Plane geometry

Plane geometry
Raphael · Public domain · source
Name	Plane geometry
Caption	Euclidean plane with triangle, circle, and polygon
Field	Mathematics
Introduced	Ancient Greece
Notable people	Euclid, Archimedes, Apollonius of Perga, Pythagoras, René Descartes, Karl Friedrich Gauss, Isaac Newton, Leonhard Euler, Blaise Pascal, Thales of Miletus, Pierre de Fermat, Bernhard Riemann, David Hilbert, Sofia Kovalevskaya, Niels Henrik Abel, Evariste Galois, Augustin-Louis Cauchy, Johann Carl Friedrich Gauss, Joseph-Louis Lagrange, Évariste Galois, Henri Poincaré, Brahmagupta, Alhazen, Omar Khayyam, Girolamo Saccheri, Felix Klein, Ada Lovelace, Alan Turing, John Nash, Srinivasa Ramanujan, Hermann Minkowski, Arthur Cayley, James Clerk Maxwell, Gottfried Wilhelm Leibniz, Nicolas Bourbaki, Andrew Wiles, William Rowan Hamilton, Sophus Lie, Émile Borel, Paul Erdős, Grigori Perelman

Plane geometry Plane geometry is the branch of Euclidean geometry concerned with figures, positions, and relationships confined to a two-dimensional flat surface called a plane. It studies points, lines, angles, circles, and polygons using axioms and theorems developed from antiquity through the work of Euclid, Archimedes, and later contributors like René Descartes and Isaac Newton. Applications span navigation, surveying, architecture, and modern computational fields influenced by Karl Friedrich Gauss and Leonhard Euler.

Definitions and basic concepts

Plane geometry builds on primitive notions introduced by Euclid in the Elements: point, line, plane (explicitly the two-dimensional case), and angle; axioms and postulates formalized by later mathematicians including David Hilbert and debates involving Girolamo Saccheri. Terms such as collinearity, coplanarity, parallelism, perpendicularity, midpoint, and congruence are central and are tied to constructions by classical authors like Thales of Miletus and results refined by Archimedes and Apollonius of Perga. Coordinate formulations from René Descartes link plane concepts to algebra, while rigor and foundations were addressed by Bernhard Riemann and Felix Klein.

Classical theorems and results

The subject contains foundational theorems named after figures: the Pythagoras theorem, Thales of Miletus’s intercept theorem, Ceva’s theorem, Menelaus’ theorem, Euler’s line and Euler’s formula in polyhedral contexts, and Pascal’s theorem for conics. Results by Archimedes include area and circumference relations for circles; Apollonius of Perga studied loci producing conic sections. Later formal work by David Hilbert produced axiomatic treatments; counterexamples and alternatives were explored by Girolamo Saccheri and extended into non-Euclidean contexts by Bernhard Riemann and Henri Poincaré.

Figures and properties (triangles, polygons, circles)

Triangles: centers (centroid, circumcenter, incenter, orthocenter) and relations such as the Euler line and Nagel and Gergonne points studied by classical and modern researchers including Pierre de Fermat (for Fermat point) and contributors in triangle geometry like Évariste Galois era contemporaries. Polygons: regular polygons and tessellations studied by Johannes Kepler and later by Felix Klein; properties include interior and exterior angle sums, symmetry groups connected to Évariste Galois-type group theory and Sophus Lie’s transformation groups. Circles: tangent properties, power of a point, chord and arc relations formalized by Euclid and extended by Apollonius of Perga and Brahmagupta in cyclic quadrilaterals.

Measurements: area, perimeter, angles

Area and perimeter formulas for triangles, rectangles, and polygons derive from decompositions used by Archimedes and later integration concepts related to Isaac Newton and Gottfried Wilhelm Leibniz. Heron’s formula links side lengths to area; Pick’s theorem relates lattice-point geometry to area, with lattice studies influenced by Carl Friedrich Gauss and Hermann Minkowski. Angle measurement uses degrees and radians, connecting to trigonometric developments by Ptolemy and analytic trigonometry refined by Leonhard Euler and Niels Henrik Abel. Metric concepts and distance formulas are central to analytic approaches introduced by René Descartes.

Constructions and compass-and-straightedge

Classical constructions with compass and straightedge follow rules codified in Euclid’s Elements and were revisited by René Descartes, Pierre de Fermat, and later algebraic impossibility proofs by Niels Henrik Abel and Évariste Galois showing the impossibility of angle trisection, doubling the cube, and squaring the circle under Euclidean tools. Methods include bisectors, perpendiculars, regular polygon constructions (constructible n-gons tied to Carl Friedrich Gauss’s cyclotomic work) and loci problems explored by Apollonius of Perga.

Coordinate and analytic approaches

The Cartesian plane from René Descartes and algebraic geometry link plane curves to polynomial equations, as developed by Isaac Newton and systematized by Gottfried Wilhelm Leibniz and Bernhard Riemann. Conic sections studied by Apollonius of Perga acquire algebraic classifications; the study of loci, transformations, and mappings involves work by Felix Klein (Erlangen program), Hermann Minkowski (geometry of numbers), and Sophus Lie (continuous transformations). Modern computational geometry relies on algorithmic foundations influenced by John von Neumann and applications in computer graphics traced to engineers like James Clerk Maxwell.

Applications and extensions

Plane geometry underpins cartography practiced by explorers and institutions such as the Royal Geographical Society, surveying standards in civil engineering projects like those overseen historically by Institution of Civil Engineers, and design principles in architecture exemplified by works associated with Vitruvius and later architects influenced by geometric treatises. Extensions include discrete and computational geometry researched by Paul Erdős and Donald Knuth, topology links via Henri Poincaré, and educational dissemination through curricula promoted by societies like the Mathematical Association of America. Advanced directions connect to complex analysis (Riemann surfaces of Bernhard Riemann), algebraic geometry influenced by Alexander Grothendieck, and applied fields such as robotics, computer vision, and geographic information systems where foundational plane results remain essential.

Category:Geometry