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Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry is one of the two fields of pre-modern mathematics, the other being the study of numbers.

In modern times, geometric concepts have been extended. They sometimes show a high level of abstraction and complexity. Geometry now uses methods of calculus and abstract algebra, so that many modern branches of the field are not easily recognizable as the descendants of early geometry. (See areas of mathematics.) A geometer is one who works or is specialized in geometry.

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The frontispiece of Sir Henry Billingsley's first English version of Euclid's Elements, 1570

Euclid's Elements (Greek: Στοιχεῖα) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Egypt during the early 3rd century BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems) and proofs thereof. Euclid's books are in the fields of Euclidean geometry, as well as the ancient Greek version of number theory. The Elements is one of the oldest extant axiomatic deductive treatments of geometry, and has proven instrumental in the development of logic and modern science.

It is considered one of the most successful textbooks ever written: the Elements was one of the very first books to go to press, and is second only to the Bible in number of editions published (well over 1000). For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century did it cease to be considered something all educated people had read. It is still (though rarely) used as a basic introduction to geometry today.

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Euclid of Alexandria

Euclid (also referred to as Euclid of Alexandria) (Greek: Εὐκλείδης) (c. 325–c. 265 BC), a Greek mathematician, who lived in Alexandria, Hellenistic Egypt, almost certainly during the reign of Ptolemy I (323 BC–283 BC), is often considered to be the "father of geometry". His most popular work, Elements, is thought to be one of the most successful textbooks in the history of mathematics. Within it, the properties of geometrical objects are deduced from a small set of axioms, thereby founding the axiomatic method of mathematics.

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The Mathematics WikiProject is the center for mathematics-related editing on Wikipedia. Join the discussion on the project's talk page.

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One way of constructing an astroid, by tracking the path a point on the smaller circle follows as it is rolled round within the larger circle. Hence, an astroid is a hypocycloid.

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Image: User:Doctormatt

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Did you know?

...that the hyperboloid of one sheet is a doubly ruled surface?
...that as the dimension of a hypersphere tends to infinity, its "volume" (content) tends to 0?
...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
...that a regular heptagon is the regular polygon with the fewest number of sides which is not constructible with a compass and straightedge?
...that it is possible for a three-dimensional figure to have a finite volume but infinite surface area? An example of this is Gabriel's Horn.

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Topics in Geometry

Basic topics	Trigonometry	Euclidean geometry	Other geometries
point line line segment plane Hyperplane Geometric shape Polygon Polyhedron Platonic solid Archimedean solid Johnson solid List of basic geometry topics	Trigonometry Trigonometric functions Law of sines Law of cosines Law of tangents Trigonometric identity Pythagorean theorem Euler's formula De Moivre's formula List of trigonometric identities List of basic trigonometry topics List of trigonometry topics	Euclidean geometry Combinatorial Convex Discrete Plane geometry Solid geometry	Non-Euclidean geometry Conformal Elliptic Parabolic Hyperbolic Symplectic Spherical Affine Projective Riemannian Finite Non-Desarguesian planes
Differential geometry	Riemannian geometry	Algebraic geometry	Other
Differential geometry Differential geometry of curves Differential geometry of surfaces Manifold Tensor analysis Jets Differential topology Fiber bundle connection List of differential geometry topics	Riemannian geometry Riemannian manifold Levi-Civita connection Geodesic Riemannian symmetric space Riemannian submanifold Curvature of Riemannian manifolds Isometry Laplace-Beltrami operator List of formulas in Riemannian geometry	Algebraic geometry Algebraic curve Algebraic surface (list) Algebraic group Complex manifold List of algebraic geometry topics	Lie group Index theory Homogeneous space List of mathematical shapes List of geometry topics