* Arundhati Dasgupta, " The Measure in Euclidean Quantum Gravity.

* Euclidean Quantum Gravity, World Scientific ( Singapore, 1993 ); Paperback ISBN 981-02-0516-3

3, 249 – 251 ; see also Self-dual solutions to Euclidean Gravity.

* Tarski's axioms: Alfred Tarski ( 1902 – 1983 ) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basis, in contrast to Hilbert's axioms, which involve point sets.

Diffeomorphism does not respect distance and angle, so these key concepts of Euclidean geometry are lost on a smooth manifold.

Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.

The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the smaller number is subtracted from the larger number.

In Euclidean geometry, AAA ( Angle-Angle-Angle ) ( or just AA, since in Euclidean geometry the angles of a triangle add up to 180 °) does not provide information regarding the size of the two triangles and hence proves only similarity and not congruence in Euclidean space.

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.

Squared Euclidean Distance is not a metric as it does not satisfy the triangle inequality, however it is frequently used in optimization problems in which distances only have to be compared.

A global geometry is also called a topology, as a global geometry is a local geometry plus a topology, but this terminology is misleading because a topology does not give a global geometry: for instance, Euclidean 3-space and hyperbolic 3-space have the same topology but different global geometries.

Besides finding the greatest common divisor of integers a and b, as the Euclidean algorithm does, it also finds integers x and y ( one of which is typically negative ) that satisfy Bézout's identity

What does a pair of orthonormal vectors in 2-D Euclidean space look like?

* Euclidean space: The Riemann tensor of an n-dimensional Euclidean space vanishes identically, so the scalar curvature does as well.

Absolute geometry is a geometry based on an axiom system for Euclidean geometry that does not assume the parallel postulate or any of its alternatives.

It is sometimes referred to as neutral geometry ,< ref >< cite > Greenberg </ cite > cites W. Prenowitz and M. Jordan ( Greenberg, p. xvi ) for having used the term neutral geometry to refer to that part of Euclidean geometry that does not depend on Euclid's parallel postulate.

Unlike the case of real Euclidean space, in the complex case there are directions in which the artist can look which do not see the landscape ( because it does not have high enough dimension ).

Unlike a Euclidean distance matrix, the matrix does not need to be symmetric -- that is, the values x < sub > i, j </ sub > do not necessarily equal x < sub > j, i </ sub >.

In Euclidean space, three points usually determine a plane, but in the case of three collinear points this does not happen.

In particular, if the map is a continuous bijection ( a homeomorphism ), so that the two spaces have the same topology, then their i-th homotopy groups are isomorphic for all i. However, the real plane has exactly the same homotopy groups as a solitary point ( as does a Euclidean space of any dimension ), and the real plane with a point removed has the same groups as a circle, so groups alone are not enough to distinguish spaces.

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.

This does not match the 117 ° dihedral angle of a regular dodecahedron in Euclidean space, but in hyperbolic space there exist regular dodecahedra with any dihedral angle between 60º and 117º, and the hyperbolic dodecahedron with dihedral angle 72º may be used to give the Seifert – Weber space a geometric structure as a hyperbolic manifold.

In an axiomatic formulation of Euclidean geometry, such as that of Hilbert ( Euclid's original axioms contained various flaws which have been corrected by modern mathematicians ), a line is stated to have certain properties which relate it to other lines and points.

The problem of finding the Delaunay triangulation of a set of points in d-dimensional Euclidean space can be converted to the problem of finding the convex hull of a set of points in ( d + 1 )- dimensional space, by giving each point p an extra coordinate equal to | p |< sup > 2 </ sup >, taking the bottom side of the convex hull, and mapping back to d-dimensional space by deleting the last coordinate.

* 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.

A surface S in the Euclidean space R < sup > 3 </ sup > is orientable if a two-dimensional figure ( for example, 20px ) cannot be moved around the surface and back to where it started so that it looks like its own mirror image ( 20px ).

When M is a Euclidean space and all the fields are taken to be invariant by translations by the vectors of M, we get back to a situation where a tensor field is synonymous with a tensor ' sitting at the origin '.

Results obtained from the mathematically well-defined Euclidean path integral may be Wick-rotated back and give the same physical results as would be obtained by appropriate treatment of the ( potentially divergent ) Minkowskian path integral.

Further, he sent a new problem back to Van Roomen, for resolution by Euclidean tools ( rule and compass ) of the lost answer to the problem first set by Apollonius of Perga.

Formally relates Euclidean quantum gravity to ADM formalism.

It was noted by Minkowski ( 1907 ) that his space-time formalism represents a " four-dimensional non-euclidean manifold ", but in order to emphasize the formal similarity to the more familiar Euclidean geometry, Minkowski noted that the time coordinate could be treated as imaginary.

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 ).

Graphing calculators can be used to graph functions defined on the real line, or higher dimensional Euclidean space.

As suggested by the etymology of the word, one of the earliest reasons for interest in geometry was surveying, and certain practical results from Euclidean geometry, such as the right-angle property of the 3-4-5 triangle, were used long before they were proved formally.

The result can be considered as a type of generalized geometry, projective geometry, but it can also be used to produce proofs in ordinary Euclidean geometry in which the number of special cases is reduced.

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.

Nonetheless, a Euclidean model of the universe can still be used to solve many other practical problems with sufficient precision.

The Euclidean algorithm proceeds in a series of steps such that the output of each step is used as an input for the next one.

For illustration, the Euclidean algorithm can be used to find the greatest common divisor of a = 1071 and b = 462.

), and gyrovector spaces ( a geometry used in both relativity and quantum mechanics which is not Euclidean ).

In order to find the greatest common divisor, the Euclidean algorithm may be used.

When talking about null sets in Euclidean n-space R < sup > n </ sup >, it is usually understood that the measure being used is Lebesgue measure.

Vector calculus ( or vector analysis ) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3 dimensional Euclidean space The term " vector calculus " is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration.

Dedekind used the German word Schnitt ( cut ) in a visual sense rooted in Euclidean geometry.

The model for hyperbolic geometry was answered by Eugenio Beltrami, in 1868, who first showed that a surface called the pseudosphere has the appropriate curvature to model a portion of hyperbolic space, and in a second paper in the same year, defined the Klein model which models the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent, so that hyperbolic geometry was logically consistent if and only if Euclidean geometry was.

When working exclusively in two-dimensional Euclidean space, the definite article is used, so, the plane refers to the whole space.

In terms closer to those that Hilbert would have used, near the identity element e of the group G in question, we have some open set U in Euclidean space containing e, and on some open subset V of U we have a continuous mapping

In the case of R < sup > n </ sup >, the Euclidean norm is typically used.

While seldom used explicitly, the geometric view of the complex numbers is implicitly based on its structure of a Euclidean vector space of dimension 2, where the inner product of complex numbers and is given by ; then for a complex number its absolute value || coincides with its Euclidean norm, and its argument with the angle turning from 1 to.

In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs.

The extended Euclidean algorithm may be used to compute it.

This inner product is similar to the usual Euclidean inner product, but is used to describe a different geometry ; the geometry is usually associated with relativity.

In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry.