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In this paper we propose a general method for creating a new type of hierarchical structures at any level in both 2D and 3D. A simple rule based on a rotate-and-mirror procedure is introduced to achieve multi-level hierarchies. These new hierarchical structures have remarkably few degrees of freedom compared to existing designs by other methods. More importantly, these structures exhibit synchronized motions during opening or closure, resulting in uniform and easily-controllable deformations. Furthermore, a simple analytical formula is found which can be used to avoid collision of units of the structure during the closing process. The novel design concept is verified by mathematical analyses, computational simulations and physical experiments.

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A Goldberg polyhedron is a convex polyhedron made of hexagons and pentagons that have icosahedral rotational symmetry. Goldberg polyhedra have appeared frequently in art, architecture, and engineering. Some carbon fullerenes, inorganic cages, viruses, and proteins in nature exhibit the fundamental shapes of Goldberg polyhedra. According to Euler's polyhedron formula, an icosahedral Goldberg polyhedron always has exactly 12 pentagons. In Goldberg polyhedra, all pentagons are surrounded by hexagons only—this is known as the isolated pentagon rule (IPR). The present study systematically developed new families of cage-like structures derived from the initial topology of Goldberg polyhedra but with the 12 pentagons fused in five different arrangements and different densities of hexagonal faces. These families might be of great significance in biology and chemistry, where some non-IPR fullerenes have been created recently with chemical reactivity and properties markedly different from IPR fullerenes. Furthermore, this study has conducted an optimisation for multiple objectives and constraints, such as equal edge length, equal area, planarity, and spherical shape. The optimised configurations are highly desirable for architectural applications, where a structure with a small number of different edge lengths and planar faces may significantly reduce the fabrication cost and enable the construction of surfaces with flat panels.

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