3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
An illustration of Desargues' theorem, a result in Euclidean and projective geometry
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
A European and an Arab practicing geometry in the 15th century
The quadratic formula expresses concisely the solutions of all quadratic equations
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
Rubik's cube: the study of its possible moves is a concrete application of group theory
An illustration of Euclid's parallel postulate
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
A sphere is a surface that can be defined parametrically (by  or implicitly (by x2 + y2 + z2 − r2 = 0.)
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The Pythagorean theorem is a consequence of the Euclidean metric.
A page from al-Khwārizmī's Algebra
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
A tiling of the hyperbolic plane
Leonhard Euler created and popularized much of the mathematical notation used today.
Differential geometry uses tools from calculus to study problems involving curvature.
Carl Friedrich Gauss, known as the prince of mathematicians
A thickening of the trefoil knot
The front side of the Fields Medal
Quintic Calabi–Yau threefold
Discrete geometry includes the study of various sphere packings.
The Cayley graph of the free group on two generators a and b
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

Mathematics is an area of knowledge that includes such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

- Mathematics

Geometry is, with arithmetic, one of the oldest branches of mathematics.

- Geometry
3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

30 related topics

Alpha

A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.

Differential geometry

A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.
An osculating circle

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.

The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.

Manifold

The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
The surface of the Earth requires (at least) two charts to include every point. Here the globe is decomposed into charts around the North and South Poles.
Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle.
Four manifolds from algebraic curves: circles, parabola,  hyperbola,  cubic.
The chart maps the part of the sphere with positive z coordinate to a disc.
A finite cylinder is a manifold with boundary.
Möbius strip
The Klein bottle immersed in three-dimensional space
A Morin surface, an immersion used in sphere eversion

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces.

Archimedes used the method of exhaustion to calculate the area under a parabola.

Calculus

Archimedes used the method of exhaustion to calculate the area under a parabola.
Alhazen, 11th-century Arab mathematician and physicist
Isaac Newton developed the use of calculus in his laws of motion and gravitation.
Gottfried Wilhelm Leibniz was the first to state clearly the rules of calculus.
Maria Gaetana Agnesi
The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus.

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

Topological space

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.

Addition of functions: The sum of the sine and the exponential function is

Vector space

[[File: Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector

[[File: Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector

Addition of functions: The sum of the sine and the exponential function is
A typical matrix
Commutative diagram depicting the universal property of the tensor product.
The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).
An affine plane (light blue) in R3. It is a two-dimensional subspace shifted by a vector x (red).

In mathematics, physics, and engineering, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars.

Finite-dimensional vector spaces occur naturally in geometry and related areas.

The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.

Algebra

Quadratic formula.svg expresses the solution of the equation

Quadratic formula.svg expresses the solution of the equation

The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.
A page from Al-Khwārizmī's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.
Linear algebra lecture at the Aalto University
Algebraic expression notation:
 1 – power (exponent)
 2 – coefficient
 3 – term
 4 – operator
 5 – constant term
 x y c – variables/constants
The graph of a polynomial function of degree 3

Algebra is one of the broad areas of mathematics.

Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example).

A point in three-dimensional Euclidean space can be located by three coordinates.

Euclidean space

A point in three-dimensional Euclidean space can be located by three coordinates.
Positive and negative angles on the oriented plane
3-dimensional skew coordinates

Euclidean space is the fundamental space of geometry, intended to represent physical space.

Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the Euclidean plane (dimension two).

A definite integral of a function can be represented as the signed area of the region bounded by its graph.

Integral

A definite integral of a function can be represented as the signed area of the region bounded by its graph.
Riemann–Darboux's integration (top) and Lebesgue integration (bottom)
A line integral sums together elements along a curve.
Numerical quadrature methods: rectangle method, trapezoidal rule, Romberg's method, Gaussian quadrature

In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data.

Area can sometimes be found via geometrical compass-and-straightedge constructions of an equivalent square.

An illustration of Euclid's proof of the Pythagorean theorem.

Greek mathematics

An illustration of Euclid's proof of the Pythagorean theorem.
Detail of Pythagoras with a tablet of ratios, from The School of Athens by Raphael. Vatican Palace, Rome, 1509.
A fragment from Euclid's Elements (c. 300 BC), widely considered the most influential mathematics textbook of all time.
Cover of Arithmetica written by Greek Mathematician Diophantus

Greek mathematics refers to mathematics texts written during and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean.

Greek mathematics constitutes an important period in the history of mathematics: fundamental in respect of geometry and for the idea of formal proof.

A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.

Mathematical analysis

A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.
Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis.

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).