We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(alpha_0,ldots, alpha_{n-1})$, $alpha_iin (-pi,pi)$, for $iin{0,ldots, n-1}$. The problem of realizing $A$ by a polygon ca
n be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an emph{angle graph}. In 2D, we characterize sequences $A$ for which every generic polygon $Psubset mathbb{R}^2$ realizing $A$ has at least $c$ crossings, for every $cin mathbb{N}$, and describe an efficient algorithm that constructs, for a given sequence $A$, a generic polygon $Psubset mathbb{R}^2$ that realizes $A$ with the minimum number of crossings. In 3D, we describe an efficient algorithm that tests whether a given sequence $A$ can be realized by a (not necessarily generic) polygon $Psubset mathbb{R}^3$, and for every realizable sequence the algorithm finds a realization.
In the picture-hanging puzzle we are to hang a picture so that the string loops around $n$ nails and the removal of any nail results in a fall of the picture. We show that the length of a sequence representing an element in the free group with $n$ ge
nerators that corresponds to a solution of the picture-hanging puzzle must be at least $n2^{sqrt{log_2 n}}$. In other words, this is a lower bound on the length of a sequence representing a non-trivial element in the free group with $n$ generators such that if we replace any of the generators by the identity the sequence becomes trivial.
