ds2 = dt2 - dx2 - dy2 = dE2 - dpx2 - dpy2 = dm2
where ds is the proper time interval, and dm is the rest mass interval.
If one then does a sterographic projection of the hyperboliod given by
ds2 = dm2 = constant
onto a disc, the space upon that disc will be hyperbolic velocity (rapidity = R)
R = arc tanh v
The center of the disc represents zero (relative) velocity and the edge of the disc is a radial velocity at the speed of light (v=c=1).
If one then takes three particles (let's call them quarks) of equal rest mass, and constrains them to move or vibrate radially with respect to their center of mass, then their motion is described by three geodesics in this velocity space which intersect at equal angles and are centered around v=0. But to non-repeditively cover this space (a tesselation) only certain angles are allowed i.e. p/2n where n=2,3,4,5,...
In addition, if one then sets a fixed energy, greater than their rest mass, that is described by horocircles on the plane. But there is only one horocircle which intersects these geodesics at the same angle.
If one then does an inversion of each particle with respect to the other and with the selected energy level, one gets this construct in which this basic structure is quantized and reappears repeditively at at larger velocities (Energies).
© November 1980
Mario Giannella
Cornell University
Ithaca, NY
Now located at:
Spallation Neutron Source
Oak Ridge, TN 37830