Non-Euclidean geometry is a branch of geometry that does not follow Euclid’s fifth postulate, called the parallel postulate. The Egyptian Euclid lived around 300 B.C. His parallel postulate was part of his most famous work, “Elements”. He used other mathematicians of his time and the past to complete his work. Examples of non-Euclidean geometry are hyperbolic geometry, spherical geometry, and elliptic geometry. None follow the parallel postulate, which states that given a line and a point not on the line, there exists only one line through the point that does not intersect with the first line. The line that does not intersect with the first line is parallel to it. The parallel postulate was not proven until recently.
Hyperbolic geometry is also called Lobachevsky-Bolyai-Gauss geometry. The main idea is that curved lines are allowed, even though they are not allowed in Euclid’s parallel postulate. Although the sum of the angles of a triangle is equal to one hundred eighty degrees in Euclidean geometry, the sum of the angles in hyperbolic geometry is equal to less than one hundred eighty degrees. Two models of hyperbolic geometry are the Klein-Beltrami model and the Poincare hyperbolic disk. Felix Klein in 1870 devised a formula for hyperbolic geometry points using the coordinate system (x,y) where x^2+y^2<1. It is an open disk in the complex plane. The Poincare hyperbolic disk can be used to improve the Klein-Beltrami model. It uses real numbers in the second dimension x such that the absolute value of x is less than one.
Elliptic geometry changes the parallel postulate to “there are no lines parallel to a line through a point not on that line”. Elliptic geometry also does not have the angles of a triangle equal to one hundred eighty degrees, they are greater than one hundred eighty degrees. The Euclidean lines are changed to great circles. Great circles are sections of a sphere that contain the diameter of the sphere. This is any circle that is on the surface of the sphere going completely around it. The axiom on betweenness cannot be used.
Spherical geometry studies spherical triangles and spherical polygons. Spherical triangles are formed by three great circular arcs intersecting in three vertices. The sum of the angles of a spherical triangle is between one hundred eighty degrees and five hundred forty degrees. Two lines meet in two points instead of one as in plane geometry. This is because lines are defined as great circles. The angles between two lines are determined by the planes of the great circles. Parallel lines do not exist.
