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Acta Cryst. (2014). A70, C1280
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Authors have developed proper polyhedron models to enable people to learn the concept of three-dimensional symmetry. Touching and operating the symmetry elements of the proper polyhedron enables people to understand symmetry. In this study, authors made three-dimensional tessellation models. Certain polyhedra can be stacked in a regular periodic pattern to fill three-dimensional space completely. Figures show our models. The cube (Fig. (a)) is the only regular polyhedron to fill three-dimensional space completely. The cube is a Voronoi region of the simple cubic lattice (sc). The truncated octahedron (Fig. (b)) is the only Archimedean solid to fill three-dimensional space completely. The truncated octahedron is a Voronoi region of the body-centered cubic lattice (bcc). The rhombic dodecahedron (Fig. (c)) is the only Catalan solid (or Archimedean dual) to fill three-dimensional space completely. The rhombic dodecahedron is a Voronoi region of the face-centered cubic lattice (fcc). Figs. (a), (b), and (c) show three kinds of their aggregate respectively. In each of left-hand aggregate, there is a two-fold rotational axis along a vertical direction. In each of central aggregate, there is a three-fold rotational axis along a vertical direction. In each of right-hand aggregate, there is a four-fold rotational axis along a vertical direction. Fig. (d) is a nontrivial polyhedron to fill three-dimensional space completely. The external shape of the polyhedron was designed as a tree shape. We call such a model three-dimensional Escher shape (3DES) [1]. This can be stacked in a regular periodic pattern too.

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Acta Cryst. (2014). A70, C1423
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Packing problems are an important aspect of crystallography. In particular, sphere packings have played an important role in improving our understanding of crystal structures. Cylinder packings are also important for the same reason and have been investigated in the fields of both science and engineering. In the field of science, the complex structure of garnet has been explained on the basis of cylinder packing to be a periodic structure with a cubic <111> four-way cylinder packing [1a]. In the field of engineering, cylinder packings are important for determining the fiber packings of composite materials. Some regular fiber packing structures have been designed. Motivated by structures of composite materials, periodic cubic <110> six-way cylinder packing structures have also been investigated [1b]. The known <110> six-way cylinder packings can be classified into three categories on the basis of packing density: (√2)π/9 ≍ 0.494 (Type-I), (√2)π/18 ≍ 0.247 (Type-II), and (351√2 + 108√6)π/1936 ≍ 0.376 (Type-III). Recently, Teshima and Matsumoto studied the space group of the Type-III structure [2]. And Moore reported another type of periodic cubic <110> six-way cylinder packing structure (packing density ≍ 0.133) [3a,b]. In this study, authors consider a general description of periodic cubic <110> six-way cylinder packing structures.
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