Abstract Several magnetic materials consisting of dipoles owe their properties to the specific nature of the dipole-dipole interaction. In the present work, we study systems of dipoles where the particles are arranged on various types of three-dimensional structures. However, these solids are not arbitrary. They constitute the well-known Platonic, Archimedean and Catalan solids. We systematically study them in order to fill a gap in the literature that does not contemplate this interaction in the previous solids, despite the fact that they are encountered in many different physical systems. In particular, in the regime of strong dipole moments where a classical treatment is possible, we shall provide not only the minimum energy but also the precise orientations of all their dipoles. We will numerically obtain the minimum energy configuration where all vertices possess the same classic dipole, either electric or magnetic.

中文翻译：

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柏拉图、阿基米德和加泰罗尼亚固体结构的偶极-偶极最小能量配置

摘要 几种由偶极子组成的磁性材料由于偶极子-偶极子相互作用的特殊性质而具有它们的特性。在目前的工作中，我们研究偶极子系统，其中粒子排列在各种类型的三维结构上。然而，这些固体不是任意的。它们构成了著名的柏拉图、阿基米德和加泰罗尼亚固体。我们系统地研究它们以填补文献中的空白，尽管它们在许多不同的物理系统中都会遇到，但在以前的固体中没有考虑这种相互作用。特别是，在可以进行经典处理的强偶极矩范围内，我们不仅要提供最小能量，还要提供所有偶极子的精确方向。