essay on geometry

We use Euclidean and Non-Euclidean geometry in our everyday use. In many ways they are similar and different. There are similarities and differences in Euclidean geometry and spherical geometry, Euclid’s fifth postulate applies to both forms, and it is used every day in astronomy. Euclidean geometry is the study of flat space, and can be easily drawn on a piece of paper. Non-Euclidean geometry is any form of geometry that uses a postulate that is equivalent to the negation of Euclidean parallel postulate. In Euclidean geometry 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 Euclidean geometry 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 Euclidean geometry the definition of the fifth postulate is “Through any given point not on a given line there exists exactly one line parallel to the given line." But in spherical geometry, it states that “through any given point not on a given line, there exist no lines parallel to the given line." So, the fifth postulate does not apply to both forms of geometry, because there are no parallel lines in spherical geometry. However, “through any two great circles drawn on the sphere intersect in two places.” This is only way the fifth postulate can apply in any way to spherical geometry. In Euclidean geometry the fifth postulate is actually relevant and can be applied. In this diagram below it shows how the fifth postulate is applied to Euclidean geometry. There are many real world applications of non-Euclidean geometries. Astronomy is a good