Layouts of spatial motion structures
One of the most important aspect in design of large spatial motion structures is the identification of a suitable layout. As demonstrated in Chapters 5 and 6, in a chosen layout, the building blocks, often based on a known mechanism, are repeatedly used leading to a generic solution for a type of motion structure. The number of building blocks can be altered depending on the practical size requirement but the mobility of each block is always retained. Since most of the building blocks, though three dimensional, can be represented by two dimensional polygons schematically, a convenient method for the design of layouts is to utilise a mathematical tool known as tiling, also frequently referred to as tessellation. A plane tiling is a countable family of closed sets which cover the plane without gaps or overlaps. The closed sets are called tiles of the tiling. The layout of tiles, termed as a pattern in tiling, is a design which repeats some motif in a more or less systematic manner. The art of designing tilings and patterns is clearly extremely old and well developed (Beverley, 1999; Evans, 1931; Rossi, 1970). By contrast, the science of tilings and patterns, which means the study of their mathematical properties, is comparatively recent and many parts of the subject have yet to be explored in depth. The most methodological study of tilings and patterns can be found in Grünbaum and Shephard (1986). Only a brief introduction is provided here. In mathematics tilings by regular polygons are usually represented by the number of sides of the polygons around any cross point in the clockwise or anti-clockwise order. For instance, (36) is a tiling in which each of the points is surrounded by six triangles, ‘3’ is the number of the sides of a triangle and superscript ‘6’ is the number of triangles. Similarly, (33.42) means three triangles and two squares around a cross-point. And (36;32.62) represents a two-uniform tiling in which there are two types of points, one type is surrounded by six triangles whereas the other type is surrounded by two triangles and two hexagons. The tilings accommodating regular polygons can be classified into four types: regular and uniform tilings,
k-uniform tilings, equitransitive and edge-transitive tilings, and tilings that are not edge-to-edge (Grünbaum and Shephard, 1986). The only edge-to-edge monohedral tilings by regular polygons are the three regular tilings shown in Figure 7.1. The basic tiles are identical equilateral triangles, squares and regular hexagons, respectively. There exist precisely eleven distinct edge-to-edge uniform tilings by more than one type of regular polygons such that all vertices are of the same type. They are (36), (34.6), (33.42), (188.8.131.52), (184.108.40.206), (220.127.116.11), (3.122), (44), (4.6.12), (4.82) and (63). An edge-to-edge tiling by regular polygons is called k-uniform if its vertices form precisely k transitivity classes with respect to the group of symmetries of the tilings. Denote K(k) as the number of distinct k-uniform tilings. K(1) = 11, K(2) = 20, K(3) = 39, K(4) = 33, K(5) = 15, K(6) = 10, K(7) = 7 and K(k) = 0 when k ≥ 8. So the total number of distinct k-uniform tilings is 135. These tilings can be modified into many more tilings and patterns with methods such as transformation of symmetry, transitivity and regularity, tilings that are not edge-to-edge and patterns with overlap motifs. The regular and uniform tilings, though simple, can be extended by allowing for each tile itself containing a pattern that differs from the polygonal
shape of the tile. If a building block, consisting of a single mobile element or a mobile assembly of a number of interconnected elements, has a planar schematic representation that can be inscribed by a triangular, square or hexagonal tile, the corresponding tiling instantly gives a suitable layout provided that the patterns within a tile can be seamlessly connected to those in neighbouring tiles. Figure 7.2 shows a number of possibilities. The remaining task is to ensure that the kinematic requirements can be met in a particular tiling. In the following sections, the above approach is to be applied to motion structures based on the Bennett, Myard and Bricard linkages.