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[[Image:Cartesian-coordinate-system.svg|right|thumb|300px|Bi-dimensional Cartesian coordinate system]]
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'''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 (geometry)|plane]].
 
In [[physics]] and [[mathematics]], a [[real coordinate space|sequence of ''n'' real numbers]] can be understood as a [[point (geometry)|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 [[area]]s, 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 (geometry)|point]] uniquely in a [[plane (geometry)|plane]] by a pair of [[number|numerical]] '''coordinates''', which are the [[positive and negative numbers|signed]] distances from the point to two fixed [[perpendicular]] directed lines, measured in the same [[unit length|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 (mathematics)|origin]]'', usually at ordered pair (0,&#8239;0). The coordinates can also be defined as the positions of the [[orthogonal projection|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.<ref>{{cite encyclopedia|title=analytic geometry|encyclopedia=Encyclopædia Britannica|year=2008|edition=Encyclopædia Britannica Online|accessdate=2 August 2008}}</ref> 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.<ref>{{harvnb|Burton|2011|loc=p. 374}}</ref>
 
Later, the plane was thought of as a [[field (mathematics) |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).<ref>Wessel's memoir was presented to the Danish Academy in 1797; Argand's paper was published in 1806. (Whittaker & Watson, 1927, p. 9)</ref> Argand diagrams are frequently used to plot the positions of the [[pole (complex analysis)|poles]] and [[Zero (complex analysis)|zeroes]] of a [[mathematical function|function]] in the complex plane.
 
==Two-dimensional geometry==
===Polytopes===
{{main|Polygon}}
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 polygon|regular ''p''-gon]].
 
{| class="wikitable" style="text-align:center;"
|- bgcolor="#e0e0e0" valign="top"
!Name
![[Equilateral triangle|Triangle]]<br />([[Simplex|2-simplex]])
![[Square (geometry)|Square]]<br />([[Cross-polytope|2-orthoplex]])<br />([[Hypercube|2-cube]])
![[Pentagon]]
![[Hexagon]]
![[Heptagon]]
![[Octagon]]
|- bgcolor="#ffe0e0"
![[Schläfli]]
|{3}
|{4}
|{5}
|{6}
|{7}
|{8}
|-
!Image
|[[Image:Regular triangle.svg|75px]]
|[[Image:Regular quadrilateral.svg|75px]]
|[[Image:Regular pentagon.svg|75px]]
|[[Image:Regular hexagon.svg|75px]]
|[[Image:Regular heptagon.svg|75px]]
|[[Image:Regular octagon.svg|75px]]
|-
!Name
![[Nonagon]]
![[Decagon]]
![[Hendecagon]]
![[Dodecagon]]
![[Triskaidecagon]]
![[Tetradecagon]]
|- bgcolor="#ffe0e0"
!Schläfli
|{9}
|{10}
|{11}
|{12}
|{13}
|{14}
|-
!Image
|[[Image:Regular nonagon.svg|75px]]
|[[Image:Regular decagon.svg|75px]]
|[[Image:Regular hendecagon.svg|75px]]
|[[Image:Regular dodecagon.svg|75px]]
|[[Image:Regular tridecagon.svg|75px]]
|[[Image:Regular tetradecagon.svg|75px]]
|-
!Name
![[Pentadecagon]]
![[Hexadecagon]]
![[Heptadecagon]]
![[Octadecagon]]
![[Enneadecagon]]
![[Icosagon]]
|...[[Regular polygon|n-gon]]
|- bgcolor="#ffe0e0"
!Schläfli
|{15}
|{16}
|{17}
|{18}
|{19}
|{20}
|{''n''}
|-
!Image
|[[Image:Regular pentadecagon.svg|75px]]
|[[Image:Regular hexadecagon.svg|75px]]
|[[Image:Regular heptadecagon.svg|75px]]
|[[Image:Regular octadecagon.svg|75px]]
|[[Image:Regular enneadecagon.svg|75px]]
|[[Image:Regular icosagon.svg|75px]]
|}
 
====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 [[sphere|2-sphere]] or a [[torus|2-torus]].
 
{| class="wikitable" style="text-align:center;"
|- bgcolor="#e0e0e0" valign="top"
!Name
|[[Henagon]]
|[[Digon]]
|- bgcolor="#ffe0e0"
![[Schläfli]]
|{1}
|{2}
|-
!Image
|[[Image:Henagon.svg|75px]]
|[[Image:Digon.svg|75px]]
|}
 
====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 polygon]]s and share the same [[vertex arrangement]]s 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]].
 
{| class="wikitable" style="text-align:center;"
|- bgcolor="#e0e0e0"
!Name
|[[Pentagram]]
| colspan="2" | [[Heptagram]]s
|[[Octagram]]
| colspan="2" | [[Enneagram (geometry)|Enneagrams]]
|[[Decagram (geometry)|Decagram]]
|...[[star polygon|n-agrams]]
|- bgcolor="#ffe0e0"
![[Schläfli]]
|{5/2}
|{7/2}
|{7/3}
|{8/3}
|{9/2}
|{9/4}
|{10/3}
|{''n/m''}
|-
!Image
|[[Image:Star polygon 5-2.svg|75px]]
|[[Image:Star polygon 7-2.svg|75px]]
|[[Image:Star polygon 7-3.svg|75px]]
|[[Image:Star polygon 8-3.svg|75px]]
|[[Image:Star polygon 9-2.svg|75px]]
|[[Image:Star polygon 9-4.svg|75px]]
|[[Image:Star polygon 10-3.svg|75px]]
|&nbsp;
|}
 
===Hypersphere===
{{main|Circle}}
[[File:CIRCLE 1.svg|200px|right]]
The [[hypersphere]] in 2 dimensions is a [[circle]], sometimes called a 1-sphere (''S''<sup>1</sup>) because it is an one-dimensional [[manifold]]. In a Euclidean plane, it has the length 2&pi;''r'' and the [[area]] of its [[interior (topology)|interior]] is
:<math>A = \pi r^{2}</math>
where <math>r</math> is the radius.
 
==Coordinate systems in two-dimensional spaces<!-- now correct: if a sphere included, then there is no unique 2d space -->==
{{main|Coordinate system}}
 
The most popular coordinate systems are the [[Cartesian coordinate system]], the [[polar coordinate system]] and the [[geographic coordinate system]].
 
<gallery>
Image:Coord XY.svg|[[Cartesian coordinate system]]
Image:Coord Circular.svg|[[Polar coordinate system]]
Image:Coord LatLong.svg|[[Geographic coordinate system]]
</gallery>
 
==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 (mathematics)|graph]] that can be [[graph embedding|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.<ref>{{cite book|last=Trudeau|first=Richard J.|title=Introduction to Graph Theory|year=1993|publisher=Dover Pub.|location=New York|isbn=978-0-486-67870-2|pages=64|url=http://store.doverpublications.com/0486678709.html|edition=Corrected, enlarged republication.|accessdate=8 August 2012|quote=Thus a planar graph, when drawn on a flat surface, either has no edge-crossings or can be redrawn without them.}}</ref> 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.  
 
== References ==
<references />
 
==See also==
* [[Three-dimensional space]]
* [[Two-dimensional graph]]
 
{{Dimension topics}}
 
[[Category:Dimension]]
[[Category:Multi-dimensional geometry]]

Revision as of 23:41, 2 March 2014

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