NEW COORDINATE SYSTEMS

Discussion in 'Math Research' started by Lubomir Vlcek, Aug 9, 2011.

  1. THE NEW COORDINATE SYSTEMS
    Take the minimal number of identical particles with a globe-like form
    and forming the nearest organized configuration. This configuration is
    a disfenoid at the vertices with four particles (the  - particle has 4
    nucleons). The origin of our new coordinate system is put into the
    center of gravity of the configuration. This origin and the centres of
    the particles determine the semi-lines-semi-axes of the coordinate
    system. This coordinate system divides the space into four
    quartespaces. The pairs of semi-lines (s,t), (s,u), (s,v), (t, u),
    (t,v), (u,v) determine angles
    For angle  it precisely holds:

    To imagine better the coordinate system (s,t,u,v), we can use the
    cube. The centre of the cube is the center of gravity of the disfenoid
    and also the origin of the coordinate system (see fig. 1).
    In order to facilitate the transformation to the cartesian coordinate
    system, this will be somewhat re-arranged:
    semi-axes x,y,z will have the same marks, semi-axes (-x),(-y),(-z)
    will be marked  and so cartesian coordinate system (x,y,z) in the new
    marking will be revealed as a system ( ) formed by semi-axes  . These,
    regarding to the coordinate system (s,t,u,v), will be determined as
    follows:
    semi-axis x is the symmetral of the angle  (see fig. 2)
    semi-axis y is the symmetral of the angle  in "  - plane" (t,u)
    semi-axis z is the symmetral of the angle  in "  - plane" (u,v)
    semi-axis  is the symmetral of the angle  in "  - plane" (t,v)
    semi-axis  is the symmetral of the angle  in "  - plane" (s,v)
    semi-axis  is the symmetral of the angle  in "  - plane" (s,t).

    Fig. 1. The coordinate system (s,t,u,v)

    Fig. 2. The semi - axis x is the symmetral of the angle
    After drawing both coordinate system we will achieve fig. 3.
    The cartesian coordinate system divides the space into 8 octants:  .
    The trinities of "  - planes" determine four equal quarter-spaces:
    (s,t,u), (s,t,v), (s,u,v), (t,u,v). It is impossible to divide the
    space into equal parts using less than 4 semi-axes. It means that
    these quarter-spaces are the largest possible parts of the space
    formed by the minimal number of semi-axes.

    Fig. 3. Both coordinate system (s,t,u,v) and ( )
    The values of coordinates will be read in two ways:
    a) The straight lines placed from an arbitrary point parallel to the
    axes s,t,u,v determine coordinates s,t,u,v, fig. 4.
    Zero in the contained coordinate means that the point is placed in the
    quarter space determined by coordinates other than zero.
    See the following transformation equation between (s,t,u,v) and ( ):
    (s,t,u)


    (s,u,v)


    (s,t,v)


    (t,u,v)



    Fig. 4. The coordinates (s,t,u,v)
    (s,t,u): (s,u,v):

    (s,t,v): (t,u,v):

    The distance between two points (s1,t1,u1,v1) and (s2,t2,u2,v2) is
    determined as follows:

    b) The planes placed from an arbitrary point perpendicular to the axis
    s,t,u,v determine coordinates s*,t*,u*,v*, see fig. 5.

    Fig. 5. The coordinates s*,t*,u*,v*
    See the following transformation equations between s*,t*,u*,v*,
    and  :

    (s*,t*,u*): (s*,u*,v*):

    (s*,t*,v*): (t*,u*,v*):

    Quadrate of distance between two points (s1*,t1*, u1*,v1*) and
    (s2*,t2*, u2*,v2*) is determined by this equation:

    See the following transformation equation between (s,t,u,v) and
    (s*,t*,u*,v*):
    (s,t,u): (s,t,v):

    (s,u,v): (t,u,v):

    (s,t,u): (s,t,v):

    (s,u,v): (t,u,v):

    Rotation around axis x,y,z - the angle of rotation is  - are
    invariant. They perform the disfenoid into equivalent positions. The
    rotations around the axis s,t,u,v - the angle of rotation is  - are
    the invariant ones. E,s,s-1,t,t-1,x,u,u-1,y,v,v-1,z form the group of
    rotation, see Tab. 1.
    Table 1.
    columns - Acts as the first, rows - Acts as the second
    E s s-1 t t-1 u u-1 v v-1 x y z
    E E s s-1 t t-1 u u-1 v v-1 x y z
    s-1 s-1 E s z u y v x t u-1 t-1 v-1
    s s s-1 E v-1 y t-1 x u-1 z v u t
    t-1 t-1 z v E t x s y u v-1 s-1 u-1
    t t u-1 y t-1 E v-1 z s-1 x u v s
    u-1 u-1 y t x v E u z s s-1 v-1 t-1
    u u v-1 x s-1 z u-1 E t-1 y t s v
    v-1 v-1 x u y s z t E v t-1 u-1 s-1
    v v t-1 z u-1 x s-1 y v-1 E s t u
    x x u v-1 v u-1 s t-1 t s-1 E z y
    y y t u-1 s v-1 v s-1 u t-1 z E x
    z z v t-1 u s-1 t v-1 s u-1 y x E

    REFERENCES
    [1] MAYER, M. G.: Phys. Rev. 74, 235, (1948)
    [2] FEJES Tóth, L.: Am. Math. Month. 56, 330 (1949)
    [3] HABICHT W., van der WAERDEN, B. L.: Math. Ann. 123, 223 (1951)
    [4] WHYTE, L. L.: Am. Math. Month. 59, 606 (1952)
    [5] LEECH, J.: Math. Gaz. 41, 81 (1957)
    [6] BEREZIN, A. A.: Nature (London) 317, 208 (1985)
    [7] BEREZIN, A. A.: J. Math. Phys. 27(6), 1533 (1986)

    See you please
    L. Vlcek  : New Trends in Physics, Slovak Academic Press, Bratislava
    1996
    ISBN 80-85665-64-6. Presentation on  European Phys. Soc. 10th Gen.
    Conf. ­
    Trends in Physics  ( EPS 10) Sevilla , E

    http://www.trendsinphysics.info/
     
    Lubomir Vlcek, Aug 9, 2011
    #1
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