# What is Euclidean: Definition and 211 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

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. ### B Minkowski Spacetime vs Euclidean Spacetime

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4. ### MHB Euclidean Geometry - Demonstration Exercise

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5. ### MHB Proving Non-Degeneracy of Euclidean Inner Products

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7. ### Euclidean geometry: main theorems, formulas and concepts

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8. ### MHB How do I find the Euclidean Coordinate Functions of a parametrized curve?

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31. ### Euclidean Algorithm terminates in at most 7x the digits of b

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32. ### A Solving BTZ Black Hole w/ Euclidean Method

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33. ### Euclidean Methods for BTZ black Hole

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34. ### Self-dual solutions to Maxwell's equations, Euclidean space

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35. J

### B How to do the calculations showing the Universe is flat?

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37. ### Euclidean and non Euclidean geometries problems

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38. ### I QFT in Euclidean or Minkowski Spacetime

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39. ### Invariance of length of curve under Euclidean Motion

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40. ### A Euclidean action and Hamiltonian

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41. ### I Does mobius transformation assume 3-D Euclidean space?

Are the assumptions in mobius transformation valid in Newtonian physics?
42. ### I How is a manifold locally Euclidean?

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43. ### B Exploring a Micro Singularity in Euclidean Space Time

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44. ### A Ground state wave function from Euclidean path integral

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45. ### I Euclidean differential number counts of supernovae

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46. ### I GR vs SR: Is a Connection Necessary?

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47. ### MHB Result of Euclidean division

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48. ### Can someone help me with these 2 Euclidean geometry questions?

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49. ### A Euclidean signature and compact gauge group

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50. ### I Partitions of Euclidean space, cubic lattice, convex sets

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