Euclidean Definition and 214 Threads

Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the Euclidean plane (dimension two). It was introduced by the Ancient Greek mathematician Euclid of Alexandria, and the qualifier Euclidean is used to distinguish it from other spaces that were later discovered in physics and modern mathematics.
Ancient Greek geometers introduced Euclidean space for modeling the physical universe. Their great innovation was to prove all properties of the space as theorems by starting from a few fundamental properties, called postulates, which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (parallel postulate).
After the introduction at the end of 19th century of non-Euclidean geometries, the old postulates were re-formalized to define Euclidean spaces through axiomatic theory. Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article.In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space.
There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are isomorphic. Therefore, in many cases, it is possible to work with a specific Euclidean space, which is generally the real n-space





R


n


,


{\displaystyle \mathbb {R} ^{n},}
equipped with the dot product. An isomorphism from a Euclidean space to





R


n




{\displaystyle \mathbb {R} ^{n}}
associates with each point an n-tuple of real numbers which locate that point in the Euclidean space and are called the Cartesian coordinates of that point.

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  1. E

    Surface Volume in 4-d graph: Euclidean Geometry Question

    "Surface Volume" in 4-d graph: Euclidean Geometry Question Suppose you have a smooth parametrically defined volume V givin by the following equation. f(x,y,z,w)= r(u,s,v) = x(u,v,s)i + y(u,v,s)j +z(u,v,s)k + w(u,v,s)l Consider the vectors ru=dr/du, where dr/du is the partial...
  2. E

    How Does Parametric Volume Calculation Work in Four-Dimensional Euclidean Space?

    Suppose you have a smooth parametrically defined volume V givin by the following equation. f(x,y,z,w)= r(u,s,v) = x(u,v,s)i + y(u,v,s)j +z(u,v,s)k + w(u,v,s)l Consider the vectors ru=dr/du, where dr/du is the partial derivitive of r with respect to the parameter u. Similarly, rv =...
  3. C

    Why Minkowski spacetime and not Euclidean spacetime?

    Why does everyone use +---/-+++ Minkowski spacetime over ++++ Euclidean spacetime? Minkowski spacetime preserves spacetime intervals under Lorentz transformations but so does Euclidean spacetime under equivalent rotational transformations from which SR can also be deduced. (someone show me how...
  4. CarlB

    Is Euclidean Quantum Gravity the Future of Nonperturbative Physics?

    Stephen Hawking's latest preprint on Arxiv uses "Euclidean Quantum Gravity". In fact, he says: "I adopt the Euclidean approach [5], the only sane way to do quantum gravity nonperturbatively." http://www.arxiv.org/abs/hep-th/0507171 Any comments? Carl
  5. H

    Flat torus embedding in euclidean space?

    hi, for most of you this might be a simple question: Is it possible to embed the flat torus in Euclidean space? If we, for example, take a rectangle and identify the left and the right sides we get a cylinder shell, that can be embedded easily in R^3. If we construct the...
  6. M

    Prove/Disprove Euclidean Domains: Unique q & r Exist?

    this seems to be a very fundamental problem...but i need help... prove or disprove : let D be a euclidean domain with size function d, then for a,b in D, b != 0, there exist unique q,r in D such that a= qb+r where r=0 or d(r) < d(b). first of all, what is size function? next...do we only...
  7. I

    Proving AB is Not Equal to CD in Euclidean Geometry

    If you had line AB is parallel to BC and BC is parallel to CD, is AB parallel to CD? ----> Not if AB=CD since a line (at least in Euclidean Geometry) cannot be parallel to itself. How would you prove that AB is not line CD? PLEASE NOTE: Base all your input in the realm of Euclidean...
  8. O

    Spacetime and Euclidean Geometry-Jacobson

    Ted Jacobson has been developing a theorum using only the principle of Relativity and Euclidean Geometry:http://arxiv.org/abs/gr-qc?0407022 Having followed his papers for some time I have to say its quite amazing(this paper more so!), but I have found a flaw in this evolution paper.
  9. Loren Booda

    Curvature of reciprocal Euclidean space

    A triangle in Euclidean space can be described as having a hypotenuse of one, and legs of Lorentz parameters \beta and \gamma. What spatial curvature underlies a triangle with hypotenuse one, and legs 1/ \beta and 1/ \gamma?
  10. C

    Using the Euclidean Algorithm to Find Values for x and y in Linear Combinations?

    I need to be able to plug in appropriate x and y values for: 154x + 260y = 4 I guess this is done by working the euclidean algorithm backwards. But how do you do that?
  11. T

    Is Non-Euclidean Geometry the Key to Unlocking New Mathematical Discoveries?

    Hi everyone, i have to do a general math project for my math course. It is nothng special, just a proof of a theorem on my choice and a bit of history and interesting facts. I kind of decided to do it on non-euclidean geometry, because it is fun. Now my question: does anybody know a good...
  12. A

    Understanding the Euclidean Algorithm: Solving for Relatively Prime Numbers

    I am little confused on this issue, actually the thing that is confusing me is my book. We all know the formula m=qn+r and 0<r<n In the formula we go until r=0, then you will know relativley prime. When given a problem, for example, (862,347) we have m=862 while n=347 so we have the...
  13. S

    Proving Bisection of Angles with Euclidean Geometry

    I was at brunch this morning and I met a young man of about 35 years who is a musician but studied mathematics in college. He mentioned to me that one of the great problems of mathematics is the problem of trisecting the angle. He taught me how to bisect an angle, and kept emphasizing that...
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