Non-Euclidean geometry is any form of geometry that is based on axioms, or postulates, different from those of Euclidean geometry. These geometries were developed by mathematicians to find a way to prove Euclid’s fifth postulate as a theorem using his other four postulates. They were not accepted until around the nineteenth century. These geometries are based on a curved plane, whether it is elliptic or hyperbolic. There are no parallel lines in non-Euclidean geometry, and the angles of triangles do not have a sum of 180 degrees. Overall, non-Euclidean geometry follows almost all of the same postulates as Euclidean geometry. The main difference is non-Euclidean involves the study of curved surfaces, while Euclidean geometry involves the study of flat space. Around 1830, the Hungarian mathematician János Bolyai and a Russian mathematician named Nikolai Ivanovich Lobachevsky separately published studies on hyperbolic geometry. Both mathematicians spent years working with the fifth postulate. Neither of them gained public recognition for the work they put into their geometric discoveries. Hyperbolic geometry is a type of non-Euclidean geometry that uses the statement “If l is any line and P is any point not on l, then there exists at least two lines through P that are parallel to l” or any statement equivalent to this statement as its parallel postulate. It is the study of saddle shaped space. It applies to areas of science such as determining the orbit of objects within intense gradational fields, space travel, and astronomy. Einstein’s theory of relativity also involves hyperbolic geometry. There are many major differences between hyperbolic geometry and Euclidean geometry. In hyperbolic geometry, the sum of the angles of a triangle is less than 180 degrees and triangles with the same angles have the same area, but there are no similar triangles. In 1854, Bernhard Riemann founded Riemannian geometry, which elliptic geometry was a part of....

...Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclideangeometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p.
Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which asserts that there is exactly one line parallel to L passing through p. In elliptic geometry, there...

...Maddux Willingham
Mr. Warfle
Geometry Honors
26 September 2011
Euclidean & Non-EuclideanGeometry Paper
Isn’t it amazing that we still study the same geometry as people did back nearly twenty-three centuries ago? Euclidean and Non-Euclideangeometry communicates to us through mathematical equations immense amounts of significant information....

...segment PQ:
In Euclideangeometry the perpendicular distance between the rays
remains equal to the distance from P to Q as we move to the right.
However, in the early nineteenth century two alternative geometries
were proposed. In hyperbolic geometry (from the Greek hyperballein,
"to exceed") the distance between the rays increases. In elliptic
geometry (from the Greek elleipein, "to fall short") the distance decreases and...

...Geometry (Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science...

...When it comes to EuclideanGeometry, Spherical Geometry and Hyperbolic Geometry there are many similarities and differences among them. For example, what may be true for EuclideanGeometry may not be true for Spherical or Hyperbolic Geometry. Many instances exist where something is true for one or two geometries but not the other geometry. However, sometimes a...

...HYPERBOLIC GEOMETRY AND OMEGA TRIANGLES
Hyperbolic geometry was first discovered and explored by Omar Khayyam in the 9th century and Giovanni Gerolamo Saccheri in the 15th century. Both were attempting to prove Euclid’s parallel postulate by proving the concept of hyperbolic geometry to be inconsistent, and ironically they discovered it to be a new type of geometry. It wasn’t until the 19th century that it became fully developed...

... We use Euclidean and Non-Euclideangeometry in our everyday use. In many ways they are similar and different. There are similarities and differences in Euclideangeometry and spherical geometry, Euclid’s fifth postulate applies to both forms, and it is used every day in astronomy. Euclideangeometry is the study of flat space, and can be easily drawn on a piece of...

...through a point parallel to a given line. His decision to create this postulate enabled him to create what is now called, EuclideanGeometry, taking name after him. Not until the 19th century, was this postulate dropped and non-euclideangeometries were beginning to be studied.
Euclid's elements are divided into 13 books. The first six books are based upon just plane geometry. They give out properties of...