Geometry: Euclidean and Non-Euclidean
Geometry is simply the study of space. There are Euclidean and Non-Euclidean Geometries. Euclidean geometry is the most common and is the basis for other Non-Euclidean types of geometry. Euclidean geometry is based on five main rules, or postulates. Differences in these rules are what make new kinds of geometries. There is Euclidean, Elliptic, and Hyperbolic Geometry. Euclidean geometry is the study of flat space and was invented by Euclid, a mathematician from Alexandria, in 330 B.C. Euclid described his new ideas and general rules in the book Elements (Bradley 30). The foundation of Euclidean geometry is the concept of a few undefined terms: points, lines, and planes. In essence, a point is an exact position or location on a surface. A point has no actual length or width. A line shows infinite distance and direction but absolutely no width. A line has at least two points lying on it. Euclid’s first postulate is that only one unique straight line can be drawn between any two points. Line segments are lines that have a set length and do not go on forever. Euclid’s second postulate is that a finite straight line, or line segment, can be extended continuously into a straight line. The last of Euclid’s undefined terms is a plane, a flat surface similar to a table top or floor. However, a plane’s area is infinite. It has never ending length and width but has no depth. Lines can intersect each other or they can be parallel. Intersecting lines can be perpendicular, meaning they cross at a right angle. Lines in a plane that do not intersect or touch at a point and have a constant, unchanging distance between each other are called parallel lines. Line segments can be used to create different polygons. As in Euclid’s third postulate, with any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. All the angles in a triangle add up to 180 degrees. An acute angle is less than 90 degrees. A right angle is 90 degrees; all right
Cited: Bradley, Micheal J. The Birth of Mathematics: ancient times to 1300. “Euclid of Alexandria.” New York: Chelsea, 2006. Print. “Elliptic geometry.” Encyclopedia kids. Kids.Net.Au, 2012. Web. 8 Sept. 2012. <http://encyclopedia.kids.net.au/page/el/Elliptic_geometry> Roberts, Donna. “Euclidean and Non-Euclidean Geometry.” Geometry Lesson Page. Oswego City School. District Regents Exam Prep Center, 1998-2012. Web. 8 Sept. 2012. <http://regentsprep.org/Regents/math/geometry/GG1/Euclidean.htm> Weisstein, Eric. “Hyperbolic Geometry.” Wolfram Mathworld. Wolfram Research, Inc., 1999-2012. Web. 8 Sept. 2012. <http://mathworld.wolfram.com/HyperbolicGeometry.html>