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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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All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

The trigonometric functions are functions of an angle; they are most important when studying triangles and modeling periodic phenomena, among many other applications. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers.

The study of trigonometric functions dates back to Babylonian times, and a considerable amount of fundamental work was done by ancient Greek, Indian and Arab mathematicians.

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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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The above shows an example of doubly ruled surface – the hyperboloid of one sheet. Although the wires are straight lines, they are lying within the surface. Through any point on this surface pass two straight lines, so it is doubly ruled.

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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