.

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Versor

In mathematics, a versor is a quaternion of norm one (a unit quaternion).

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2 31 polytope

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.

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3

3 (three) is a number, numeral, and glyph.

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3-sphere

In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere.

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Euclidean Space, Euclidean manifold, Euclidean n-space, Euclidean space as a manifold, Euclidean spaces, Euclidean vector space, Euclidian space, Finite dimensional Euclidean space, N-dimensional Euclidean space.

References

[1] https://en.wikipedia.org/wiki/Euclidean_space

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