The singularity of a black hole with a quantum index b was previously shown to be consisted of b concentric spherical shells of a constant diametral step size A, corresponding to a fundamental resolution interval in nature. An interior shell j was found to be a condensation of a Primordial Stem Particles (PSP) of mass m = h/Ac, at excitation state j and momentum jmc. The spatial uncertainty of the constituents at this excitation state was found to be jA, therefore, confining the particles to the very surface of the shell of diameter jA. The number of constituents N_1 = A^2/(4L_p^2) of the innermost shell j = 1 was found to be the quanta of particle count in the singularity, such that particle count on a shell of index j was N_j = (3j^2 − 3j + 1) N_1. Invoking Tammes’s conjecture, we have shown in this article that the microstructure of the black hole singularity is consisted of a distinct set of curved hexagonal lattices that are wrapped into the spherical shells. It is shown that the distance of the constituents on the innermost shell is d_1 = √ (8π/√3) Lp. The microstructure gets progressively finer towards the outer shells. In the outermost shell of the singularity of the supermassive black holes, the lower limit d_infty is approaching to √(8π/3) Lp. It is also noted that a centered hexagonal tessellation naturally emerges from N_j law which is a remarkable hint of consistency in the proposed theory of black holes.