Geometry

1. Non-Euclidean Geometry
   1. Overview of Non-Euclidean Geometry
      1. Definition and historical context
         1. What defines Non-Euclidean Geometry
         2. Departure from Euclidean postulates
         3. Historical figures involved (e.g., Gauss, Bolyai, Lobachevsky)
      2. The significance in mathematical thought
         1. Impact on geometry and broader mathematics
         2. Philosophical implications
   2. Types of Non-Euclidean Geometry
      1. Hyperbolic Geometry
         1. Basic concepts
            1. Curved space model (negative curvature)
            2. Unique properties and axioms
         2. Models of hyperbolic planes
            1. Poincaré disk model
            2. Half-plane model
            3. Klein model
         3. Comparison with Euclidean geometry
            1. Parallel postulate differences
            2. Angle sum in triangles
         4. Applications
            1. Art and architecture
            2. Theoretical physics and cosmology
      2. Elliptic Geometry
         1. Basic concepts
            1. Spherical model (positive curvature)
            2. Axiomatic differences from Euclidean geometry
         2. Models of elliptic space
            1. Spherical model
            2. Projective models
         3. Key differences from Euclidean and hyperbolic geometry
            1. No parallel lines
            2. Great circles as lines
            3. Angle sum in triangles exceeds 180 degrees
         4. Applications
            1. Navigation and geodesy
            2. Astrophysics and general relativity
   3. Representation and Visualization
      1. Visual models of Non-Euclidean spaces
         1. How to depict in 2D and 3D
         2. Geometric and artistic representations
      2. Virtual reality applications
         1. Simulating non-Euclidean spaces
         2. Educational uses and research
   4. Mathematical Properties and Theorems
      1. Triangles and polygons in Non-Euclidean spaces
         1. Triangle angle sum variations
         2. Polygon tessellations
      2. Curvature and its implications
         1. Gaussian curvature
         2. Impact on geometric measurements and concepts
      3. Geodesics and distance
         1. Definition and properties of geodesics
         2. Distance measurement and implications
   5. Advanced Topics in Non-Euclidean Geometry
      1. Non-Euclidean Trigonometry
         1. Trigonometric identities in non-Euclidean geometry
         2. Calculating distances and angles
      2. Group theory and symmetry in Non-Euclidean spaces
         1. Role of transformations
         2. Symmetry groups and their applications
      3. Relation to modern physics
         1. The role of Non-Euclidean geometry in general relativity
         2. Measurement and perception of space-time
   6. Pedagogical Approaches to Teaching Non-Euclidean Geometry
      1. Curriculum design and challenges
      2. Use of technology in teaching
      3. Interactive tools and resources for learning
   7. Philosophical and Logical Implications
      1. Re-examining the nature of mathematical truth
      2. The logical framework of geometrical systems
      3. Impacts on philosophy and epistemology