A report on GeometryMathematics and Manifold

An illustration of Desargues' theorem, a result in Euclidean and projective geometry
3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
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.
A European and an Arab practicing geometry in the 15th century
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.
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.
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
The quadratic formula expresses concisely the solutions of all quadratic equations
Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle.
An illustration of Euclid's parallel postulate
Rubik's cube: the study of its possible moves is a concrete application of group theory
Four manifolds from algebraic curves: circles, parabola,  hyperbola,  cubic.
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
The chart maps the part of the sphere with positive z coordinate to a disc.
A sphere is a surface that can be defined parametrically (by  or implicitly (by x2 + y2 + z2 − r2 = 0.)
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
A finite cylinder is a manifold with boundary.
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.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
Möbius strip
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
A page from al-Khwārizmī's Algebra
The Klein bottle immersed in three-dimensional space
A tiling of the hyperbolic plane
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 Morin surface, an immersion used in sphere eversion
Differential geometry uses tools from calculus to study problems involving curvature.
Leonhard Euler created and popularized much of the mathematical notation used today.
A thickening of the trefoil knot
Carl Friedrich Gauss, known as the prince of mathematicians
Quintic Calabi–Yau threefold
The front side of the Fields Medal
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".

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

- Geometry

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

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

- Manifold

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.

- Manifold

This implies that surfaces can be studied intrinsically, that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry.

- Geometry

Manifold theory, the study of shapes that are not necessarily embedded in a larger space

- Mathematics
An illustration of Desargues' theorem, a result in Euclidean and projective geometry

6 related topics with Alpha

Overall

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

Euclidean space

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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).

This way of defining coordinates extends easily to other mathematical structures, and in particular to manifolds.

Topological space

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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.

Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.

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

Differential geometry

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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.

It is intimately linked to the development of geometry more generally, of the notion of space and shape, and of topology, especially the study of manifolds.

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.

Linear algebra is flat differential geometry and serves in tangent spaces to manifolds.

Plane equation in normal form

Plane (geometry)

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Plane equation in normal form
Three parallel planes.
Vector description of a plane
Two intersecting planes in three-dimensional space

In mathematics, a plane is a flat, two-dimensional surface that extends indefinitely.

Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional space, often in the plane.

Differential geometry views a plane as a 2-dimensional real manifold, a topological plane which is provided with a differential structure.

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

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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).

Differential geometry, the application of calculus to specific mathematical spaces known as manifolds that possess a complicated internal structure but behave in a simple manner locally.