What is Euclidean space: Definition and 58 Discussions

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




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



{\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. cianfa72

    I Homeomorphism linear subspace

    Hi, consider the Euclidean space ##\mathbb R^8## and the projection map ##\pi## over the first 4 coordinates, i.e. ##\pi : \mathbb R^8 \rightarrow \mathbb R^4##. I would show that the restriction of ##\pi## to the linear subspace ##A## (endowed with the subspace topology from ##\mathbb R^8##)...
  2. cianfa72

    I Definition of tangent vector on smooth manifold

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  3. SH2372 General Relativity (1): Euclidean space and coordinate systems

    SH2372 General Relativity (1): Euclidean space and coordinate systems

  4. Trysse

    I Five points in space with rational distances that are not co-linear

    Hi there, experts on three-D space! while thinking about (physical) space, I have come up with the following (geometry) question: Is it possible to define five points (A, B, C, D, E) in Euclidian space, so that all distances (AB, AC, AD, AE, BC, BD, BE, CD, CE, DE) can be expressed in rational...
  5. T

    I Dot product in Euclidean Space

    Hello As you know, the geometric definition of the dot product of two vectors is the product of their norms, and the cosine of the angle between them. (The algebraic one makes it the sum of the product of the components in Cartesian coordinates.) I have often read that this holds for Euclidean...
  6. Eclair_de_XII

    I How do you define unboundedness in Euclidean space?

    I read in my textbook Calculus on Manifolds by Spivak that a set ##A\subset \mathbb{R}^n## is bounded iff there is a closed n-rectangle ##D## such that ##A\subset D##. It should be plain that if I wanted to define unboundedness, I should just say something along the lines of: "A set ##A\subset...
  7. jk22

    B How to know if a Euclidean space is not a 3-sphere?

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  8. M

    Dot product and basis vectors in a Euclidean Space

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  9. JTC

    I Coordinate systems vs. Euclidean space

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  10. nomadreid

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  11. L

    I Does mobius transformation assume 3-D Euclidean space?

    Are the assumptions in mobius transformation valid in Newtonian physics?
  12. T

    B Exploring a Micro Singularity in Euclidean Space Time

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  13. DavideGenoa

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  14. M

    I Partitions of Euclidean space, cubic lattice, convex sets

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  15. J

    Unique smooth structure on Euclidean space

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  16. C

    Propagator in 2D Euclidean space

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  17. lucasLima

    Finding all vectors <x,z>=<y,z>=0

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  18. B

    Is R^n Euclidean Space a vector space too?

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  19. OrthoJacobian

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  20. Tony Stark

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  21. Rumo

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  22. G

    Continuing to Euclidean Space Justified in Path Integral?

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  23. R

    Euclidean space: dot product and orthonormal basis

    Dear All, Here is one of my doubts I encountered after studying many linear algebra books and texts. The Euclidean space is defined by introducing the so-called "standard" dot (or inner product) product in the form: (\boldsymbol{a},\boldsymbol{b}) = \sum \limits_{i} a_i b_i With that one...
  24. arpon

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  25. rjbeery

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  26. Sudharaka

    MHB Euclidean Space of Polynomials

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  27. Sudharaka

    MHB Orthogonal Transformation in Euclidean Space

    Hi everyone, :) Here's one of the questions that I encountered recently along with my answer. Let me know if you see any mistakes. I would really appreciate any comments, shorter methods etc. :) Problem: Let \(u,\,v\) be two vectors in a Euclidean space \(V\) such that \(|u|=|v|\). Prove that...
  28. BruceW

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  29. L

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  30. B

    Completeness of a set of basis vectors in 3D Euclidean space.

    Homework Statement The problem is Exercise 2 in the picture http://postimage.org/image/3ou3x1sh7/ Homework Equations The hint says: can you express and three-dimensional vector in terms of just two linearly independent vectors? The Attempt at a Solution I have no idea where...
  31. andrewkirk

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  32. G

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  33. J

    Wormholes in Euclidean Space?

    As it would appear the universe is spatially flat, a Euclidean Plane. If this is true then how could black holes exist? Doesn't this necessitate that if black holes are embedded in flat space that the mean curvature must be zero and thus all black holes are minimal surfaces? So, if black holes...
  34. D

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  35. L

    What is a realization of this surface in Euclidean space?

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  36. L

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  37. M

    Thought - Euclidean Space R^(-n)

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  38. Rasalhague

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  39. L

    Hypersurfaces of Euclidean space

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  40. G

    Can Complex Euclidean Space Be Defined?

    Homework Statement Euclidean space is the set of n-tuples with some operations and norm. I suddenly wonder if complex euclidean space can be defined. Is it also defined? Homework Equations The Attempt at a Solution
  41. B

    Calculate Killing Vectors in 3-D Euclidean Space

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  42. F

    Euclidean space, euclidean topology and coordinate transformation

    Hi, I have some doubts about the precise meaning of Euclidean space. I understand Euclidean space as the metric space (\mathbb{R}^n,d) where d is the usual distance d(x,y)=\sqrt{\sum_i(x_i-y_i)^2}. Now let's supose that we have our euclidean space (in 3D for simplicity) (\mathbb{R}^3,d)...
  43. V

    Length Contraction Euclidean Space

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  44. M

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    Hello my friends! My textbook has the following statement in one of its chapters: Chapter 8:Topology of R^n If you want a more abstract introduction to the topology of Euclidean spaces, skip the rest of this chapter and the next, and begin Chapter 10 now. Chapter 10 covers topological...
  45. M

    Our perception of Euclidean space

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  46. P

    Differentiation on Euclidean Space (Calculus on Manifolds)

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  47. K

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    I opted to not use the template because this is a pretty general question. I am not understanding how to find out if a set is an open subset of a Euclidean space. For example, {(x,y) belongs R2 | x squared + y squared < 1} The textbook is talking about open balls, greatly confusing me.
  48. M

    Properties of a volume in 3D Euclidean space

    Hello, I am writing a small report and trying to be mathematically accurate in my terminology- I am trying to describe an arbitrary volume of gas, but this volume must (1) not have any holes (or bubbles where the gas cannot go) in it, and (2) must be one single volume, so a gas molecule from...
  49. J

    Canonical measure on an infinite dimensional Euclidean space R^N

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  50. J

    What is an Example of a Closed Set with an Empty Interior in Euclidean Space?

    Give an example of a closed set S in R^2 such that the closure of the interior of S does not equal to S (in set notation). I have no idea where to start...any help would be nice! Thanks!