# Two-dimensional space

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.

## Two-dimensional geometry

### Polytopes

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
Name Nonagon Decagon Hendecagon Dodecagon Triskaidecagon Tetradecagon
Schläfli {9} {10} {11} {12} {13} {14}
Image
Schläfli {15} {16} {17} {18} {19} {20} {n}
Image

#### 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 Schläfli Henagon Digon {1} {2}

#### 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 Schläfli Image Pentagram Heptagrams Octagram Enneagrams Decagram ...n-agrams {5/2} {7/2} {7/3} {8/3} {9/2} {9/4} {10/3} {n/m}

### Hypersphere

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

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