The Penrose mosaic as the minimum representative of quasicrystals is discussed in terms of generalized planar lattice models. The role of these models is played by Cayley's tree graphs which, in the general case, are characterized by quasi-random branching. A three-level golden alphabet is defined, and a Penrose mosaic is synthesized with the aid of its highest level. The algebras of the suggested grammar are formulated in an explicit form. It is shown that the statistics of a Penrose mosaic at the level of golden rhombuses belongs to the class of Zipf-Mandelbrot distributions. The algorithm for mapping a Penrose mosaic into Cayley's tree graphs based on the [2q * 2p] alphabet is also formulated. The problem of the entropy percolation for quasistochastic Cayley's trees of Penrose mosaics is solved. The entropy percolation of these trees is characterized by an obvious minimum periodicity and, on average, by the invariance principle of the golden entropy.
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