Peter Jon Pearce illustrated the same subject in 1978 but mentioned only twenty-three tessellations. Keith Critchlow ( 1969) and Robert Williams ( 1979) published books in which this matter was illustrated but failed Footnote 4 to consider the whole set of Andreini’s tessellations. Figure 7 illustrates one of his beautiful stereoscopic photographs. In 1905, Alfredo Andreini published “On the regular and semiregular nets of polyhedra and on the corresponding correlative nets” with a systematic study of the 25 possibilities to close-pack polyhedra in uniform tessellations that he considered complete (although missing four and listing one mistakenly as uniform, according to Grünbaum ( 1994: 49)). ![]() Footnote 3 Given that distinct polyhedra must “have matching parallel faces in common” (Pearce 1978: 43), we may conclude that, to uniformly fill space with convex polyhedra, only cubes, tetrahedra, octahedra, cuboctahedra, truncated tetrahedra, truncated octahedra, truncated cubes, rhombicuboctahedra, rhombitruncated cuboctahedra and triangular, hexagonal, octagonal and dodecagonal prisms can perform efficiently as cells. Triangular semiregular prisms are capable of filling space as well, but they have to change orientation to do so. From convex uniform polyhedra, cubes, truncated octahedra and hexagonal prisms are space-fillers, as we have seen. In uniform solid tessellations, all the cells are convex uniform polyhedra and all vertices are transitive, that is, equally surrounded and forming “one orbit under symmetries” (Grünbaum 1994: 50). In dashed lines, examples of other producible non-regular tessellations are shown. Whether regular, quasiregular Footnote 2 or irregular, every plane tessellation shown results from sectioning each honeycomb outlined by the representatives of the six convex parallelohedra with planes passing through some of its vertices. 1, 2, 3, 4, 5 and 6 have, implicit, a regular plane tessellation (illustrated in continuous lines). Furthermore, most of the solid tessellations Footnote 1 in Figs. Solid and plane tessellations are intrinsically connected, since any section cut through the former “always produces a tessellation of some kind” (Pugh 1976: 48). As an example, we recall polar zonohedra (Towle 1996) and the dome in Bruno Taut’s Glass Pavilion built in 1914. In any case, its space-filling capabilities are preserved (Lalvani 1992) and this possibility, denotes Kappraff, “gives zonohedra an advantage over geodesic domes as building structures … since it enables polyhedral structures to be built which fit form and function” ( 1990: 375–376). The angles between vectors may also be changed, distorting the faces of polyhedra. As well as every other zonohedron, parallelohedra can be extended or shortened, by changing the length of any vector of its star. For every primary parallelohedron, there is a spatial set of segments with a common midpoint and the direction of its edges, known as the vector star (Coxeter 1973: 27), that categorizes primary parallelohedra as zonohedra. To these, in 1960 Stanko Bilinski added a second rhombic dodecahedron, additionally proving it as (the last) convex isozonohedron (Grünbaum 2010: 5). The crystallographer Evgraf Fedorov listed, in 1885, the cube (representative of all rhombohedra), the semiregular hexagonal prism, the rhombic dodecahedron, the elongated dodecahedron and the truncated octahedron (Grünbaum 2010: 4) as all the possible combinatorial types of convex polyhedra that fill space in monohedral tessellations, without changing orientation. 1, 2, 3, 4, 5 and 6 are convex polyhedra with centrally symmetric faces that fill space by translation of their replicas. The six primary parallelohedra illustrated in Figs. ![]() The search goes on, but here we will focus on primary parallelohedra, convex uniform tessellations and some topological interlocking assemblies. ![]() In 1980, two types of asymmetrical convex polyhedra with thirty-eight faces, each of which fill space monohedrally, were discovered by Peter Engel (Grünbaum and Shepard 1980: 965). The enumeration of polyhedra that fill space in infinite replicas (in other words, plesiohedra, whose centroids outline lattice points), Grünbaum and Shepard denote, “has no finite answer” ( 1980: 966) and remains an open problem in mathematics.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |