  Planar/flat configurations of fixed-angle chains and trees are well studied
in the context of polymer science, molecular biology, and puzzles. In this
paper, we focus on a simple type of fixed-angle linkage: every edge has unit
length (equilateral), and each joint has a fixed angle of $90^\circ$
(orthogonal) or $180^\circ$ (straight). When the linkage forms a path (open
chain), it always has a planar configuration, namely the zig-zag which
alternating the $90^\circ$ angles between left and right turns. But when the
linkage forms a cycle (closed chain), or is forced to lie in a box of fixed
size, we prove that the flattening problem -- deciding whether there is a
planar noncrossing configuration -- is strongly NP-complete.
  Back to open chains, we turn to the Hydrophobic-Hydrophilic (HP) model of
protein folding, where each vertex is labeled H or P, and the goal is to find a
folding that maximizes the number of H-H adjacencies. In the well-studied HP
model, the joint angles are not fixed. We introduce and analyze the fixed-angle
HP model, which is motivated by real-world proteins. We prove strong
NP-completeness of finding a planar noncrossing configuration of a fixed-angle
orthogonal equilateral open chain with the most H--H adjacencies, even if the
chain has only two H vertices. (Effectively, this lets us force the chain to be
closed.)
