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Article: Geometry


Definition, Importance, and Scope of Geometry

  • Branch of mathematics concerned with properties of space
  • Focuses on distance, shape, size, and relative position of figures
  • One of the oldest branches of mathematics, along with arithmetic
  • Used to model the physical world and has applications in various sciences
  • Also applied in art, architecture, and graphics-related activities
  • Originally devoted to Euclidean geometry
  • Carl Friedrich Gauss' Theorema Egregium expanded the scope of geometry
  • Non-Euclidean geometries developed in the 19th century
  • Geometry underlies general relativity
  • Split into subfields such as differential geometry, algebraic geometry, and computational geometry

History and Contributions of Geometry

  • Earliest beginnings in ancient Mesopotamia and Egypt
  • Developed for practical needs in surveying, construction, and astronomy
  • Egyptian Rhind Papyrus and Moscow Papyrus are early texts on geometry
  • Greek mathematicians Thales of Miletus and Pythagoras made significant contributions
  • Euclid's Elements introduced mathematical rigor and is still taught today
  • Nubians established a system of geometry, including sun clocks
  • Indian mathematicians made important contributions, such as the Pythagorean theorem
  • Bakhshali manuscript contains geometric problems and a decimal place value system
  • Aryabhata and Brahmagupta made significant contributions in geometry
  • Medieval Islam contributed to the development of algebraic geometry

Key Figures in Geometry

  • Thales of Miletus used deductive reasoning in geometry
  • Pythagoras established the Pythagorean School and proved the Pythagorean theorem
  • Euclid's Elements provided a logical framework for geometry
  • Archimedes made calculations using the method of exhaustion and approximations of pi
  • Omar Khayyam found geometric solutions to cubic equations

Main Concepts in Geometry

  • Axioms and postulates in Euclidean geometry
  • Points as fundamental objects in geometry
  • Lines and their various definitions in different geometries
  • Planes and their use in different areas of geometry
  • Angles and their role in studying polygons and trigonometry

Objects, Surfaces, and Measures in Geometry

  • Points and their definition as elements of a set called space
  • Lines and their definitions in different geometries
  • Planes and their different descriptions in various contexts
  • Angles and their formation by two rays sharing a common endpoint
  • Curves and their definition as 1-dimensional objects
  • Surfaces and their description by patches or polynomial equations
  • Solids and their three-dimensional nature bounded by closed surfaces
  • Manifolds as generalizations of curves and surfaces
  • Differentiable manifolds and their use in physics
  • Length, area, and volume as measures in geometry
  • Formulas for calculating length, area, and volume
  • Metrics and measures in geometry
  • Similarity and congruence in geometry
  • Compass and straightedge constructions
  • Rotation and orientation in geometry
  • Dimension as a broader topic in mathematics

Geometry Data Sources

Reference URL
Knowledge Graph

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