Geometry
| 200px Fundamental shapes and concepts in geometry. | |
| Overview | |
|---|---|
| Branch of | Mathematics |
| Core Concepts | Points, Lines, Angles, Shapes, Solids, Space |
| Subfields | Euclidean geometry, Non-Euclidean geometry, Differential geometry, Algebraic geometry |
| Key Figures | Euclid, Archimedes, René Descartes, Bernhard Riemann, Henri Poincaré |
| Applications | Architecture, Engineering, Physics, Computer graphics, Astronomy |
Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher-dimensional analogues. One of the oldest sciences, geometry originated in practical problems of measurement and surveying, but has since evolved into a rich and abstract field with deep connections to many areas of mathematics and science.
Core Concepts[edit]
Points, Lines, and Angles[edit]
- **Point:** A point is a zero-dimensional object that represents a location in space. It has no size, only position.
- **Line:** A line is a one-dimensional object that extends infinitely in two directions. It has length but no width.
- **Angle:** An angle is formed by two rays (half-lines) sharing a common endpoint (vertex). Angles are measured in degrees or radians.
Shapes and Solids[edit]
- **Plane Figures (2D):** Triangles, circles, quadrilaterals (squares, rectangles, parallelograms, trapezoids), polygons.
- **Solids (3D):** Cubes, spheres, cylinders, cones, pyramids, prisms.
- **Properties:** Area (for 2D shapes), surface area and volume (for 3D solids), perimeter, circumference.
Theorems and Relationships[edit]
- **Pythagorean Theorem:** In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. (a² + b² = c²)
- **Congruence and Similarity:** Congruent figures have the same shape and size; similar figures have the same shape but not necessarily the same size.
- **Parallel Postulate:** In Euclidean geometry, through a point not on a line, there is exactly one line parallel to the given line.
Branches of Geometry[edit]
Euclidean Geometry[edit]
Named after the ancient Greek mathematician Euclid, this is the geometry of flat space, based on five postulates (axioms). It is the geometry we learn in school and is still widely used in everyday applications. The Elements by Euclid is one of the most influential works in the history of mathematics.
Non-Euclidean Geometry[edit]
Developed in the 19th century by mathematicians like Gauss, Lobachevsky, and Bolyai, non-Euclidean geometries reject the parallel postulate.
- **Hyperbolic Geometry:** In this geometry, there are infinitely many lines through a point that are parallel to a given line.
- **Elliptic Geometry:** In this geometry, there are no parallel lines. The surface of a sphere is a model of elliptic geometry.
Differential Geometry[edit]
This branch studies the geometry of curves and surfaces using calculus. It is essential for understanding the shape of the Earth (geodesy), general relativity (spacetime), and computer graphics (surface modeling).
Algebraic Geometry[edit]
Algebraic geometry studies geometric objects defined by polynomial equations. It is a powerful area connecting algebra, geometry, and number theory.
History of Geometry[edit]
- **Ancient Geometry:** The Egyptians and Babylonians used geometry for practical purposes like land surveying and construction. The Greeks, starting with Thales and Pythagoras, transformed it into a theoretical science. Euclid's Elements (c. 300 BCE) systematized all known geometry.
- **Renaissance and Modern Geometry:** The development of analytic geometry (Descartes and Fermat) linked algebra and geometry. The 19th century saw the rise of non-Euclidean geometries, which challenged the absolute nature of Euclidean space. The 20th century saw the development of topology and fractal geometry.
See Also[edit]
References[edit]
- Euclid (c. 300 BCE). *The Elements*.
- Hilbert, D. (1902). *The Foundations of Geometry*. Open Court Publishing.
- Coxeter, H. S. M. (1961). *Introduction to Geometry*. Wiley.
- Stillwell, J. (2010). *Mathematics and Its History*. Springer.