A report on Mathematics and Geometry

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, 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 with Alpha

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

Integral

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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.

In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations.

Linear algebra

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In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations.

Linear algebra is the branch of mathematics concerning linear equations such as:

For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations.

The combined area of these three shapes is approximately 15.57 squares.

Area

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Quantity that expresses the extent of a region on the plane or on a curved surface.

Quantity that expresses the extent of a region on the plane or on a curved surface.

The combined area of these three shapes is approximately 15.57 squares.
This square and this disk both have the same area (see: squaring the circle).
A square metre quadrat made of PVC pipe.
Although there are 10 mm in 1 cm, there are 100 mm2 in 1 cm2.
The area of this rectangle is lw.
A diagram showing how a parallelogram can be re-arranged into the shape of a rectangle.
A parallelogram split into two equal triangles.
A circle can be divided into sectors which rearrange to form an approximate parallelogram.
Archimedes showed that the surface area of a sphere is exactly four times the area of a flat disk of the same radius, and the volume enclosed by the sphere is exactly 2/3 of the volume of a cylinder of the same height and radius.
Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).
The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions

In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.

Detail from Raphael's The School of Athens presumed to represent Donato Bramante as Euclid

Euclid

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Detail from Raphael's The School of Athens presumed to represent Donato Bramante as Euclid
Euclidis quae supersunt omnia (1704)
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.
Euclid's construction of a regular dodecahedron.
Construction of a dodecahedron by placing faces on the edges of a cube.
19th-century statue of Euclid by Joseph Durham in the Oxford University Museum of Natural History

Euclid ( Eὐkleídēs; 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".

His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century.

Axiom

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Statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

Statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms".

The ancient Greeks considered geometry as just one of several sciences, and held the theorems of geometry on par with scientific facts.

A collection of circles and the corresponding unit disk graph

Discrete geometry

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A collection of circles and the corresponding unit disk graph
Graphs are drawn as rods connected by rotating hinges. The cycle graph C4 drawn as a square can be tilted over by the blue force into a parallelogram, so it is a flexible graph. K3, drawn as a triangle, cannot be altered by any force that is applied to it, so it is a rigid graph.
Seven points are elements of seven lines in the Fano plane, an example of an incidence structure.

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects.

In mathematics, tessellations can be generalized to higher dimensions.

Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0.

Measure (mathematics)

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Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0.
Countable additivity of a measure μ: The measure of a countable disjoint union is the same as the sum of all measures of each subset.

In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events.

An example of change ringing (with six bells), one of the earliest nontrivial results in graph theory.

Combinatorics

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An example of change ringing (with six bells), one of the earliest nontrivial results in graph theory.
Five binary trees on three vertices, an example of Catalan numbers.
A plane partition.
Petersen graph.
Hasse diagram of the powerset of {x,y,z} ordered by inclusion.
Self-avoiding walk in a square grid graph.
Young diagram of a partition (5,4,1).
Construction of a Thue–Morse infinite word.
An icosahedron.
Splitting a necklace with two cuts.
Kissing spheres are connected to both coding theory and discrete geometry.

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.

Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas.

A page from The Compendious Book on Calculation by Completion and Balancing by Al-Khwarizmi

Mathematics in medieval Islam

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A page from The Compendious Book on Calculation by Completion and Balancing by Al-Khwarizmi
Omar Khayyám's "Cubic equations and intersections of conic sections" the first page of the two-chaptered manuscript kept in Tehran University
To solve the third-degree equation x3 + a2x = b Khayyám constructed the parabola x2 = ay, a circle with diameter b/a2, and a vertical line through the intersection point. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the x-axis.
Engraving of Abū Sahl al-Qūhī's perfect compass to draw conic sections.
The theorem of Ibn Haytham.

Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta).

Important progress was made, such as full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra, and advances in geometry and trigonometry.

A typical example of a complex space is the complex projective line. It may be viewed either as the sphere, a smooth manifold arising from differential geometry, or the Riemann sphere, an extension of the complex plane by adding a point at infinity.

Complex geometry

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A typical example of a complex space is the complex projective line. It may be viewed either as the sphere, a smooth manifold arising from differential geometry, or the Riemann sphere, an extension of the complex plane by adding a point at infinity.
A real two-dimensional slice of a quintic Calabi–Yau threefold
Moment polytope describing the first Hirzebruch surface.

In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers.