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sauravshakya Group Title

Place four points on the surface of the sphere with radius 1m such that all four points are separated by equal distance from each other.... Find the distance between any two points.

  • one year ago
  • one year ago

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  1. hba Group Title
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    @sauravshakya Do you have a diag ?

    • one year ago
  2. Kainui Group Title
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    My intuition tells me that they will be separated such that they will form a regular tetrahedron by connecting the dots. When you say distance between any two points, do you mean in terms of traversing on the surface of the sphere or going straight between two points?

    • one year ago
  3. sauravshakya Group Title
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    traversing through the surface

    • one year ago
  4. sauravshakya Group Title
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    I mean on the surface

    • one year ago
  5. Kainui Group Title
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    It might be difficult, but I suggest drawing a picture and finding out for yourself. The angles should be something like 109.5 degrees. At this point it's just finding the arc length by knowing that this ratio is always true \[\frac{ \theta }{ 2 \pi }=\frac{ arc length }{ 2 \pi r }\]

    • one year ago
  6. sauravshakya Group Title
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    But how did u get angle between any two point is 109.5 degree

    • one year ago
  7. sauravshakya Group Title
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    I got this question in the chemistry class

    • one year ago
  8. Kainui Group Title
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    It's not very easy to draw out, but basically you have 4 arms coming outwards from the center and the angle between each one in 3 dimensions is 109.5 degrees. Only two are ever flat at one point. |dw:1355919400604:dw|

    • one year ago
  9. sauravshakya Group Title
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    I also know it is 109.5 degree but how to calculate it geometrically.

    • one year ago
  10. sauravshakya Group Title
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    I mean how to show the angle must be 109.5 degree

    • one year ago
  11. Kainui Group Title
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    Alright let me work on it real quick.

    • one year ago
  12. Kainui Group Title
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    |dw:1355920398348:dw| cut diagonally on the cube with the tetrahedral angle in the center. |dw:1355920471025:dw|

    • one year ago
  13. Kainui Group Title
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    Keep in mind, this is all pythagorean theorem up to this point, the sides of the cube all being 1 to keep things simple. The reason the diagonal through the center of the cube is sqrt(3) is because 1^2+(sqrt2)^2=3 because one edge of it is the hypotenuse of a triangle with legs of 1. Now let's look at what we can do here: |dw:1355921319840:dw|

    • one year ago
  14. Kainui Group Title
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    I hope this explains it well, it's fairly difficult to draw out accurately without being a total piece of garbage.

    • one year ago
  15. UnkleRhaukus Group Title
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    i always wondered where that number 109.5... came from

    • one year ago
  16. Kainui Group Title
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    I'm glad this could answer it! =D

    • one year ago
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