7 Layouts of spatial motion
structures
7.1 Tilings and patterns
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, (3
6
) 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, (3
3
.4
2
)
means three triangles and two squares around a cross-point. And (3
6
;3
2
.6
2
)
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,
structures
7.1 Tilings and patterns
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, (3
6
) 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, (3
3
.4
2
)
means three triangles and two squares around a cross-point. And (3
6
;3
2
.6
2
)
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,