The results of this thesis concern folding of one-dimensional objects
in two dimensions: planar linkages. More precisely, a planar linkage consists
of a collection of rigid bars (line segments) connected at their endpoints.
Foldings of such a linkage must preserve the connections at endpoints, preserve
the bar lengths, and (in our context) prevent bars from crossing. The main
result of this thesis
is that a planar linkage
forming a collection of polygonal arcs and cycles can be folded so that all
outermost arcs (not enclosed by other cycles) become straight and all outermost
cycles become convex. A complementary result of this thesis
is that once a cycle becomes convex, it can be folded into any
other convex cycle with the same counterclockwise sequence of bar lengths.
Together, these results show that the configuration space of all possible
foldings of a planar arc or cycle linkage is connected.
These results fall into the broader context of folding and unfolding
k-dimensional objects in n-dimensional space,
k ≤ n.
Another contribution of this thesis is a survey of research in this field.
The survey revolves around three principal aspects
that have received extensive study:
linkages in arbitrary dimensions (folding one-dimensional objects
in two or more dimensions, including protein folding),
paper folding (normally, folding two-dimensional
objects in three dimensions), and folding and unfolding polyhedra
(two-dimensional objects embedded in three-dimensional space).