A report on GeometryMathematics and Euclidean space

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)
A point in three-dimensional Euclidean space can be located by three coordinates.
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.
Positive and negative angles on the oriented plane
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
3-dimensional skew coordinates
An illustration of Euclid's parallel postulate
Rubik's cube: the study of its possible moves is a concrete application of group theory
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.
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.
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.
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
A page from al-Khwārizmī's Algebra
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.
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

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

- Euclidean 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).

- Euclidean space

One of the oldest such discoveries is Gauss' Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space.

- Geometry

The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane (plane geometry) and the (three-dimensional) Euclidean space.

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

4 related topics with Alpha

Overall

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.

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

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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.

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

Vector space

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[[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.

Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional Euclidean space.

This Togliatti surface is an algebraic surface of degree five. The picture represents a portion of its real locus.

Algebraic geometry

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This Togliatti surface is an algebraic surface of degree five. The picture represents a portion of its real locus.
Sphere and slanted circle

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.

For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R3 could be defined as the set of all points (x,y,z) with