In geometry, topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. Length is the long Dimension of any object The length of a thing is the distance between its ends its linear extent as measured from end to end Thus, a point is a 0-dimensional object. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it Because of their nature as one of the simplest geometric concepts, they are often used in one form or another as the fundamental constituents of geometry, physics, vector graphics, and many other fields. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Vector graphics is the use of geometrical primitives such as points lines, Curves and shapes or Polygon (s which are all based
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Points in Euclidean geometry
Points are most often considered within the framework of Euclidean geometry, where they are one of the fundamental objects. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Euclid originally defined the point vaguely, as "that which has no part". Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry In two dimensional Euclidean space, a point is represented by an ordered pair, (x,y), of numbers, where the first number conventionally represents the horizontal and is often denoted by x, and the second number conventionally represents the vertical and is often denoted by y. In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry A convention is a set of agreed, stipulated or generally accepted Standards norms social norms or criteria, often taking the form of This idea is easily generalized to three dimensional Euclidean space, where a point is represented by an ordered triplet, 
, with the additional third number representing depth and often denoted by z. Further generalizations are represented by an ordered tuplet of n terms,
where n is the dimension of the space in which the point is located.
Many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness This is usually represented by a set of points; As an example, a line is an infinite set of points of the form L={
}, where 
through 
and 
are constants and n is the dimension of the space. Similar constructions exist that define the plane, line segment and other related concepts. 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 end points
In addition to defining points and constructs related to points, Euclid also postulated a key idea about points; he claimed that any two points can be connected by a straight line, this is easily confirmed under modern expansions of Euclidean geometry, and had grave consequences at the time of its introduction, allowing the construction of almost all the geometric concepts of the time. However, Euclid's axiomatization of points was neither complete nor definitive, as he occasionally assumed facts that didn't follow directly from his axioms, such as the ordering of points on the line or the existence of specific points, but in spite of this, modern expansions of the system have since removed these assumptions.
Points in Branches of Mathematics
A point in point-set topology is defined as a member of the underlying set of a topological space. In Mathematics, general topology or point-set topology is the branch of Topology which studies properties of Topological spaces and structures Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.
See Also
External links
- Definition of Point with interactive applet
- Points definition pages With interactive animations that are also useful in a classroom setting. In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated" In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. For a different notion of boundary related to Manifolds see that article Math Open Reference
- Point on PlanetMath
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