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Objects with Prime Symmetry -23- Objects with Prime Symmetry The purpose of this chapter is to describe the objects used as exam- ples in Chapter 22. The Three Lattices The symmetries of the three lattices are easily found. Since the bc lattice consists of all points 0, 1, 2, 3 and doesn’t distinguish + from −, its symmetries can achieve the full dihedral group of per- mutations of 0, 1, 2, 3 and also the interchange of + and −.They therefore constitute the doubled octad group (8◦:2). The nc lattice consists just of 0 and 2, whose symmetries effect (02), (13), (+−) , determining the negative double tetrad group (4−:2). Finally, the symmetries of the fc lattice, whose points are those colored 0, effect only (13), (+−) ,soformthenegative double dyad group (2−:2). The three cubic lattices: The bc lattice has symmetry 8◦:2, the nc has 4−:2,andthefchas 2−:2. (opposite page) The fascinating propeller-hedron has group 4◦◦. 327 © 2016 by Taylor & Francis Group, LLC 328 23. Objects with Prime Symmetry Each of the three lattices actually gives five groups, one for each point group: point group ∗432 432 3∗2 ∗332 332 bc lattice 8◦:2 8+◦ 8−◦ 4◦:2 4◦◦ nc lattice 4−:2 4◦− 4− 2◦:2 2◦ fc lattice 2−:2 2◦− 2− 1◦:2 1◦ The additional groups in the table are obtained by restricting to the given point groups. Assemblies with these as symmetries are easily obtained by surrounding the lattice points by finite objects that force the appropriate point group and are all parallel to each other. One such assembly gives the densest known packing of regular dodecahedra (see Figure 23.1): these dodecahedra are arranged much like the rhombic ones we will see shortly in Figure 23.4, so alternate cubes of the cubic lattice are inscribed in them. The packing has density ρ = .9405, the unused space consisting of curious polyhedra we call endo-dodecahedra, which like regular dodecahedra have 12 pentagonal faces with all edges of the same length although neither the faces nor the polyhedron are convex. Since the dodecahedra are centered on the fc lattice and force the pyritohedral group 3∗2,the symmetry group is 2−. Although the cubic tessellation has group 4−:2, a model of it made with the Zome System construction set (a proprietary system Downloaded by [University of Bergen Library] at 04:55 26 October 2016 Figure 23.1. A packing of regular dodecahedra. This half-cubic lattice of dodecahedra has point group 3∗2 and symmetry 2−. The endo-dodecahedral holes are shown in orange. © 2016 by Taylor & Francis Group, LLC Voronoi Tilings of the Lattices 329 Figure 23.2. A bicubic assembly of regular tetrahedra with symmetry 4◦:2. also known as “Zometool”) can only have the subgroup 4−, obtained by restricting the point group to 3∗2, when examined in fine detail. This is because the edges, as shown at the right, have rectangular cross-section and are assembled pyritohedrally into the balls at the vertices. For the same reason, the largest of the 35 prime groups that can be exactly modeled with Zometool is 8−◦. Figure 23.2 shows a bicubic assembly of parallel regular tetra- hedra, so (if we ignore the colors) its symmetry group (obtained by restricting the bc group to ∗332)is4◦:2. The cyclic order is forced by the way the vertices of the tetrahedra point to each other: yel- low → red → blue → green → yellow. If we pay attention to the colors, the lattice drops to fc with the same point group 332 and so the group decreases to 1◦, otherwise known as the smallest plenary group p. Downloaded by [University of Bergen Library] at 04:55 26 October 2016 Voronoi Tilings of the Lattices Any lattice determines a tiling of space into polyhedra called its Voronoi cells (“vocells”). The vocell of any lattice point consists of all points of space that are nearer to it than they are to any other lattice point. For the normal cubic (nc) lattice, the Voronoi tiling is obviously the decomposition of space into cubes (Figure 23.3, left). For the body-centered or bicubic (bc) lattice, it is the tiling (“truncoctahedrille”) into truncated octahedra (Figure 23.3, right) © 2016 by Taylor & Francis Group, LLC 330 23. Objects with Prime Symmetry Figure 23.3. The Voronoi cells of the normal cubic lattice are cubes; those of the bicubic lattice are truncated octahedra. described at the start of Chapter 21. The centers of the tiles are the nodes 0, 1, 2, 3 and their vertices are the nodes + and −. Finally, for the face-centered or half-cubic (fc) lattice, the vocells are the rhombic dodecahedra of “rhombohedrille.” This tiling can be obtained as follows: To obtain the vocells of the half-cubic lattice, divide alternate cubes of the lattice into six square pyramids (Fig- ure 23.4, left) by joining their centers to their faces, and then join these square pyramids to the adjacent, undivided cubes, resulting in rhombic dodecahedra. Their centers are, say, the nodes 0, and their vertices are the other three digital nodes 1, 2, and 3 (red, blue, and green, respectively, on the right in Figure 23.4). The vertices (here 2) that have the same parity as the centers are 4-valent while those (1 and 3) of the other parity are 3-valent. Downloaded by [University of Bergen Library] at 04:55 26 October 2016 Figure 23.4. The Voronoi cells of the fc are rhombic dodecahedra. © 2016 by Taylor & Francis Group, LLC Salt, Diamond, and Bubbles 331 Salt, Diamond, and Bubbles Many simple crystals are easily described in terms of one or other of these lattices. For example, the atoms of a salt crystal will form a normal cubic lattice, but it does not have all the symmetries of the nc lattice, since the atoms alternate between sodium (Na) and chlorine (Cl)—say 1 = Na and 3 = Cl. So, we can no longer move 1 or 3, and the symmetries effect only (02), (+−) , showing that a salt crystal has the negative double dyad group 2−:2. Salt. The diamond net has the same symmetry as the catoptric tiling by obcubes (Chapter 21). A diamond crystal is made of carbon atoms situated at the 0 and 1 nodes. Its symmetries effect1 (01)(23), (+−) , so the group is now the positive double dyad group 2+:2. The diamond net is Downloaded by [University of Bergen Library] at 04:55 26 October 2016 formed by the blue and green balls on the left in the figure above and the lines joining them. (Beware! The polyhedra between them are oblique cubes or “obcubes,” not cubes!) The points at maximal distance from these (our red and yellow nodes on the right above, at the centers of the obcubes) would form another diamond net; the two together form what we call the “Double Diamond” with group 4+:2. 1We cannot interchange just 0 with 1 or just 2 with 3, since the permutations (01) and (23) are not in the group of possible effects. © 2016 by Taylor & Francis Group, LLC 332 23. Objects with Prime Symmetry Voronoi cells of atoms in a diamond. The Voronoi cells of the atoms in a diamond are triakis truncated tetrahedra formed by erecting small pyramids on the triangular faces of a truncated tetrahedron. The Voronoi tiling is closely related to the Architectonic “trunctetrahedrille” tessellation of space into the truncated tetrahedra obtained by removing these pyramids and the ordinary tetrahedra obtained by gluing them together in sets of four. Both tessellations have the same group, 2+:2, as the diamond. In 1887 Lord Kelvin asked what was the most efficient way to fill space with bubbles of equal volume, and he conjectured that it was the arrangement shown on the left in Figure 23.5, in which the bubbles are “relaxed” versions of our truncated octahedra (“relaxed” because the area of the faces is reduced by allowing them to be slightly curved). We shall call these the Scottish bubbles.2 Their centers are 0, 1, 2, and 3, so the group is again the double octad Downloaded by [University of Bergen Library] at 04:55 26 October 2016 group 8◦:2 of symmetries of the bc lattice. The record stood until 1993, when Weaire and Phelan found what we call the Irish bubbles, shown on the right in Figure 23.5, whose centers are all the points 0, 1, 2, 3, and +, by relaxing the Voronoi cells determined by these points. Their group is therefore the pure octad group 8◦,since+and− can no longer be interchanged. 2Lord Kelvin was actually born in Belfast: however, he did most of his work at Glasgow University and in 1892 became Baron Kelvin of Largs, the nearby town where he lived. © 2016 by Taylor & Francis Group, LLC Infinite Platonic Polyhedra 333 Figure 23.5. Efficient bubbles: At left, the Scottish bubbles, considered by Kelvin in 1887; at right, the more efficient Irish Bubbles, found by Phelan and Weaire in 1993. The actual bubbles are slightly “bulgy” versions of the polyhedra illustrated here. Infinite Platonic Polyhedra In 1923, the 16-year-old schoolboys H. S. M. Coxeter and J. F. Petrie found themselves confined to the same sanatorium and started to discuss mathematics. Later they published a paper about three new regular polyhedra [10]. These Coxeter-Petrie polyhedra don’t seem to have received individual names in their first eight decades; since they are multiple covers of the first three Platonic solids, we shall call them the multiplied tetrahedron, multiplied cube,andmultiplied oc- tahedron and slangily shorten these to mucube (μC), muoctahedron (μO), and mutetrahedron (μT ), respectively.
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