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 are no parallel lines at all. Elliptic geometry has other unusual properties. For example, the sum of the angles of any triangle is always greater than 180°.
The simplest model of elliptic geometry is that of spherical geometry, where points are points on the sphere, and lines are great circles through those points. On the sphere, such as the surface of the Earth, it is easy to give an example of a triangle that requires more than 180°: For two of the sides, take lines of longitude that differ by 90°. These form an angle of 90° at the North pole. For the third side, take the equator. The angle of any longitude line makes with the equator is again 90°. This gives us a triangle with an angle sum of 270°, which would be impossible in Euclidian geometry.
Elliptic geometry is sometimes called Riemannian geometry, in honor of Bernhard Riemann, but this term is usually used for a vast generalization of...

...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. Without the study of geometry, many people would be unemployed. Euclidean and Non-Euclideangeometry have several similarities, however they also have numerous differences, as well as their historical aspects.
To begin with, these mathematical concepts have many similarities. For example, both studies of geometry include perpendicular lines, the drawings for these lines may be different, but they still make 90 degree angles. And both geometries are used in physics by thousands of people daily. Euclideangeometry and Non-Euclideangeometries are both hard to grasp and have a lot of theorems and postulates that say a lot about each individual mathematical study. Both have quite a few similarities, but they have even more differences.
Next, there are several differences between Euclidean and...

...2
1[1
Introduction
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 the rays eventually meet. These non-Euclideangeometries were later incorporated in a much more general geometry developed by C. F. Gauss and G. F. B. Riemann (it is this more general
geometry that is used in Einstein's general theory of relativity).1
We will concentrate on Euclidean and hyperbolic geometries in this
book. Hyperbolic geometry requires a change in only one of Euclid's
axioms, and can be as easily grasped as high school geometry. Elliptic
geometry, on the other hand, involves the new topological notion of
"nonorientability," since all the points of the elliptic plane not on a
given line lie on the same side of that line. This geometry cannot easily
be approached in the spirit of Euclid. I have therefore made only brief
comments about elliptic geometry in the body of the text, with...

...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 property is true for all three geometries. These points bring us to the purpose of this paper. This paper is an opportunity for me to demonstrate my growing understanding about EuclideanGeometry, Spherical Geometry, and Hyperbolic Geometry.
The first issue that I will focus on is the definition of a straight line on all of these surfaces. For a Euclidean plane the definition of a "straight line" is a line that can be traced by a point that travels at a constant direction. When I say constant direction I mean that any portion of this line can move along the rest of this line without leaving it. In other words, a "straight line" is a line with zero curvature or zero deviation. Zero curvature can be determined by using the following symmetries. These symmetries include: reflection-in-the-line symmetry, reflection-perpendicular-to-the-line symmetry, half-turn symmetry, rigid-motion-along-itself...

...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 emerging in the West as early as Thales (6th Century BC). By the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclideangeometry—set a standard for many centuries to follow.[1] Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. Both geometry and astronomy were considered in the classical world to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.
History of geometry
The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the...

...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 with help from Karl Friedrich Gauss, Janos Bolyai, and Nikolai Ivanovich Labachevsky. Later on, Eugenio Beltrami developed models of it and used these to prove that hyperbolic geometry is consistent if Euclideangeometry is.
Hyperbolic geometry is a form of non-Euclideangeometry. It upholds all of Euclid’s principles except the parallel postulate that says that if given a line [pic]and a point[pic] not on [pic], there is exactly one line through[pic] that doesn’t intersect [pic]. Hyperbolic geometry instead has the following modified postulate: “given any line [pic], and point[pic]not on [pic], there are exactly two lines through[pic]which are hyperparallel to [pic], and an infinite number of lines through[pic]ultraparallel to [pic]” (Wikipedia). Hyperbolic geometry has become well-understood in two dimensions; however, not much is known about it in three...

... 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 paper. Non-Euclideangeometry is any form of geometry that uses a postulate that is equivalent to the negation of Euclidean parallel postulate.
In Euclideangeometry you can draw on a flat piece of paper, where in spherical geometry you have to draw on a curved surface. Or, Euclidean is assuming a flat plane and spherical is on curved surfaces. Also, in Euclideangeometry a triangles angles add up to 180 degrees, and in spherical geometry a triangle angles do not equal 180 degrees. Another big difference between the two geometries is that Euclidean has parallel lines, and in spherical geometry there are no such things as lines. These “lines” are known as geodesics. In both geometries however, the angles of a triangle have to be no less than 180 degrees.
In...

...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 triangles, parallelograms, parallels, rectangles and squares. They also deal with problems with circles, and circles in general. Books seven through nine explain the number theory. In particular book seven is a self-contained introduction to number theory and contains the Euclidean algorithm for finding the greatest common divisor of two numbers. Book eight talks about geometrical progressions. The tenth book explains the theory of irrational numbers. It is mainly based upon the work of Theaetetus. Euclid had to change many of the proofs written by Eudoxus. From book eleven through thirteen, describes the geometries of three-dimensional shapes. More than one thousand editions of this book have been printed since its first edition in 1482. Euclid also wrote many other books. Data, On Divisions, Optics, and Phenomena are all other books that have survived. The ones that have been lost are Surface Loci, Porisms, Conics, Fallacies and Elements of Music.
Euclid has enabled us today, the ability to...