  Molecular robotics is challenging, so it seems best to keep it simple. We
consider an abstract molecular robotics model based on simple folding
instructions that execute asynchronously. Turning Machines are a simple 1D to
2D folding model, also easily generalisable to 2D to 3D folding. A Turning
Machine starts out as a line of connected monomers in the discrete plane, each
with an associated turning number. A monomer turns relative to its neighbours,
executing a unit-distance translation that drags other monomers along with it,
and through collective motion the initial set of monomers eventually folds into
a programmed shape. We provide a suite of tools for reasoning about Turning
Machines by fully characterising their ability to execute line rotations:
executing an almost-full line rotation of $5\pi/3$ radians is possible, yet a
full $2\pi$ rotation is impossible. Furthermore, line rotations up to $5\pi/3$
are executed efficiently, in $O(\log n)$ expected time in our continuous time
Markov chain time model. We then show that such line-rotations represent a
fundamental primitive in the model, by using them to efficiently and
asynchronously fold shapes. In particular, arbitrarily large zig-zag-rastered
squares and zig-zag paths are foldable, as are $y$-monotone shapes albeit with
error (bounded by perimeter length). Finally, we give shapes that despite
having paths that traverse all their points, are in fact impossible to fold, as
well as techniques for folding certain classes of (scaled) shapes without
error. Our approach relies on careful geometric-based analyses of the feats
possible and impossible by a very simple robotic system, and pushes
conceptional hardness towards mathematical analysis and away from molecular
implementation.
