A spatial point is an entity with a location in space but no extent (volume, area or length). In geometry, a point therefore captures the notion of location; no further information is captured. Points are used in the basic language of geometry, physics, vector graphics (both 2d and 3d), and many other fields. In mathematics generally, particularly in topology, any form of space is considered as made up of points as basic elements. Volume, also called capacity, is a quantification of how much space an object occupies. ... Area is a physical quantity expressing the size of a part of a surface. ... Table of Geometry, from the 1728 Cyclopaedia. ... A Superconductor demonstrating the Meissner Effect. ... Steam Locomotive 7646 as a vector, originally Windows Metafile (converted to GIF for display here). ... Euclid, detail from The School of Athens by Raphael. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...

Points in Euclidean geometry

A point in Euclidean geometry has no size, orientation, or any other feature except position. Euclid's axioms or postulates assert in some cases that points exist: for example, they assert that if two lines on a plane are not parallel, there is exactly one point that lies on both of them. Euclid sometimes implicitly assumed facts that did not follow from the axioms (for example about the ordering of points on lines, and occasionally about the existence of points distinct from a finite list of points). Therefore the traditional axiomatization of point was not entirely complete and definitive. Euclid Euclidean geometry is a mathematical system due to the Hellenistic mathematician Euclid of Egypt. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... For the algebra software named Axiom, see Axiom computer algebra system. ... Two intersecting planes in R3 In mathematics, a plane is a fundamental two-dimensional object. ... Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ... In mathematics, axiomatization is the process of defining the basic axiomatic systems from which mathematical theories can be derived. ...

Points in Cartesian geometry

Intuitively one can understand a location in the Cartesian 3D space. This location could be described with three real number coordinates: for instance Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...

P = (2, 6, 9).

But one can also describe points in 1, 2 or more than 3 dimensions. The description of a point is quite similar to the description of a spatial vector, which also can exist in space with dimensions from one to many. In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ...

The conceptual difference between these notions is significant, though: a point indicates a location, while a vector indicates a direction and length. If a distinguished point (the origin) is given, one can describe a location by giving the direction and distance from the origin to that point. In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ...

One could argue that in this world it makes no sense to say that a point is in a one or two dimensional space, because we experience space in 3 dimensions, where one or two dimensions exists within this space, thus forcing 1d and 2d points to actually be 3d points. This way one could say that the only real spatial points are 3d points. And one could also argue that by giving more than 3 coordinates one starts to describe features which are not related to space (how would you describe the fourth dimension in spatial terms?) This is really a question about what we mean by space. The word space has many meanings, including: Physics The definition of space in physics is contentious. ...