NONEUCLIDEAN GEOMETRY Mathematics（非欧几里得几何,数学）.pdf

NON-EUCLIDEAN GEOMETRY NON-EUCLIDEAN GEOMETRY: A HISTORY AND A BRIEF LOOK Lisa K. Clayton 1. INTRODUCTION High school students are first exposed to geometry starting with Euclids classic postulates: 1. It is possible to draw a straight line from any one point to another point. 2. It is possible to create a finite straight line continuously on a straight line. 3. It is possible to describe a circle of any center and distance. 4. That all right angles are equal to one another 5. That if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Postulate #5, the so-called “parallel postulate” has always been a sticking point for mathematicians. Historically, mathematicians encountering Euclids beautiful work wonder why #5 is a postulate instead of a proven theorem. There have been many, many attempts to prove #5; there have been many, many failures. One such failure was that of Jesuit priest and mathematician, Girolamo Saccheri. His failure sparked ideas that led to what is now known as Non-Euclidean geometry, a branch of geometry that discards #5 and finds out where the geometries lead them. In particular, two Non-Euclidean branches will be discussed: that of Nikolai Lobachevsky and Bernhard Riemann. 2. LEARNING OBJECTIVES AND MATERIALS REQUIRED There are two main objectives: one, to introduce the concept of non-Euclidean geometries to high school geometry students who have examined Euclidean geometry at length, including some basic worksheets so they can study the concept for themselves. Two, to introduce students to the rich history of mathematics and mathematical ideas. To accomplish this, the lesson will include: 2.1 A 15 to 20 minute Po