I point out that the phase transitions of the $d+1$ Gross-Neveu and $CP^{N-1}$ models at finite temperature and imaginary chemical potential can be mapped to transformations of regular hexagonal and regular triangular lattices to square lattice. The duality elements of two continuous models of fermions and bosons and two discrete lattice models make their appearance offering a new view of their phase transitions. I also show that the fermion-boson map in odd dimensions at finite temperature and imaginary chemical potential has a generalization for arbitrary $d$ that gives an expression of the transfer momentum of fundamental particles that behave like Bloch waves. These particles are travelling inside a periodic potential and scattering from specific surfaces (hexagonal and triangular kind) with a specific ordered construction based on golden ratio formula $ϕ=\frac{1}ϕ+1$ and its generalization. I further argue that this transfer momentum gives us a modified Bragg Law equation which it has a large $d$ limit to the well known expression for the transfer momentum when the scattering lattice is square. Interestingly these surfaces make a family of some first Brillouin zones that interact with particle beams and the maximum amount of momentum of the beam is transferred to them for specific angles related to their construction. Their construction is based on the golden ratio $ϕ$ and the Riemann $ζ(n)$ functions. The zeros and extrema of the Bloch-Wigner-Ramakrishnan $D_d(z)$ functions and Clausen $Cl_d(θ) $ functions play an important role to the analysis since they allow us not only to study the lattice transformations but also to study the fermionic theory deep inside the strong coupling regime as the dimension of the theory increases.
