A report on Geometry

An illustration of Desargues' theorem, a result in Euclidean and projective geometry
A European and an Arab practicing geometry in the 15th century
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
An illustration of Euclid's parallel postulate
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
A sphere is a surface that can be defined parametrically (by  or implicitly (by x2 + y2 + z2 − r2 = 0.)
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 Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
A tiling of the hyperbolic plane
Differential geometry uses tools from calculus to study problems involving curvature.
A thickening of the trefoil knot
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.

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

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

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

Mathematics

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3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
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 quadratic formula expresses concisely the solutions of all quadratic equations
Rubik's cube: the study of its possible moves is a concrete application of group theory
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
A page from al-Khwārizmī's Algebra
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.
Leonhard Euler created and popularized much of the mathematical notation used today.
Carl Friedrich Gauss, known as the prince of mathematicians
The front side of the Fields Medal
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).

Detail from Raphael's The School of Athens featuring a Greek mathematician – perhaps representing Euclid or Archimedes – using a compass to draw a geometric construction.

Euclidean geometry

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Detail from Raphael's The School of Athens featuring a Greek mathematician – perhaps representing Euclid or Archimedes – using a compass to draw a geometric construction.
The parallel postulate (Postulate 5): If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
A proof from Euclid's Elements that, given a line segment, one may construct an equilateral triangle that includes the segment as one of its sides: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.
An example of congruence. The two figures on the left are congruent, while the third is similar to them. The last figure is neither. Congruences alter some properties, such as location and orientation, but leave others unchanged, like distance and angles. The latter sort of properties are called invariants and studying them is the essence of geometry.
Congruence of triangles is determined by specifying two sides and the angle between them (SAS), two angles and the side between them (ASA) or two angles and a corresponding adjacent side (AAS). Specifying two sides and an adjacent angle (SSA), however, can yield two distinct possible triangles unless the angle specified is a right angle.
A sphere has 2/3 the volume and surface area of its circumscribing cylinder. A sphere and cylinder were placed on the tomb of Archimedes at his request.
René Descartes. Portrait after Frans Hals, 1648.
Squaring the circle: the areas of this square and this circle are equal. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge.
A disproof of Euclidean geometry as a description of physical space. In a 1919 test of the general theory of relativity, stars (marked with short horizontal lines) were photographed during a solar eclipse. The rays of starlight were bent by the Sun's gravity on their way to Earth. This is interpreted as evidence in favor of Einstein's prediction that gravity would cause deviations from Euclidean geometry.
The pons asinorum or bridge of asses theorem states that in an isosceles triangle, α = β and γ = δ.
The triangle angle sum theorem states that the sum of the three angles of any triangle, in this case angles α, β, and γ, will always equal 180 degrees.
The Pythagorean theorem states that the sum of the areas of the two squares on the legs (a and b) of a right triangle equals the area of the square on the hypotenuse (c).
Thales' theorem states that if AC is a diameter, then the angle at B is a right angle.
A surveyor uses a level
Sphere packing applies to a stack of oranges.
A parabolic mirror brings parallel rays of light to a focus.
Geometry is used in art and architecture.
The water tower consists of a cone, a cylinder, and a hemisphere. Its volume can be calculated using solid geometry.
Geometry can be used to design origami.

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

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.

Archimedes Thoughtful
by Domenico Fetti (1620)

Archimedes

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Approximation of pi; defining and investigating the spiral that now bears his name; and devising a system using exponentiation for expressing very large numbers.

Approximation of pi; defining and investigating the spiral that now bears his name; and devising a system using exponentiation for expressing very large numbers.

Archimedes Thoughtful
by Domenico Fetti (1620)
The Death of Archimedes (1815) by Thomas Degeorge
Cicero Discovering the Tomb of Archimedes (1805) by Benjamin West
The Archimedes' screw can raise water efficiently.
Archimedes may have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse.
Archimedes calculates the side of the 12-gon from that of the hexagon and for each subsequent doubling of the sides of the regular polygon.
A proof that the area of the parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure from Quadrature of the Parabola.
Frontpage of Archimedes' Opera, in Greek and Latin, edited by David Rivault (1615).
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases.
Ostomachion is a dissection puzzle found in the Archimedes Palimpsest.
In 1906, the Archimedes Palimpsest revealed works by Archimedes thought to have been lost.
The Fields Medal carries a portrait of Archimedes.
Artistic interpretation of Archimedes' mirror used to burn Roman ships.
Painting by Giulio Parigi, c. 1599.
Bronze statue of Archimedes in Berlin

Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including: the area of a circle; the surface area and volume of a sphere; area of an ellipse; the area under a parabola; the volume of a segment of a paraboloid of revolution; the volume of a segment of a hyperboloid of revolution; and the area of a spiral.

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.

If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side.

Parallel postulate

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If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side.
Euclidean, elliptical and hyperbolic geometry. The Parallel Postulate is satisfied only for models of Euclidean geometry.
The converse of the parallel postulate: If the sum of the two interior angles equals 180°, then the lines are parallel and will never intersect.

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry.

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

Calculus

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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.

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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Branch of mathematics, classically studying zeros of multivariate polynomials.

Branch of mathematics, classically studying zeros of multivariate polynomials.

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

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.

Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.

Analytic geometry

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Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
The distance formula on the plane follows from the Pythagorean theorem.
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In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system.

The red and blue lines on this graph have the same slope (gradient); the red and green lines have the same y-intercept (cross the y-axis at the same place).

Line (geometry)

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The red and blue lines on this graph have the same slope (gradient); the red and green lines have the same y-intercept (cross the y-axis at the same place).
A representation of one line segment.
Ray

In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.