Modern Geometry – mth405 (3 credits)

This course explores geometry from heuristic, axiomatic, and computational angles. Students examine ancient results, Euclid, non-Euclidean geometry via the Poincaré disk, and transformational geometry.

This undergraduate-level course is 5 weeks. This course is available to take individually or as part of a degree or certificate program. To enroll, speak with an Enrollment Representative.

Transformational Geometry

  • Analyze the compositions of linear transformations.
  • Identify the invariants of matrix transformations.
  • Represent geometric objects and linear transformations using matrices.
  • Represent direct and indirect isometries using matrix methods.
  • Demonstrate properties of isometries: reflections, rotations, translations, and glide reflections.

Non-Euclidean Geometry

  • Prove Simple theorems of hyperbolic geometry.
  • Summarize contributions of pivotal geometers.
  • Explain the significance of denying the fifth axiom of Euclid's Playfair's Postulate.
  • Construct objects in the Poincaré Disk.
  • Calculate distances, angle measures, and areas in the Poincaré Disk.

Symmetry and Fractal Geometry

  • Demonstrate properties of types of finite plane symmetry groups.
  • Prove elementary results involving finite plane symmetry groups.
  • Identify characteristics of self-similar geometric objects.
  • Calculate the similarity dimension of geometric objects.

Axiomatic Systems and Ancient and Neutral (Finite) Geometries

  • Solve elementary problems from Western and Non-Western sources.
  • Demonstrate properties of components of axiomatic systems.
  • Prove elementary theorems in neutral geometry.

Euclidean Plane Geometry and Constructions

  • Prove theorems in Euclidean Geometry.
  • Perform basic constructions with straight-edge and compass.

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