This choice gives us two alternative forms of geometry in which the interior angles of a triangle add up to exactly 180 degrees or less, respectively, and are known as Euclidean and hyperbolic geometries.

Angles are usually presumed to be in a Euclidean plane, but are also defined in non-Euclidean geometry.

This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on axioms and theorems to derive truth.

This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates ( x, y, z ).

These definitions are designed to be consistent with the underlying Euclidean geometry.

In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.

Richardson had believed, based on Euclidean geometry, that a coastline would approach a fixed length, as do similar estimations of regular geometric figures.

In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't require the hypothesis ( in the same way that it is desirable that statements in Euclidean geometry be proved using only the axioms of neutral geometry, i. e. no parallel postulate.

A circle is a simple shape of Euclidean geometry that is the set of points in the plane that are equidistant from a given point, the

The invention of Cartesian coordinates in the 17th century by René Descartes ( Latinized name: Cartesius ) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.

The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, although they still resemble the Euclidean space at each point " infinitesimally ", i. e. in the first order of approximation.

These are the closest analogues to the " ordinary " plane and space considered in Euclidean and non-Euclidean geometry.

In Euclidean geometry, the ellipse is usually defined as the bounded case of a conic section, or as the set of points such that the sum of the distances to two fixed points ( the foci ) is constant.

In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms.

Today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century.

* Euclidean geometry

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

For over two thousand years, the adjective " Euclidean " was unnecessary because no other sort of geometry had been conceived.

Euclidean geometry is an axiomatic system, in which all theorems (" true statements ") are derived from a small number of axioms.

Most of Nunes ' achievements were possible because of his profound understanding of spherical trigonometry and his ability to transpose Ptolemy's adaptations of Euclidean geometry to it.

In mathematics, the Borsuk – Ulam theorem, named after Stanisław Ulam and Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.

The Bolzano – Weierstrass theorem gives an equivalent condition for sequential compactness when considering subsets of Euclidean space: a set then is compact if and only if it is closed and bounded.

That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine – Borel theorem.

The following theorem connects Euclidean relations and equivalence relations:

It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization ( which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations ), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

( This doesn't violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.

Principal ideal domains are thus mathematical objects which behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements ( so an analogue of the fundamental theorem of arithmetic holds ); any two elements of a PID have a greatest common divisor ( although it may not be possible to find it using the Euclidean algorithm ).

The Heine – Borel theorem implies that a Euclidean n-sphere is compact.

This classical Kelvin – Stokes theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂ Σ:

The division algorithm ( see Euclidean division ) is a theorem expressing the outcome of division in the natural numbers and more general rings.

In mathematics, specifically in real analysis, the Bolzano – Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space R < sup > n </ sup >.

* The Borsuk – Ulam theorem: any continuous map from the n-sphere to Euclidean n-space identifies at least one pair of antipodal points.

In the study of complicated geometries, we call this ( most common ) type of distance Euclidean distance, as it is derived from the Pythagorean theorem, which does not hold in Non-Euclidean geometries.

For example, in the " game " of Euclidean geometry ( which is seen as consisting of some strings called " axioms ", and some " rules of inference " to generate new strings from given ones ), one can prove that the Pythagorean theorem holds ( that is, you can generate the string corresponding to the Pythagorean theorem ).

* Invariance of domain, a theorem in topology about homeomorphic subsets of Euclidean space

* The Euclidean spaces R < sup >< var > n </ var ></ sup > ( and in particular the real line R ) are locally compact as a consequence of the Heine – Borel theorem.

In particular, as follows from the Whitney embedding theorem, any m-dimensional Riemannian manifold admits an isometric C < sup > 1 </ sup >- embedding into an arbitrarily small neighborhood in 2m-dimensional Euclidean space.

** More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ƒ: U → R < sup > m </ sup >, where U is an open set in R < sup > n </ sup >, is almost everywhere differentiable.

In Euclidean geometry and some other geometries the triangle inequality is a theorem about distances.

In Euclidean geometry, for right triangles it is a consequence of the Pythagorean theorem, and for general triangles a consequence of the law of cosines, although it may be proven without these theorems.

This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the L < sup > p </ sup > spaces ( p ≥ 1 ), and inner product spaces.

It may be computed using the extended Euclidean algorithm or using Fermat's little theorem as k < sup > q − 2 </ sup > mod q.

The Euclidean distance formula is simply the Pythagorean theorem.