The study of the polyhedra described in this paper is relevant to the icosahedral symmetry in the assembly of various spherical molecules, biomolecules and viruses. A symmetry-breaking mechanism is applied to the family of polytopes constructed for each type of dominant point λ. Here a polytope is considered as a dual of a polytope obtained from the action of the Coxeter group H₃ on a single point . The H₃ symmetry is reduced to the symmetry of its two-dimensional subgroups H₂, A₁ × A₁ and A₂ that are used to examine the geometric structure of polytopes. The latter is presented as a stack of parallel circular/polygonal orbits known as the `pancake' structure of a polytope. Inserting more orbits into an orbit decomposition results in the extension of the structure into various nanotubes. Moreover, since a polytope may contain the orbits obtained by the action of H₃ on the seed points (a, 0, 0), (0, b, 0) and (0, 0, c) within its structure, the stellations of flat-faced polytopes are constructed whenever the radii of such orbits are appropriately scaled. Finally, since the fullerene C₂₀ has the dodecahedral structure of , the construction of the smallest fullerenes C₂₄, C₂₆, C₂₈, C₃₀ together with the nanotubes C_20+6N, C_20+10N is presented.

中文翻译：

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非晶群H3对偶多面体的对称破缺

本文描述的多面体研究与各种球形分子、生物分子和病毒组装中的二十面体对称性有关。对称破坏机制应用于为每种类型的主导点 λ 构建的多胞体族。此处多胞体被认为是由 Coxeter 群H _{3 作用}于单个点而获得的多胞体的对偶。的ħ ₃对称性被减少到其二维子组的对称ħ ₂，阿₁  × 阿₁和阿₂被用于检查的几何结构多胞体。后者表现为一堆平行的圆形/多边形轨道，称为多胞体的“煎饼”结构。在轨道分解中插入更多的轨道会导致结构扩展到各种纳米管中。此外，由于多胞体可能包含通过H ₃在其结构内的种子点 ( a , 0, 0)、(0,  b , 0) 和 (0, 0,  c )上的作用获得的轨道，只要适当地缩放此类轨道的半径，就会构建平面多胞体。最后，由于富勒烯 C ₂₀具有 的十二面体结构，最小富勒烯 C ₂₄的构造、C ₂₆、C ₂₈、C ₃₀与纳米管C _{20+6 N}、C _{20+10 N 一起}呈现。