The clique graphs of the hexagonal lattice -- an explicit construction and a short proof of divergence

We present a new, explicit and very geometric construction for the iterated clique graphs of the hexagonal lattice $\mathrm{H}\mathrm{e}\mathrm{x}$ which makes apparent its clique-divergence and sheds light on some previous observations, such as the boundedness of the degrees and clique sizes of $k^n\mathrm{H}\mathrm{e}\mathrm{x}$ as $n\to \mathrm{\infty }$.