Department of Applied Mathematics, Research School of Physics and Engineering, Australian National University, Canberra ACT 2601, Australia
Department of Applied Mathematics, Research School of Physics and Engineering, Australian National University, Canberra ACT 2601, Australia.
Proc Natl Acad Sci U S A. 2018 Jul 3;115(27):6905-6910. doi: 10.1073/pnas.1720307115. Epub 2018 Jun 20.
We derive more than 80 embeddings of 2D hyperbolic honeycombs in Euclidean 3 space, forming 3-periodic infinite polyhedra with cubic symmetry. All embeddings are "minimally frustrated," formed by removing just enough isometries of the (regular, but unphysical) 2D hyperbolic honeycombs [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] to allow embeddings in Euclidean 3 space. Nearly all of these triangulated "simplicial polyhedra" have symmetrically identical vertices, and most are chiral. The most symmetric examples include 10 infinite "deltahedra," with equilateral triangular faces, 6 of which were previously unknown and some of which can be described as packings of Platonic deltahedra. We describe also related cubic crystalline packings of equal hyperbolic discs in 3 space that are frustrated analogues of optimally dense hyperbolic disc packings. The 10-coordinated packings are the least "loosened" Euclidean embeddings, although frustration swells all of the hyperbolic disc packings to give less dense arrays than the flat penny-packing even though their unfrustrated analogues in [Formula: see text] are denser.
我们从二维双曲蜂巢在欧几里得 3 空间中的 80 个嵌入中推导出 3 个周期的无限多面体，具有立方对称性。所有的嵌入都是“最小受挫的”，通过去除足够数量的（规则的，但非物理的）二维双曲蜂巢[公式：见正文]、[公式：见正文]、[公式：见正文]、[公式：见正文]和[公式：见正文]的等距变换，允许在欧几里得 3 空间中嵌入。几乎所有这些三角化的“单纯多面体”都具有对称相同的顶点，并且大多数是手性的。最对称的例子包括 10 个无限的“三角面体”，具有等边三角形的面，其中 6 个是以前未知的，其中一些可以被描述为柏拉图三角面体的堆积。我们还描述了相关的三维立方晶体堆积，其中等距的双曲圆盘在 3 空间中是最优密集的双曲圆盘堆积的受挫类似物。10 配位的堆积是最“放松”的欧几里得嵌入，尽管受挫会使所有的双曲圆盘堆积膨胀，形成比平坦的便士堆积更稀疏的排列，尽管它们在[公式：见正文]中的无受挫类似物更密集。
