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




{\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. bobrubino

    B Minkowski Spacetime vs Euclidean Spacetime

    Which one would you use in order to map out a black hole and its connection to a white hole?
  2. N

    I Do AEST (Absolute Euclidean Spacetime) models work?

    I was reading a paper by J.M.C Montanus which was published in <low quality journal reference removed> in which he claims under AEST the new gravitational dynamics and electrodynamics are reformulated in close correspondence with classical physics, and subsequently leads to the correct...
  3. M

    Use the Euclidean algorithm to find integers ## a, b, c ##

    Let ## a, b, c ## and ## d ## be integers such that ## 225a+360b+432c+480d=3 ##. Then ## 75a+120b+144c+160d=1 ##. Applying the Euclidean algorithm produces: ## gcd(75, 120)=15, gcd(120, 144)=24 ## and ## gcd(144, 160)=16 ##. Now we see that ## 15x+24y+16z=1 ##. By Euclidean algorithm, we have...
  4. S

    MHB Euclidean Geometry - Demonstration Exercise

    (a) Let be m a line and the only two semiplans determined by m. (i) Show that: If are points that do not belong to such , so and are in opposite sides of m. (ii) In the same conditions of the last item, show: and . (iii) Determine the union result , carefully justifying your answer...
  5. S

    MHB Proving Non-Degeneracy of Euclidean Inner Products

    I have the following (small) problem: Let $ ( , ):V \times V \rightarrow \mathbb{R} $ be a real-valued non-degenerate inner product on the real vector space $V$. Given, for all $v \in V$ we have $(v,v) \geq 0$ Now prove that if $(x,x)=0$ then $x=0$ for $x \in V$, that is, prove that the inner...
  6. greg_rack

    Euclidean geometry: main theorems, formulas and concepts

    Hi guys, Hopefully, no geometry-enthusiasts are going to read these next few lines, but if that's the case, be lenient :) I have always hated high-school geometry, those basic boring theorems about triangles, polygons, circles, and so on, and I have always "skipped" such classes, studying...
  7. banananaz

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

    I've been given a curve α parametrized by t : α (t) = (cos(t), t^2, 0) How would I go about finding the euclidean coordinate functions for this curve? I know how to find euclidean coord. fns. for a vector field, but I am a bit confused here. (Sorry about the formatting)
  8. SchroedingersLion

    A Is the Chain Rule Applicable to the Euclidean Norm in Calculating Derivatives?

    Greetings, suppose we have 3d vectors ##\mathbf{x}_k, \mathbf{y}_k, \mathbf{b}## for ##k=1,...,N## and a 3x3 matrix ##\mathbf{W}## with real elements ##w_{i,j}##. Are the following two results correct? $$ \frac{\partial}{\partial \mathbf{b}} \sum_k ||\mathbf{Wx}_k+\mathbf{b}-\mathbf{y}_k||² =...
  9. 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...
  10. binis

    B Is the Euclidean postulate a theorem?

    Consider a point A outside of a line α. Α and α define a plane.Let us suppose that more than one lines parallels to α are passing through A. Then these lines are also parallels to each other; wrong because they all have common point A.
  11. LittleRookie

    B Before understanding theorems in elementary Euclidean plane geometry

    Before looking at the proof of basic theorems in Euclidean plane geometry, I feel that I should draw pictures or use other physical objects to have some idea why the theorem must be true. After all, I should not just plainly play the "game of logic". And, it is from such observations in real...
  12. T

    I Does Euclidean geometry require initial fine-tuning?

    https://en.wikipedia.org/wiki/Flatness_problem The flatness problem (also known as the oldness problem) is a cosmological fine-tuning problem within the Big Bang model of the universe. The fine-tuning problem of the last century was solved by introducing the theory of inflation which flattens...
  13. 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...
  14. jk22

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

    If we suppose the following 8-dimensional manifold given by $$a_1=cos(x)cos(y)cos(z)$$ $$a_2=cos(x)cos(y)sin(z)$$ $$a_3=cos(x)sin(y)cos(z)$$ $$a_4=cos(x)sin(y)sin(z)$$ $$a_5=sin(x)cos(y)cos(z)$$ $$a_6=sin(x)cos(y)sin(z)$$ $$a_7=sin(x)sin(y)cos(z)$$ $$a_8=sin(x)sin(y)sin(z)$$ Then obviously...
  15. Cerenkov

    B Is the Hawking-Hartle Euclidean approach confirmed by CMBR data?

    Hello. I have three questions about a claim made by Stephen Hawking in his book, 'My Brief History' and I would be grateful to receive some help concerning it please. Here is a .pdf version of it...
  16. ali PMPAINT

    B Proving congruent with Euclidean axioms

    So, given one, you can prove the others, but I don't know how to prove one with using the five axioms.
  17. R

    Modified Euclidean Algorithm proof

    gcd(f_n,f_{n-1}) gcd[f_{n-1},f_n - f_{n-1}] gcd[(f_n - f_{n-1}), (f_{n-2} - f_{n-1})] gcd[(f_{n-2} - f_{n-1}),f_{n-3} - f_{n-2})] gcd[(f_{n-2} - f_{n-3}),(f_{n-4} - f_{n-3})] . . . gcd(f_2,f_1), where f_2 = 1, f_1 = 1 I assume LateX is not working yet. Not sure if I am on point here or not...
  18. R

    Continuity of a function under Euclidean topology

    Homework Statement Let ##f:X\rightarrow Y## with X = Y = ##\mathbb{R}^2## an euclidean topology. ## f(x_1,x_2) =( x^2_1+x_2*sin(x_1),x^3_2-sin(e^{x_1+x_2} ) )## Is f continuous? Homework Equations f is continuous if for every open set U in Y, its pre-image ##f^{-1}(U)## is open in X. or if...
  19. FritoTaco

    Finding the Inverse of 2 (mod 17): Euclidean Extended Algorithm

    Homework Statement Hi, I'm doing a problem by solving congruences but my question is simply trying to find the inverse of 2 \enspace (mod\enspace 17) from 2x \equiv 7(mod \enspace 17). Homework Equations It's hard to find a definition that makes sense but if you check my upload images you...
  20. M

    Dot product and basis vectors in a Euclidean Space

    Homework Statement I am asked to write an expression for the length of a vector V in terms of its dot product in an arbitrary system in Euclidean space. Homework EquationsThe Attempt at a Solution The dot product of a vector a with itself can be given by I a I2. Does that expression only apply...
  21. S

    I What's the difference between Euclidean & Cartesian space?

    What's the difference between Euclidean & Cartesian space?
  22. Math Amateur

    MHB Definition of a Euclidean Domain ....

    In the book "The Basics of Abstract Algebra" Bland defines a Euclidean Domain using two conditions as follows: In the book "Abstract Algebra"by Dummit and Foote we find that a Euclidean Domain is defined using only one of Bland's conditions ... as follows:What are the consequences of these...
  23. RoboNerd

    Finding GCD with Fibonacci: Base Case

    Homework Statement Suppose that m divisions are required to find gcd(a,b). Prove by induction that for m >= 1, a >= F(m+2) and b>= F(m+1) where F(n) is the Fibonacci sequence. Hint: to find gcd(a,b), after the first division the algorithm computes gcd(b,r). Homework Equations Fibonacci...
  24. Math Amateur

    MHB The Euclidean Norm is Lipschitz Continuous .... D&K Example 1.3.5 .... ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Example 1.3.5 ... ... The start of Duistermaat and Kolk's Example 1.3.5 reads as...
  25. A

    Contractions of the Euclidean Group ISO(3) = E(3)

    Homework Statement Consider the contractions of the 3D Euclidean symmetry while preserving the SO(2) subgroup. In the physics point of view, explain the resulting symmetries G(2) (Galilean symmetry group) and H(3) (Heisenberg-Weyl group for quantum mechanics) and give their Lie algebras...
  26. JTC

    I Coordinate systems vs. Euclidean space

    Good Morning I am having some trouble categorizing a few concepts (I made the one that is critical to this post to be BOLD) Remote parallelism: the ability to move coordinate systems and frames around in space. Euclidean Space Coordinate systems: Cartesian vs. cylindrical I am aware that if...
  27. Vicol

    A Let's remove one axiom from Euclidean geometry

    I'm wondering what could happen if we remove one axiom from Euclidean geometry. What are the conseqences? For example - how would space without postulate "To describe a cicle with any centre and distance" look like?
  28. PsychonautQQ

    I Is R_d x R Locally Euclidean of Dimension 1?

    Let us look at the topological space R_d x R where R_d is the set of real numbers with the discrete toplogy and R the euclidean topology. This set is not second countable, because R_d has no countable basis. I am wondering if this space is locally euclidean, and if so, of what dimension? Given...
  29. evinda

    MHB Number of steps of euclidean algorithm

    Hello! (Wave) I am looking at the following exercise: Let $b=r_0, r_1, r_2, \dots$ be the successive remainders in the Euclidean algorithm applied to $a$ and $b$. Show that after every two steps, the remainder is reduced by at least one half. In other words, verify that $$r_{i+2}< \frac{1}{2}...
  30. T

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

    Homework Statement please see the image Homework Equations I'm not sure if this is relevant: ##r_2 \leq \frac{1}{2}r_1## ... ##r_n \leq (\frac{1}{2})^nr_1## The Attempt at a Solution i have shown that ##r_{i+2} < r_i## by showing the ##r_{i+2} - r_i## is negative, but how do I show that the...
  31. C

    A Solving BTZ Black Hole w/ Euclidean Method

    I know this is some kind of exercise problem, but it isnot widely discussed in general general relativity textbook. Sorry to post it here. I want to calculate the mass and entropy of non-rotating BTZ black hole using Euclidean method. When I calculate the Euclidean action, I always get an...
  32. C

    Euclidean Methods for BTZ black Hole

    This is an exercise from Hartman's lecture 6th. Using the Euclidean method to calculate the BTZ black hole mass entropy. The BTZ metric is given by $$ ds^2=(r^2-8M)d\tau^2 +\frac{dr^2}{r^2-8M}+r^2d\phi^2$$ and ##\tau \sim \tau+\beta, \beta=\frac{\pi}{\sqrt{2M}}##. Then we calculate the...
  33. nomadreid

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

    I am attempting to understand a question posed to me by an acquaintance, who asked me if I could refer him to literature freely available on the Internet on "self-dual solutions to Maxwell's equations on Euclidean space, or pseudo-Euclidean space, not Minkowski space (where there are none)" and...
  34. J

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

    I've been trying to understand how we know that the observable universe is flat, and I'm having difficulty finding any sources that explain exactly how the calculations were done. On this WMAP website (https://map.gsfc.nasa.gov/mission/sgoals_parameters_geom.html), it says: "A central feature of...
  35. A

    A Boundary conditions on the Euclidean Schwarzschild black hole

    This question is based on page 71 of Thomas Hartman's notes on Quantum Gravity and Black Holes (http://www.hartmanhep.net/topics2015/gravity-lectures.pdf). The Euclidean Schwarzschild black hole $$ds^{2} = \left(1-\frac{2M}{r}\right)d\tau^{2} + \frac{dr^{2}}{1-\frac{2M}{r}} +...
  36. N

    Euclidean and non Euclidean geometries problems

    So I was reading this book, "Euclidean and non Euclidean geometries" by Greenberg I solved the first problems of the first chapter, and I would like to verify my solutions 1. Homework Statement Homework Equations [/B] Um, none that I can think of? The Attempt at a Solution (1) Correct...
  37. LarryS

    I QFT in Euclidean or Minkowski Spacetime

    Forgetting for the moment about curved spacetime, does the relativistic QFT in use today by experimental physicists live in Euclidean spacetime or Minkowski spacetime. Thanks in advance.
  38. M

    Invariance of length of curve under Euclidean Motion

    Homework Statement Show that the length of a curve γ in ℝn is invariant under euclidean motions. I.e., show that L[Aγ] = L[γ] for Ax = Rx + a Homework Equations The length of a curve is given by the arc-length formula: s(t) = ∫γ'(t)dt from t0 to tThe Attempt at a Solution I would imagine I...
  39. ShayanJ

    A Euclidean action and Hamiltonian

    Yesterday I was asking questions from someone and in between his explanations, he said that the Euclidean action in a QFT is actually equal to its Hamiltonian. He had to go so there was no time for me to ask more questions. So I ask here, is it true? I couldn't find anything on google. If its...
  40. L

    I Does mobius transformation assume 3-D Euclidean space?

    Are the assumptions in mobius transformation valid in Newtonian physics?
  41. FallenApple

    I How is a manifold locally Euclidean?

    So if I pick any 2 points on a 2d manifold, say x1 and x2, then the distance between these two points is a secant line that passes through 3 space that isn't part of the manifold. So no matter what, there doesn't exist an point epsilon, e , where ||e ||>||0 || and ||x2-x1||<|| e || No matter...
  42. T

    B Exploring a Micro Singularity in Euclidean Space Time

    What if the LHC produced a mini black hole and as a result a micro singularity was produced. If you were using Euclidean space time what equations and factors do you think would be most relevant?
  43. ShayanJ

    A Ground state wave function from Euclidean path integral

    From the path integral approach, we know that ## \displaystyle \langle x,t|x_i,0\rangle \propto \int_{\xi(0)=x_i}^{\xi(t_f)=x} D\xi(t) \ e^{iS[\xi]}##. Now, using ## |x,t\rangle=e^{-iHt}|x,0\rangle ##, ## |y\rangle\equiv |y,0\rangle ## and ## \sum_b |\phi_b\rangle\langle \phi_b|=1 ## where ## \{...
  44. resurgance2001

    I Euclidean differential number counts of supernovae

    Hi I am working on an assignment which is has asked us to derive an expression for a differential number count of supernovae in a euclidean flat non-expanding space. I am bit perplexed by this question and am wondering whether it is a trick question. We are allowed to do research to find an...
  45. G

    I GR vs SR: Is a Connection Necessary?

    Hi, When I started learning about GR I wondered if it emerged from SR (which the name suggests) or if the connection between the two is mere technical. GR describes the behaviour of the metric of space-time, which is locally Minkowskian and therefore SR applies. But is a curvature-based theory...
  46. evinda

    MHB Result of Euclidean division

    Hello! (Wave) I have applied a lot of times the euclidean division of $x^6-1$ with $x^2- \alpha^{a+1} (\alpha+1)x+ \alpha^{2a+3}, a \geq 0$, $\alpha$ a primitive $6$-th root of unity. But I don't get the right result... (Sweating) We are over $\mathbb{F}_7$. I got that $x^6-1=(x^2-...
  47. Ameer Bux

    Can someone help me with these 2 Euclidean geometry questions?

    Homework Statement write the proof Homework Equations none The Attempt at a Solution I've tried 5 times, got nowhere
  48. Einj

    A Euclidean signature and compact gauge group

    Hello everyone, I have been reading around that when performing the analytic continuation to Euclidean space (t\to-i\tau) one also has to continue the gauge field (A_t\to iA_4) in order to keep the gauge group compact. I already knew that the gauge field had to be continued as well but I didn't...
  49. M

    I Partitions of Euclidean space, cubic lattice, convex sets

    If the Euclidean plane is partitioned into convex sets each of area A in such a way that each contains exactly one vertex of a unit square lattice and this vertex is in its interior, is it true that A must be at least 1/2? If not what is the greatest lower bound for A? The analogous greatest...