The Planck and other natural numbers are used for units of forces. They arise also as weights of Gleason operators, defined by 3-dimensional spin-like base triples GF and their weigths. The spin lengths are the spin GF weights for instance. The measuring GF operator triples arise by projective duality from 1-dimensional force vectors in projective to R5 extended Hilbert space H4. Color charges are set as a separate force, using a G-compass (figure 2). For the universes evolution after a big bang several maps are introduced, mostly belonging to the gravity field quantum rgb-graviton. It presents the neutral color charge of nucleons. Orthogonal projections of H4, also in spiralic and angular form, central or stereographic projective maps belong to them. They project also the S³ factor of the strong interation geometry S³xS5 down to the SU(2) geometry S³ of the Hopf map. Fiber bundle maps are added also to S5 with the same fiber S1 to the base space CP² for nucleons and atomic kernels. In octonian coordinates, listed by indices, 01234567, there are three projections from the energy space 123456 of SI to complex quaternionic 2x2-matrix presentations of spacetime 1234, of CP² as 3456 and of GR with mass and rgb-gravitons 1256. GR and CP² are projected into 1234 as the universes spacetime, observable as bubbles for atoms and matter 3456 and GR potentials and actions about and for mass carrying systems 1256.
HOPF MAP AND QUANTIZATION ON SPHERE
