What is Fundamental: Definition and 946 Discussions

Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits.
There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells.

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  1. B

    Fundamental Frequency: Odd Harmonic?

    Is the fundamental frequency considered an odd harmonic? Thanks for any clarification
  2. P

    Fundamental Frequency (String Resonance) - stumped

    Fundamental Frequency (String Resonance) - velocity problem! Homework Statement Hey guys, i just started working on Fundamental Frequency but am getting confused! A string that is 6.0m long is vibrating with three loops in it. The frequency of the source is 16.5Hz. A)What is the...
  3. A

    [Cosmology] Fundamental Plane to find distance

    I have been given data for the Coma and S639 clusters, including their velocity distances. For part a I calibrated the fundamental plane using the Coma data, and got: log(Re) = 1.24log(\sigma) - .82log<I> - .259 I am now supposed to use the S639 data along with the above fit to get a distance...
  4. H

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    Homework Statement prove that the limit of anxn + an-1xn-1 + ... + a1x1 + a0 as x goes to infinity equals infinity ** I forgot to mention that n is an odd number and this is for n>0 otherwise yes your counter example would be correct** Thanks for the quick responses by the way Otherwise, that...
  5. M

    Algebraic Topology: Fundamental group of a cube

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  6. bcrowell

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  7. C

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  8. N

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  9. C

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  10. O

    Fundamental question about conditional Expectation

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  11. H

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  12. L

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  13. K

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  14. Z

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  15. N

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  16. E

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  17. C

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  18. N

    How to find fundamental frequency

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  19. K

    What's the fundamental group of a punctured torus?

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  20. J

    Fundamental Theorem of Arithmetic

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  21. R

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  22. C

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  23. T

    Why isn't CHARGE a fundamental quantity rather than CURRENT?

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  24. Z

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  25. P

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  26. J

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  27. D

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  28. D

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  29. E

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  30. B

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  31. O

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  32. D

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  33. S

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  34. R

    Is c1 + c2 x^.5 a Solution of y y'' + (y')^.5 = 0?

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  35. S

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  36. T

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  37. L

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  38. S

    Fundamental groups of subsets of S^3

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  39. L

    Fundamental group to second homology group

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  40. Pythagorean

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  41. G

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  42. M

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  43. M

    Fundamental Group of (X,p): D^2\{(x,0) : 0<=x<=1}

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  44. MTd2

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  45. Z

    Understanding the Fundamental Group: Exploring Pi1(X.x) and Its Definition

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  46. D

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  47. D

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  48. J

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  49. A

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  50. L

    Calculate Fundamental Groups of X_1-X_5

    I need to calculate the fundamental group of the following spaces: X_1 = \{ (x,y,z) \in \mathbb{R}^3 | x>0 \} X_2 = \{ (x,y,z) \in \mathbb{R}^3 | x \neq 0 \} X_3 = \{ (x,y,z) \in \mathbb{R}^3 | (x,y,z) \neq (0,0,0) \} X_4 = \mathbb{R}^3 \backslash \{ (x,y,z) \in \mathbb{R}^3 | x=0,y=0...
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