Euclidean in a sentence

This insight had the corollary that non-Euclidean geometry was consistent if and only if Euclidean geometry was, putting Euclidean and non-Euclidean geometries on the same footing, and ending all controversy surrounding non-Euclidean geometry.

Euclidean transformations The Euclidean transformations or Euclidean motions are the ( bijective ) mappings of points of the Euclidean plane to themselves which preserve distances between points.

It is important to note that a particular Euclidean function f is not part of the structure of a Euclidean domain: in general, a Euclidean domain will admit many different Euclidean functions.

Ideas and terminology from Euclidean geometry (both traditional and analytic) are pervasive in modern mathematics, where other geometric objects share many similarities with Euclidean spaces, share part of their structure, or embed Euclidean spaces.

If a field is not norm-Euclidean then that does not mean the ring of integers is not Euclidean, just that the field norm does not satisfy the axioms of a Euclidean function.

If we consider that its length is actually the distance from its tail to its tip, it becomes clear that the Euclidean norm of a vector is just a special case of Euclidean distance: the Euclidean distance between its tail and its tip.

A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function.

All Euclidean spaces are affine, but there are affine spaces that are non-Euclidean.

An implication of Albert Einstein 's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only where the gravitational field is weak.

Ball, p. 485 Since non-Euclidean geometry is provably relatively consistent with Euclidean geometry, the parallel postulate cannot be proved from the other postulates.

Carnap, R, An introduction to the philosophy of science, p. 148 Since Euclidean geometry is simpler than non-Euclidean geometry, he assumed the former would always be used to describe the 'true' geometry of the world.

Discovery of non-Euclidean geometry The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry.

Euclidean method The Euclidean method was first introduced in Efficient exponentiation using precomputation and vector addition chains by P.D Rooij.

Euclidean spaces have finite dimension. citation Intuitive overview One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle.

Euclidean structure These are distances between points and the angles between lines or vectors, which satisfy certain conditions (see below), which makes a set of points a Euclidean space.

For example, the Klein bottle is a surface, which cannot be represented in the three-dimensional Euclidean space without introducing self-intersections (it cannot be embedded in the three dimensional Euclidean space).

Further formulas for general Euclidean triangles seeAlso The formulas in this section are true for all Euclidean triangles.

Geometrically, one studies the Euclidean plane ( two dimensions ) and Euclidean space ( three dimensions ).

However, if there is no "obvious" Euclidean function, then determining whether R is a PID is generally a much easier problem than determining whether it is a Euclidean domain.

In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space ).