1.
Dimension
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In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one only one coordinate is needed to specify a point on it – for example. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces, in classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, Minkowski space first approximates the universe without gravity, the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to string theory, and the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects, high-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics, in mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n, when trying to generalize to other types of spaces, one is faced with the question what makes En n-dimensional. One answer is that to cover a ball in En by small balls of radius ε. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, for example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces, a tesseract is an example of a four-dimensional object. The rest of this section some of the more important mathematical definitions of the dimensions. A complex number has a real part x and an imaginary part y, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, complex dimensions appear in the study of complex manifolds and algebraic varieties. The dimension of a space is the number of vectors in any basis for the space. This notion of dimension is referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension

2.
Abscissa and ordinate
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In mathematics, an abscissa is the number whose absolute value is the perpendicular distance of a point from the vertical axis. Usually this is the coordinate of a point in a two-dimensional rectangular Cartesian coordinate system. The term can refer to the horizontal axis of a two-dimensional graph. An ordered pair consists of two terms—the abscissa and the ordinate —which define the location of a point in two-dimensional rectangular space and we know no earlier use of the word abscissa in Latin originals. Maybe the word descends from translations of the Apollonian conics, where in Book I, Chapter 20 there appears ἀποτεμνομέναις, for which there would hardly be as an appropriate Latin word as abscissa. In a somewhat obsolete variant usage, the abscissa of a point may refer to any number that describes the points location along some path. Used in this way, the abscissa can be thought of as an analog to the independent variable in a mathematical model or experiment. For the point,2 is called the abscissa and 3 the ordinate, for the point, −1.5 is called the abscissa and −2.5 the ordinate.3 or later. The dictionary definition of abscissa at Wiktionary

3.
Cartesian coordinate system
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half

4.
Degrees of freedom (physics and chemistry)
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In physics, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all dimensions of a system is known as a phase space, a degree of freedom of a physical system is an independent parameter that is necessary to characterize the state of a physical system. In general, a degree of freedom may be any property that is not dependent on other variables. The location of a particle in space requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. On the other hand, a system with an object that can rotate or vibrate can have more than six degrees of freedom. In statistical mechanics, a degree of freedom is a scalar number describing the microstate of a system. The specification of all microstates of a system is a point in the phase space. In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer and it is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a function to the energy of the system. In three-dimensional space, three degrees of freedom are associated with the movement of a particle, a diatomic gas molecule has 7 degrees of freedom. This set may be decomposed in terms of translations, rotations, the center of mass motion of the entire molecule accounts for 3 degrees of freedom. In addition, the molecule has two degrees of motion and two vibrational modes. The rotations occur around the two axes perpendicular to the line between the two atoms, the rotation around the atom–atom bond is not a physical rotation. This yields, for a molecule, a decomposition of. In special cases, such as adsorbed large molecules, the degrees of freedom can be limited to only one. As defined above one can also count degrees of freedom using the number of coordinates required to specify a position. This is done as follows, For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space, thus its degree of freedom in a 3-D space is 3