the equation of rotation is fully radical sqrts

[hgjf2]

If determining the ortogonal matrixes (sum[j=1..n](T(a,j)^2-T(b,j)^2)=0 and sum[j=1..n](2*T(a,j)*T(b,j))=0 whatever a,b natural numbers 1<a<b<n.
Having properties:
two dimensions:

                   T11                 |                 T12
              ______________________________________
                  +T12             |      -+  T11    
 three dimensions:

                T11         |          T12       |       T13
           _________________________________________
                T21         |          T22       |       T23
           _________________________________________
                T31         |          T32       |       T33

where T22 = (-T11*T12*T21+T13*sqrt((T11^2+T12^2+T13^2)*(T12^2+T13^2-T21^2)))/(T12^2+T13^3)
and T23= (-T11*T13*T21+T12*sqrt((T11^2+T12^2+T13^2)*(T12^2+T13^2-T21^2)))/(T12^2+T13^3)
and T31 = +sqrt(T12^2+T13^2-T21^2)
and T32=  (-T11*T12*(+sqrt(T12^2+T13^2-T21^2))+T13*T21*sqrt(T11^2+T12^2+T13^2))/(T12^2+T13^2)
and T33= (-T11*T13*(+sqrt(T12^2+T13^2-T21^2))+T12*T21*sqrt(T11^2+T12^2+T13^2))/(T12^2+T13^2)
the radicals is and answer why the polynomial mandelbulb formula contain least a sqrt.

four dimensions:

                T11         |          T12       |       T13      |       T14    
           ___________________________________________________________
                T21         |          T22       |       T23      |       T24
           ___________________________________________________________
                T31         |          T32       |       T33      |       T34
           ___________________________________________________________
                T41         |          T42       |       T43      |       T44


where T23 = (-T11*T21*T13-T12*T22*T13+T14*sqrt(-(T13*T21)^2-(T14*T21)^2-(T13*T22)^2-(T14*T22)^2-(T11*T21)^2-2*T11*T12*T21*T22-(T12*T22)^2+(T11*T13)^2+(T11*T14)^2+(T12*T13)^2+(T12*14)^2+T13^4+2*(T13*T14)^2+T14^4))/(T13^2+T14^2))
and T24=  (-T11*T21*T14-T12*T22*T14+T13*sqrt(-(T13*T21)^2-(T14*T21)^2-(T13*T22)^2-(T14*T22)^2-(T11*T21)^2-2*T11*T12*T21*T22-(T12*T22)^2+(T11*T13)^2+(T11*T14)^2+(T12*T13)^2+(T12*14)^2+T13^4+2*(T13*T14)^2+T14^4))/(T13^2+T14^2))
and T32 = (-T11*T12*T31-T21*T22*T31+sqrt(T12^2-T21^2+T13^2-T31^2+T14^2)*sqrt(-(T13*21)^2-(T14*T21)^2-(T13*T22)^2-(T14*T22)^2-(T11*T21)^2-2*T11*T12*T21*T22-(T12*T22)^2+(T11*T13)^2+(T11*T14)^2+(T12*T13)^2+(T12*14)^2+T13^4+
2*(T13*T14)^2+T14^4))/(T13^2+T14^2))
and T41 = +sqrt(T12^2-T21^2+T13^2-T31^2+T14^2)
and T42 = (-T11*T12*sqrt(T12^2-T21^2+T13^2-T31^2+T14^2)-T21*T22*sqrt(T12^2-T21^2+T13^2-T31^2+T14^2)+T31*sqrt(-(T13*T21)^2-(T14*T21)^2-(T13*T22)^2-(T14*T22)^2-(T11*T21)^2-2*T11*T12*T21*T22-(T12*T22)^2+(T11*T13)^2+(T11*T14)^2+(T12*T13)^2+(T12*14)^2+T13^4+2*(T13*T14)^2+T14^4))/(T13^2+T14^2))
and many terms.