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首页> 外文期刊>Proceedings of the National Academy of Sciences of the United States of America >New family of tilings of three-dimensional Euclidean space by tetrahedra and octahedra
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New family of tilings of three-dimensional Euclidean space by tetrahedra and octahedra

机译：四面体和八面体的三维欧氏空间的新平铺族

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摘要

It is well known that two regular tetrahedra can be combined with a single regular octahedron to tile (complete fill) three-dimensional Euclidean space R～3. This structure was called the "octet truss" by Buckminster Fuller. It was believed that such a tiling, which is the Delaunay tessellation of the face-centered cubic (fcc) lattice, and its closely related stacking variants, are the only tessellations of R～3 that involve two different regular polyhedra. Here we identify and analyze a unique family comprised of a noncountably infinite number of periodic tilings of R～3 whose smallest repeat tiling unit consists of one regular octahedron and six smaller regular tetrahe dra. We first derive an extreme member of this unique tiling family by showing that the "holes" in the optimal lattice packing of octa hedra, obtained by Minkowski over a century ago, are congruent tetrahedra. This tiling has 694 distinct concave (i.e., nonconvex) repeat units, 24 of which possess central symmetry, and hence is distinctly different and combinatorically richer than the fcc tetrahe dra-octahedra tiling, which only has two distinct tiling units. Then we construct a one-parameter family of octahedron packings that continuously spans from the fcc to the optimal lattice packing of octahedra. We show that the "holes" in these packings, except for the two extreme cases, are tetrahedra of two sizes, leading to a family of periodic tilings with units composed four small tetrahedra and two large tetrahedra that contact an octahedron. These tilings generally possess 2,068 distinct concave tiling units, 62 of which are centrally symmetric.
机译：众所周知，两个规则的四面体可以与一个规则的八面体组合在一起，以平铺（完全填充）三维欧几里得空间R〜3。该结构被Buckminster Fuller称为“八角形桁架”。可以认为，这种面心立方（fcc）格的Delaunay镶嵌及其紧密相关的堆积形式是唯一的R〜3涉及两个不同规则多面体的镶嵌。在此我们确定并分析了一个独特的族，该族由R〜3的无限数量的周期性平铺组成，其最小重复平铺单元由一个规则的八面体和六个较小的规则的四面体组成。我们首先通过证明Minkowski一个多世纪前获得的八面体最佳晶格堆积中的“孔”是一致的四面体，来得出这个独特的平铺家族的极端成员。该平铺具有694个不同的凹面（即非凸面）重复单元，其中有24个具有中心对称性，因此与fcc四面体八面体平铺（仅具有两个不同的平铺单元）明显不同且组合上更丰富。然后，我们构建了一个从八面体FCC到八面体最佳晶格堆积连续的八面体堆积的单参数族。我们显示，除了两个极端情况外，这些填料中的“孔”是两种大小的四面体，从而导致一个周期性的平铺族，其单元由与小八面体接触的四个小四面体和两个大四面体组成。这些平铺通常具有2,068个不同的凹面平铺单元，其中62个是中心对称的。

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  • 作者

    John H. Conway; Yang Jiao; Salvatore Torquato;

  • 作者单位

    Department of Mathematics, Princeton University, Princeton, NJ 08544;

    Princeton Institute for the Science and Technology of Materials,Princeton University, Princeton, NJ 08544;

    Princeton Institute for the Science and Technology of Materials,Princeton University, Princeton, NJ 08544,Department of Chemistry, Department of Physics, Princeton Center for Theoretical Science,Program in Computational and Applied Mathematics, Princeton University, Princeton, NJ 08544;

  • 收录信息 美国《科学引文索引》(SCI);美国《生物学医学文摘》(MEDLINE);美国《化学文摘》(CA);
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

    space-filling; nonoverlapping solids; polytopes;

    机译：空间填充;不重叠的固体;多面体;

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