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Sum of the angles in 4D coordinates system

  1. Sep 21, 2009 #1
    Case 1
    In 2D coordinate system the sum of the angles of all quadrants=360
    degree
    in (x,y) plane
    Case 2
    In 3D coordinate system (x,y,z) I took 3 possible planes

    i.e.
    (x,y) (y,z) (x,z)
    each plane provides angle of 360 degree, so can we say that in 3D coordinate system sum of all quadrants is 1080?
    Because
    (x,y) gives 360
    (y,z) gives 360
    (x,z) gives 360
    so their sum is 1080 degree
    Case 3
    And in 4d spacetime the distance formula according to Minkowski formulation is
    s^2=x^2+y^2+z^2-(ct)^2
    Where c is speed of light and t is time and ct also has the dimension length
    like other coordinates.
    so can I deal it as a 4D coordinate system
    (x,y,z,ct)
    all possible planes are
    (x,y) (y,z) (z,ct) (x,ct) (y,ct) (x,z)
    so can we say,
    sum of all quadrants in 4D coordinate system is 2160 degree?
    Because each plane provides 360 degree
    & there are 6 planes.
    Three of them belong to 3D and 3 belong to space and time.
    Can I deal these coordinates in this way.
     
  2. jcsd
  3. Sep 21, 2009 #2

    Dale

    Staff: Mentor

    The angle of a circle is 2pi radians in 2D Euclidean geometry. The solid angle of a sphere is 4pi steradians in 3D Euclidean geometry. The "solid" angle of a hypersphere is 2pi² (hyperradians?) in 4D Euclidean geometry. I don't know about a hypersphere in Minkowski geometry.
     
    Last edited: Sep 21, 2009
  4. Sep 21, 2009 #3

    Ben Niehoff

    User Avatar
    Science Advisor
    Gold Member

    A hypersphere in Minkowski geometry is going to be some higher-dimensional analogue of a hyperboloid; therefore its solid "angle" is infinite.
     
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