Polycrystalline materials have been used throughout the recorded history, and their macroscopic properties strongly depend on the defects in their crystalline structures. Unique properties of the grain boundaries (GBs) should result from nearby nano-scale space, leading to the questions regarding the type of atomic polyhedra that pack the GB region as well as the universal set of structural units. It is essential to reveal the underlying mathematical framework for the polyhedral arrangement of GB structures and identify the origin of structure-property relationships in polycrystalline materials, which is a fundamental aspect but important for designing high-performance materials. Herein, 3D atomic structures of , , and  symmetrical tilt GBs are analyzed in the face-centered cubic (fcc) crystals, where the polyhedral units are rigorously defined in contrast to the polygonal arrangement within the 2D structural unit model. It is concluded that the GB region can only be packed by the bulk or GB-type polyhedral units with minor variations. In this study, the 3D arrangement of atomic polyhedra around the GBs is quantitatively determined and systematically described by numerical analysis. Moreover, the GB hierarchy directly follows the distribution of rational numbers that is represented by the modified Farey diagram, which represents the 3D atomic structure of the symmetrical tilt GBs in an fcc crystal. The definition of singular GB is proposed as the GB packed by at the most two types of polyhedral units, and any GB can be packed by the polyhedral units that form the singular GBs.
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