Equatorial electrojet

Bi-dimensional Cartesian coordinate system

Bi-dimensional space is a geometric model of the planar projection of the physical universe in which we live. The two dimensions are commonly called length and width. Both directions lie in the same plane.

In physics and mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 2, the set of all such locations is called 2-dimensional space or bi-dimensional space, and usually is thought of as an Euclidean space.

In physics, our bi-dimensional space is viewed as a planar representation of the space in which we move, described as bi-dimensional space or two-dimensional space.

History of two-dimensional space

Euclid's Elements dealt almost exclusively with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics.

Later, the plane was described in a so-called Cartesian coordinate system, a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed 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, usually 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.

The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions, and did not publish the discovery. Both authors used a single axis in their treatments and have a variable 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 his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.

Later, the plane was thought of as a field, where any two points could be multiplied and, except for 0, divided. This was known as the complex plane. The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768–1822), although they were first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel (1745–1818). Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane.

Two-dimensional geometry

Polytopes

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. In two dimensions, there are infinitely many regular polytopes: the polygons. The first few are shown below:

Convex

The Schläfli symbol {p} represents a regular p-gon.

Name	Triangle (2-simplex)	Square (2-orthoplex) (2-cube)	Pentagon	Hexagon	Heptagon	Octagon
Schläfli	{3}	{4}	{5}	{6}	{7}	{8}
Image	File:Regular triangle.svg	File:Regular quadrilateral.svg	File:Regular pentagon.svg	File:Regular hexagon.svg	File:Regular heptagon.svg	File:Regular octagon.svg
Name	Nonagon	Decagon	Hendecagon	Dodecagon	Triskaidecagon	Tetradecagon
Schläfli	{9}	{10}	{11}	{12}	{13}	{14}
Image	File:Regular nonagon.svg	File:Regular decagon.svg	File:Regular hendecagon.svg	File:Regular dodecagon.svg	File:Regular tridecagon.svg	File:Regular tetradecagon.svg
Name	Pentadecagon	Hexadecagon	Heptadecagon	Octadecagon	Enneadecagon	Icosagon	...n-gon
Schläfli	{15}	{16}	{17}	{18}	{19}	{20}	{n}
Image	File:Regular pentadecagon.svg	File:Regular hexadecagon.svg	File:Regular heptadecagon.svg	File:Regular octadecagon.svg	File:Regular enneadecagon.svg	File:Regular icosagon.svg

Degenerate (spherical)

The regular henagon {1} and regular digon {2} can be considered degenerate regular polygons. They can exist nondegenerately in non-Euclidean spaces like on a 2-sphere or a 2-torus.

Name	Henagon	Digon
Schläfli	{1}	{2}
Image	File:Henagon.svg	File:Digon.svg

Non-convex

There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}. They are called star polygons and share the same vertex arrangements of the convex regular polygons.

In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m} = {n/(n − m)}) and m and n are coprime.

Name	Pentagram	Heptagrams		Octagram	Enneagrams		Decagram	...n-agrams
Schläfli	{5/2}	{7/2}	{7/3}	{8/3}	{9/2}	{9/4}	{10/3}	{n/m}
Image	File:Star polygon 5-2.svg	File:Star polygon 7-2.svg	File:Star polygon 7-3.svg	File:Star polygon 8-3.svg	File:Star polygon 9-2.svg	File:Star polygon 9-4.svg	File:Star polygon 10-3.svg

Hypersphere

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File:CIRCLE 1.svg

The hypersphere in 2 dimensions is a circle, sometimes called a 1-sphere (S¹) because it is an one-dimensional manifold. In a Euclidean plane, it has the length 2πr and the area of its interior is

$A=\pi r^2$

where $r$ is the radius.

Coordinate systems in two-dimensional spaces

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The most popular coordinate systems are the Cartesian coordinate system, the polar coordinate system and the geographic coordinate system.

• 
  Cartesian coordinate system
• 
  Polar coordinate system
• 
  Geographic coordinate system

Topological properties of the plane

In topology, the plane is characterized as being the unique contractible 2-manifold.

Its dimension is characterized by the fact that removing a point from the plane leaves a space that is connected, but not simply connected.

Planar graphs

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

See also

• Three-dimensional space
• Two-dimensional graph