Geometry

1. Historical Development of Geometry
   1. Ancient Geometries
      1. Egyptian Contributions
         1. Use of geometry in architecture and construction
         2. The Rhind Mathematical Papyrus and practical problems
         3. Measurement systems and approximations for area and volume
      2. Babylonian Contributions
         1. Clay tablets and geometric problem-solving
         2. Geometry in astronomical calculations
         3. Development of early geometric theorems
      3. Greek Geometry
         1. Thales and foundational principles
         2. Pythagoras and the Pythagorean theorem
         3. Euclid’s "Elements"
            1. Axiomatic approach to geometry
            2. Definitions, postulates, and propositions
            3. Influence on later mathematics
         4. Contributions of Archimedes
            1. Method of exhaustion
            2. Calculations of areas and volumes
         5. Contributions of Apollonius
            1. Study of conic sections
            2. Influence on later geometric work
         6. Contributions of Hypatia
            1. Commentary on previous works
            2. Development and teaching of geometric principles
   2. Renaissance and Modern Geometry
      1. Islamic Golden Age Contributions
         1. Preservation and enhancement of Greek works
         2. Contributions to spherical geometry
         3. Key figures: Al-Khwarizmi and Al-Biruni
      2. Descartes and Analytic Geometry
         1. Cartesian coordinate system
         2. Linking algebra and geometry
         3. Introduction to analytical methods
      3. Newton and Leibniz
         1. Development of calculus
         2. Application to geometric problems
         3. Concept of limits and infinitesimals
      4. Projective Geometry
         1. Pioneers: Desargues and Pascal
         2. Properties of projection and perspective
         3. Introduction to vanishing points and the concept of infinity
      5. Non-Euclidean Geometry
         1. Lobachevsky and Hyperbolic Geometry
         2. Riemann and Elliptic Geometry
         3. Impact on understanding of space and time
      6. Development of Topology
         1. Euler and the concept of connectivity
         2. Poincaré and the classification theorem
         3. Relationship to modern spatial concepts
      7. Twentieth Century Advancements
         1. Einstein's General Relativity and geometric interpretation
         2. Development of computational geometry
         3. Impact on computer science and graphics