Abstract

Various intersections of Boerdijk–Coxeter (BC) helices consisting of regular tetrahedra and having the same chirality are considered in a three-dimensional Euclidean space. The angles between their axes are calculated. Next, these rigorous geometric models of intersections of BC helices are subjected to physically justified distortions (merging of closely spaced vertices and formation of additional bonds between vertices spaced by a distance of about a tetrahedron-edge length). It is shown that with introduction of these distortions there are seven different types of intersection of two BC helices passing through the same icosahedron. They correspond to seven types of intersection of helices 30/11 in a polytope {3, 3, 5}. For different discretized Hopf fibrations of the set of polytope {3, 3, 5} vertices, it is demonstrated that the local structure of a polytope {3, 3, 5} in the vicinity of a Hopf circle passing through the polytope vertices can be described as a certain set of intersecting helices 30/11.

中文翻译：

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三维欧几里得空间中Boerdijk-Coxeter螺旋的交点

摘要

在三维欧几里得空间中考虑了由规则四面体组成且具有相同手性的Boerdijk-Coxeter（BC）螺旋的各种相交。计算它们的轴之间的角度。接下来，这些严格的BC螺旋相交的几何模型会受到物理上合理的扭曲（紧密间隔的顶点合并以及在顶点之间以大约四面体边缘长度的距离形成的附加键）。结果表明，引入这些畸变后，穿过同一二十面体的两个BC螺旋有七个不同类型的交点。它们对应于多面体{3，3，5}中的螺旋30/11的七种交集类型。对于多顶点{3，3，5}顶点集的不同离散Hopf纤维，