Foundations of Geometry

Course Description

A discussion of fundamental ideas and processes common to Euclidean and Non-Euclidean Geometries: projective, affine and metric geometry. Examines the interplay between inductive and deductive reasoning and formal and informal proof. Addresses uses in science (transformations, scaling), art (Escher-type tessellations, projections), architecture (three-dimensional figures) and computer science (fractals, computer-aided design).

Learning Outcomes

  • Compare and contrast projective geometry as generalizations of Euclidean geometry.
  • Compare and contrast projective geometry as developed from a set of axioms or from groups of transformations.
  • Discuss practical applications of projective geometry (including matrices).
  • Discuss the concepts of duality, harmonic sets, and involutions.
  • Discuss Brouwer’s fixed point theorem and its practical applications.
  • Discuss the map-coloring problem and the controversy of its computer proof.
  • Explain Euler’s formula and construct a Mobius strip.
  • Describe the various attempts to prove the Fifth Postulate and the social climate of the times and countries.
  • Compare and contrast Euclidean, hyperbolic geometry and elliptic geometry.
  • Discuss the concept of parallax and describe how modern scientists use non-Euclidean geometry to attempt to describe the physical nature of the universe.


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