New families of cage-like structures based on Goldberg polyhedra with non-isolated pentagons

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.