Maddux Willingham
Mr. Warfle
Geometry Honors
26 September 2011
Euclidean & Non-Euclidean Geometry Paper
Isn’t it amazing that we still study the same geometry as people did back nearly twenty-three centuries ago? Euclidean and Non-Euclidean geometry communicates to us through mathematical equations immense amounts of significant information. Without the study of geometry, many people would be unemployed. Euclidean and Non-Euclidean geometry 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. Euclidean geometry and Non-Euclidean geometries 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 Non-Euclidean geometries, such as those in Euclidean geometry the measure of all angles of a triangle add up to 180 degrees. Euclidean Geometry is the study of flat space, and there is only one form on Euclidean geometry. But there are several types of Non-Euclidean geometry, such as Riemmanian Geometry (or spherical geometry), which is the study of curved spaces. Riemmanian geometry says that in curved space, the sum of the angles of any triangle is now always greater than 180 degrees. It also says that there are no straight lines; once you start drawing a line it immediately curves due to the sphere on which it is drawn on. Another form of Non-Euclidean geometry is Hyperbolic geometry, which is the study of saddle shaped space. In Hyperbolic geometry the measurements of all angles of a triangle is less than 180 degrees...

...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...

...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,...

...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...

...Non-Euclideangeometry is any form of geometry that is based on axioms, or postulates, different from those of Euclideangeometry. 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-Euclideangeometry, and the angles of triangles do not have a sum of 180 degrees. Overall, non-Euclideangeometry follows almost all of the same postulates as Euclideangeometry. The main difference is non-Euclidean involves the study of curved surfaces, while Euclideangeometry 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...

...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...

... 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 ofpaper. 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...

...problem!
Geometry (Ancient Greek: γεωμετρία; geo- "earth", -metri "measurement") "Earth-measuring" is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclideangeometry—set a standard for many centuries to follow. 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. A mathematician who works in the field of geometry is called a geometer.
The introduction of coordinates by René Descartes and the concurrent development of algebra marked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence ofinfinitesimal calculus in the 17th century. Furthermore, the theory of perspective showed that there is more to...

...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...