Two-dimensional space

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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[edit]

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.[1] 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.[2]

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).[3] Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane.

Two-dimensional geometry[edit]

Polytopes[edit]

Main article: Polygon

In two dimensions, there are infinitely many regular polytopes: the polygons. The first few are shown below:

Convex[edit]

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 Regular triangle.svg Regular quadrilateral.svg Regular pentagon.svg Regular hexagon.svg Regular heptagon.svg Regular octagon.svg
Name Nonagon Decagon Hendecagon Dodecagon Triskaidecagon Tetradecagon
Schläfli {9} {10} {11} {12} {13} {14}
Image Regular nonagon.svg Regular decagon.svg Regular hendecagon.svg Regular dodecagon.svg Regular tridecagon.svg Regular tetradecagon.svg
Name Pentadecagon Hexadecagon Heptadecagon Octadecagon Enneadecagon Icosagon ...n-gon
Schläfli {15} {16} {17} {18} {19} {20} {n}
Image Regular pentadecagon.svg Regular hexadecagon.svg Regular heptadecagon.svg Regular octadecagon.svg Regular enneadecagon.svg Regular icosagon.svg

Degenerate (spherical)[edit]

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 Henagon.svg Digon.svg

Non-convex[edit]

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/(nm)}) 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 Star polygon 5-2.svg Star polygon 7-2.svg Star polygon 7-3.svg Star polygon 8-3.svg Star polygon 9-2.svg Star polygon 9-4.svg Star polygon 10-3.svg  

Hypersphere[edit]

Main article: Circle
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[edit]

Main article: Coordinate system

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

Topological properties of the plane[edit]

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[edit]

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.[4] 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[edit]

  1. ^ "analytic geometry". Encyclopædia Britannica (Encyclopædia Britannica Online ed.). 2008. 
  2. ^ Burton 2011, p. 374
  3. ^ Wessel's memoir was presented to the Danish Academy in 1797; Argand's paper was published in 1806. (Whittaker & Watson, 1927, p. 9)
  4. ^ Trudeau, Richard J. (1993). Introduction to Graph Theory (Corrected, enlarged republication. ed.). New York: Dover Pub. p. 64. ISBN 978-0-486-67870-2. Retrieved 8 August 2012. "Thus a planar graph, when drawn on a flat surface, either has no edge-crossings or can be redrawn without them." 

See also[edit]