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...