# Geometry

**geometricgeometricalgeometriesgeometricallyelementary geometrygeometerGeometersgeometric methodsgeometric propertiesgeometric shape**

Geometry (from the ; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.wikipedia

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### Mathematics

**mathematicalmathmathematician**

Geometry (from the ; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (mathematical analysis).

### Euclid

**Euclid of AlexandriaEuklidGreek Mathematician**

By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow.

Euclid ( – Eukleídēs, ; fl. 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".

### Mathematical analysis

**analysisclassical analysisanalytic**

By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat.

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).

### Surveying

**surveyorsurveyland surveyor**

Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts.

Surveyors work with elements of geometry, trigonometry, regression analysis, physics, engineering, metrology, programming languages, and the law.

### Manifold

**manifoldsboundarymanifold with boundary**

These include the concepts of point, line, plane, distance, angle, surface, and curve, as well as the more advanced notions of topology and 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 and understood in terms of the simpler local topological properties of Euclidean space.

### René Descartes

**DescartesCartesianRene Descartes**

By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. The first was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665).

He is credited as the father of analytical geometry, the bridge between algebra and geometry—used in the discovery of infinitesimal calculus and analysis.

### Frustum

**frustatruncated conesquare frustum**

For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum.

In geometry, a frustum (plural: frusta or frustums) is the portion of a solid (normally a cone or pyramid) that lies between one or two parallel planes cutting it.

### Area

**surface areaArea (geometry)area formula**

Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. 287–212 BC) of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi.

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.

### Thales's theorem

**Thales' theoremanother theorem with the same namespecial case**

He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem.

In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, then the angle ∠ABC is a right angle.

### Archimedes

**Archimedes of SyracuseArchimedeanArchimedes Heat Ray**

Archimedes (c.

Generally considered the greatest mathematician of antiquity and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals 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, and the area under a parabola.

### Thales of Miletus

**ThalesThalisThales Avionics**

In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore.

In mathematics, Thales used geometry to calculate the heights of pyramids and the distance of ships from the shore.

### Pi

**ππ\pi**

287–212 BC) of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi.

Because its most elementary definition relates to the circle, is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres.

### Greek mathematics

**Greek mathematicianancient Greek mathematiciansGreek**

In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. Geometry began to see elements of formal mathematical science emerging in Greek mathematics as early as the 6th century BC.

Thales is supposed to have used geometry to solve problems such as calculating the height of pyramids based on the length of shadows, and the distance of ships from the shore.

### Heron's formula

**Heron–Archimedes formulaHeron’s formula**

Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula), as well as a complete description of rational triangles (i.e. triangles with rational sides and rational areas).

In geometry, Heron's formula (sometimes called Hero's formula), named after Hero of Alexandria, gives the area of a triangle when the length of all three sides are known.

### Algebraic geometry

**complex algebraic geometryalgebraiccomputational algebraic geometry**

In the Middle Ages, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry.

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.

### Indian mathematics

**Indian mathematicianmathematicianmathematics**

Indian mathematicians also made many important contributions in geometry.

They mass-produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, and cylinders, thereby demonstrating knowledge of basic geometry.

### Playfair's axiom

**Euclid's AxiomPlayfair's formPlayfair's postulate**

The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were early results in hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including Witelo (c.

In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate):

### Babylonian mathematics

**BabyloniansBabylonian mathematiciansBabylonian**

1890 BC), the Babylonian clay tablets such as Plimpton 322 (1900 BC).

From 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems.

### Pythagoreanism

**PythagoreanPythagoreansPythagorean school**

Pythagoras established the Pythagorean School, which is credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.

Today, Pythagoras is mostly remembered for his mathematical ideas, and by association with the work early Pythagoreans did in advancing mathematical concepts and theories on harmonic musical intervals, the definition of numbers, proportion and mathematical methods such as arithmetic and geometry.

### Lambert quadrilateral

**Ibn al-Haytham–Lambert quadrilateral**

The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were early results in hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including Witelo (c.

In geometry, a Lambert quadrilateral,

### Arithmetic

**arithmetic operationsarithmeticsarithmetic operation**

Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry.

Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis.

### Thābit ibn Qurra

**Thabit ibn QurraThebitIbn Qurra**

Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry.

Thābit ibn Qurrah made important discoveries in algebra, geometry, and astronomy.

### Mathematics in medieval Islam

**mathematicianmathematicsIslamic mathematics**

In the Middle Ages, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry.

Important progress was made, such as the full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra (named for The Compendious Book on Calculation by Completion and Balancing by scholar Al-Khwarizmi), and advances in geometry and trigonometry.

### Equation

**equationsmathematical equationunknown**

The first was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665).

In geometry, equations are used to describe geometric figures.

### Felix Klein

**KleinFelix Christian KleinC. Felix Klein**

These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries).

Christian Felix Klein (25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory.