Geometry is one of the oldest fields of mathematics that goes back to the times of Euclid, Pythagoras, and other famous ancient Greek mathematicians. Initially concerned with fundamental concepts of point, line, plane, distance, angle, surface, and curve, the scope of geometry has been expanded during the last two centuries, leading to the creation of several subfields that include Riemannian geometry, algebraic geometry, etc.
Geometry is a good subject to explore the role of visual and spatial reasoning in the practice of mathematics: generating insights, problem-solving, communication, remembering, proofs, etc. In this course, students will explore a variety of topics in the exciting field of geometry, including axiomatic and analytic treatment of incident geometry, plane geometry, spherical geometry, hyperbolic geometry, and Poincaré disk model from Riemannian geometry. Important themes for each of the topics include axioms & common notions, geodesics & straightness, congruency & similarity, area & holonomy, isometries, projections, parallel postulates in different geometries, isometries with matrices & eigenvectors, etc.
Coursework will explore geometric questions through investigation, hands-on materials and dynamic geometry software such as Geometer's Sketchpad, GeoGebra 3D, GeoGebra Geometry, Desmos, Maple and also emphasize the many applications of geometry in the areas of computer-aided design, computer graphics, robotics, architecture, virtual reality, video game programming, and engineering.
Geometry also has important modern applications, in such areas as Computer-Aided Geometric Design, computer graphics, computational geometry, robotics, modern physics, biology, and engineering. How we practice Euclidean geometry (and teach geometry) is being changed by computer programs, both symbolic (such as Maple) and visual (such as Geometer's Sketchpad) and 3D printing. We will incorporate technology and look at geometric questions raised by how we might do geometry with computers.
This course is designed to further reflect on the teaching and learning of geometry. Connections to the Ontario curriculum will be made, from time to time, and students are encouraged to bring their own connections into classroom discussions. You are expected to raise your own questions and explore ways to answer them. Since a project is a critical part of the course, your own questions should strongly support the development of possible project topics. Value your questions – and ask about connections. Developing your questions and these connections is an important objective of the course.
A related goal of the course is to help students develop and expand their capacity in Spatial Reasoning – a form of reasoning which recent research has highlighted as well connected to many subject areas, as well as general performance in mathematics, and general creativity and problem-solving. An underlying goal of the course is to 'change the way we see'. How we see is key to many of the ways in which we work in geometry (and in other parts of mathematics and beyond).
