Euclidean and Non-Euclidean Geometry
MATH 635. Euclidean and Non-Euclidean Geometry
Three Credit Hours (3).
Prerequisite: Undergraduate degree in mathematics or by instructor permission.
This course will introduce students to systems of postulates in a comparison of Euclidean and Non-Euclidean geometries. Geometric structures of transformational, fractal, and projective geometry are examined together with a brief history of the development of axiomatic systems of geometry.
Detailed Description of Course
The development of Geometry as an axiomatic system starting with pre-epoch Greece, and following through to Non-Euclidean Geometries. This coverage will include the development of the School-based SMSG (School Mathematics Study Group) geometries. The geometries examined will consist of Euclidean, Neutral, and Elliptic and Hyperbolic (non-Euclidean) geometries. Euclidean Geometry is expanded using an Abstract Algebra point of view to introduce transformational geometry. Then fractal geometry is developed using recursive transformations of geometric objects. Projective geometry will be examined through perspective drawings.
Detailed Description of Conduct of Course
In addition to lecture, students will work collaboratively on assignments created to help students understand the mathematics introduced. Calculators and mathematics software (such as Geometers Sketchpad and Excel) will be used to present and work on the material presented in class. A project will be presented by the students on a topic chosen by the instructor.
Goals and Objectives of the Course
Students will develop an understanding of and appreciation for axiomatic systems and their role in problem-solving in mathematics. Students will examine the interconnections among the different types of geometry and the expansive nature of mathematical development. They will build knowledge of the role of geometry in understanding the world and of how this understanding can be developed in their own classrooms. They will extend their conceptions of geometry beyond plane figures and their properties.
Students will demonstrate content understanding through written (possibly oral) exams, written homework problems, collaborative work in class, and a project.
Other Course Information
The mathematics and ideas presented during this class will be demonstrated not only at an advanced level but also at a level comparable to that which will be applicable in their classrooms.
Review and Approval
Date Action Reviewed