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.

Bézout's lemma is a consequence of the Euclidean division defining property, namely that the division by a nonzero integer b has a remainder strictly less than | b |.

For example, the minimum spanning tree of the graph associated with an instance of the Euclidean TSP is a Euclidean minimum spanning tree, and so can be computed in expected O ( n log n ) time for n points ( considerably less than the number of edges ).

* The sum of the measures of the angles of any triangle is less than 180 ° if the geometry is hyperbolic, equal to 180 ° if the geometry is Euclidean, and greater than 180 ° if the geometry is elliptic.

Projective geometry is less restrictive than either Euclidean geometry or affine geometry.

The Frenet-Serret frame plays a key role in the differential geometry of curves, ultimately leading to a more or less complete classification of smooth curves in Euclidean space up to congruence.

The occasion of the proof by Hassler Whitney of the embedding theorem for smooth manifolds is said ( rather surprisingly ) to have been the first complete exposition of the manifold concept precisely because it brought together and unified the differing concepts of manifolds at the time: no longer was there any confusion as to whether abstract manifolds, intrinsically defined via charts, were any more or less general than manifold extrinsically defined as submanifolds of Euclidean space.

In geometric group theory, triangle groups are classified into Euclidean, spherical, and hyperbolic cases according to whether an associated sum of unit fractions is equal to one, greater than one, or less than one respectively.

However, differentiation of vector fields is less straightforward: this is a simple matter in Euclidean space, because the tangent space of based vectors at a point p can be identified naturally ( by translation ) with the tangent space at a nearby point q.

In cellular automata defined over tessellations of the hyperbolic plane, or of higher dimensional hyperbolic spaces, the counting argument in the proof of the Garden of Eden theorem does not work, because it depends implicitly on the property of Euclidean spaces that the boundary of a region grows less quickly than its volume as a function of the radius.

In mathematics, particularly linear algebra and numerical analysis, the Gram – Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space R < sup > n </ sup >.

In more than two dimensions, conformal geometry may refer either to the study of conformal transformations of " flat " spaces ( such as Euclidean spaces or spheres ), or, more commonly, to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics defined up to scale.

In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most commonly studied cases, the range will lie in a Euclidean space such as the 2-dimensional plane ( a planar curve ) or the 3-dimensional space ( space curve ).

A lattice arrangement ( commonly called a regular arrangement ) is one in which the centers of the spheres form a very symmetric pattern which only needs n vectors to be uniquely defined ( in n-dimensional Euclidean space ).

Its magnitude or length is most commonly defined as its Euclidean norm ( or Euclidean length ):

Most commonly, the vectors are elements of an Euclidean space, or are functions

In particular, the embedding diagram most commonly found in textbooks ( an isometric embedding of a constant-time equatorial slice of the Schwarzschild metric in Euclidean 3-dimensional space ) superficially resembles a gravity well.

Cells are defined in a normed space, commonly a two-dimensional Euclidean geometry, like a grid.

which can easily be proven by considering the Euclidean algorithm in base n. Another useful identity relates to the Euler's totient function:

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:

There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space ( usually a Euclidean space ) in a way that relates to the radius of curvature of circles that touch the object, and intrinsic curvature, which is defined at each point in a Riemannian manifold.

relates how John Leech about 1964 investigated close packings of spheres in Euclidean spaces of large dimension.

Similarly it relates to the infinite series of tilings with the face configurations V3. 2n. 3. 2n, the first in the Euclidean plane, and the rest in the hyperbolic plane.

Formally relates Euclidean quantum gravity to ADM formalism.

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

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

Euclid frequently used the method of proof by contradiction, and therefore the traditional presentation of Euclidean geometry assumes classical logic, in which every proposition is either true or false, i. e., for any proposition P, the proposition " P or not P " is automatically true.

The term “ Euclidean ” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, and is named for the Greek mathematician Euclid of Alexandria.

Euclid ( c. 325-265 BC ), of Alexandria, probably a student of one of Plato ’ s students, wrote a treatise in 13 books ( chapters ), titled The Elements of Geometry, in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry.

* 300 Euclid, Greek mathematician, publishes Elements, treating both geometry and number theory ( see also Euclidean algorithm ).

Named for the type of zoning code adopted in the town of Euclid, Ohio, and approved in a landmark decision of the U. S. Supreme Court, Village of Euclid, Ohio v. Ambler Realty Co. Euclidean zoning codes are the most prevalent in the United States.

While Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century.

* Euclidean zoning, a system of land use management modeled after the zoning code of Euclid, Ohio

Euclid of Alexandria authored the earliest extant axiomatic presentation of Euclidean geometry and number theory.

Since Euclid, geometry had meant the geometry of Euclidean space of two dimensions ( plane geometry ) or of three dimensions ( solid geometry ).

Single-use zoning is often called Euclidean zoning by urban planners and other professionals, a reference to the court case that established its constitutionality, Village of Euclid, Ohio v. Ambler Realty Co.

Euclid described a line as " breadthless length ", and introduced several postulates as basic unprovable properties from which he constructed the geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of nineteenth century ( such as non-Euclidean geometry, projective geometry, and affine geometry ).

The ancient Greek deiknymi, or thought experiment, " was the most ancient pattern of mathematical proof ", and existed before Euclidean mathematics, where the emphasis was on the conceptual, rather than on the experimental part of a thought-experiment.

The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory.

It gains a measure of efficiency over the ancient Euclidean algorithm by replacing divisions and multiplications with shifts, which are cheaper when operating on the binary representation used by modern computers.

Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems.

* Theodosius of Bithynia, Greek astronomer and mathematician who will write the Sphaerics, a book on the geometry of the sphere ( d. c. 100 BC ), later translated from Arabic back into Latin to help restore knowledge of Euclidean geometry to the West.

Sidis taught three classes: Euclidean geometry, non-Euclidean geometry, and trigonometry ( he wrote a textbook for the Euclidean geometry course in Greek ).