Quasicrystals, a fascinating class of materials discovered in 1982 by Israeli physicist Daniel Shechtman, have since captivated the scientific community with their exceptional properties. These structures exhibit a quasi-periodic translational sequence and possess rotational symmetry axes of 5, 8, 10, or 12—features that were previously considered unattainable in both conventional crystals and non-crystalline substances. This paper delves into the prevalent structural models of quasicrystals, spanning one-dimensional, two-dimensional, and three-dimensional representations. The one-dimensional model prominently features the quasi-periodic Fibonacci sequence, which has undergone extensive development from both experimental and theoretical standpoints. Among the various theoretical models for two-dimensional quasicrystals, this work focuses primarily on the Penrose tiling within the mosaic model—a quintessential example representing fivefold symmetric quasicrystal structures. In the realm of quasicrystals, piecework models come to the fore, characterized by the tessellation of two or more pieced blocks. The construction rules governing these models dictate that no overlap or coverage should exist between the pieced blocks, thereby forming a two-dimensional quasi-periodic structure devoid of gaps. The Penrose tiling, an iconic representative of this model, exhibits long-range order akin to crystal lattices, yet it lacks translational symmetry. It remains a widely studied quasicrystal model renowned for its distinctive fivefold symmetry.