Non-Euclidean Geometry

Non-Euclidean geometry refers to certain types of geometry which differ from plane and solid geometry which dominated the realm of mathematics for several centuries. There are other types of geometry which do not assume all of Euclid's postulates such as hyperbolic geometry, elliptic geometry, spherical geometry, descriptive geometry, differential geometry, geometric algebra, and multidimensional geometry. These geometries deal with more complex components of curves in space rather than the simple plane or solids used as the foundation for Euclid's geometry. The first five postulates of Euclidean geometry will be listed in order to better understand the changes that are made to make it non-Euclidean.

1. A straight line can be drawn from any point to any point.
2. A finite straight line can be produced continuously in a straight line.
3. A circle may be described with any point as center and any distance as a radius.
4. All right angles are equal to one another.
5. If a transversal falls on two lines in such a way that the interior angles on one side of the transversal are less than two right angles, then the lines meet on the side on which the angles are less than two right angles.

A consistent logical system for which one of these postulates is modified in an essential way is non-Euclidean geometry. Although there are different types of Non-Euclidean geometry which do not use all of the postulates or make alterations of one or more of the postulates of Euclidean geometry, hyperbolic and elliptic are usually most closely associated with the term non-Euclidean geometry.

Hyperbolic geometry is based on changing Euclid's parallel postulate, which is also referred to as Euclid's fifth postulate, the last of the five postulates of Euclidian Geometry. Euclid's parallel postulate may also be stated as one and only one parallel to a given line goes through a given point not on the line.

Elliptic geometry uses a modification of Postulate II. Postulate II allows for lines of infinite length, which are denied in Elliptic geometry, where only finite lines are assumed.