Abstract
The structures arising in spaces of various dimensions with simultaneous normal partitioning of spaces and their hierarchical fillings are considered. The conditions for the appearance of translational symmetry in these structures are investigated. It is shown that simultaneous hierarchical filling and normal tiling in three-dimensional spaces do not lead to the formation of translational symmetry. Such consistent transformations lead to many elements of translational symmetry in spaces of higher dimension. The higher the dimension of space, the more complex the emerging structure and the more symmetry the elements.
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