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