Figure Captions 13 Fig. 1: Basic cells of the WB family of hierarchical lattices, with a scaling factor b = 2. In each case, the empty circles ( µ and ν ) represent the external sites of the cell, whereas the black circles are internal sites to be decimated in the RG procedure. (a) The WB cell of fractal dimension D = (ln 5) / (ln 2) ≈ 2 . 32; (b) The WB cell of fractal dimension D = (ln 12) / (ln 2) ≈ 3 . 58; (c) The WB cell of fractal dimension D = (ln 28) / (ln 2) ≈ 4 . 81. For clearness, all the bonds of cell (c) are not shown explicitly; the external site µ is connected directly to sites 5 , 6 , 7 , and 8 (as illustrated by the bond connecting sites µ and 5, whereas site ν is connected directly to sites 1 , 2 , 3 , and 4 (as illustrated by the bond connecting sites ν and 3). Fig. 2: First three levels (levels 0 , 1 , and 2) in the generation process of the Wheatstone- Bridge hierarchical lattice with fractal dimension D ≈ 2 . 32. The lattice-generation process starts at the 0-th level with a single bond and at each step a bond is replaced by its respective basic cell. In each level, the empty circles ( µ and ν ) represent the external sites of the lattice, whereas the black circles are internal sites to be decimated in successive RG operations. Fig. 3: Phase diagram for an Ising SG on the hierarchical lattice defined by the unit cell of Fig. 1(a), with a Gaussian distribution for the couplings. Fig. 4: Phase diagrams of the Ising SG on the hierarchical lattice defined by the unit cell of Fig. 1(b): (a) Gaussian distribution for the couplings; (b) bimodal distribution for the couplings. Fig. 5: Evolution of the average of couplings with the RG step n , for the case of an initial Gaussian distribution, considering typical values of J$_{0}$/J around the critical frontier separating the phases SG and F . (a) Fixed temperature below the SG critical temperature; (b) Zero temperature.