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Topic: NEW COORDINATE SYSTEMS
Replies: 0

 vlcek Posts: 18 Registered: 11/30/06
NEW COORDINATE SYSTEMS
Posted: Aug 9, 2011 3:45 PM

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)