Erik Demaine and I also included a proof for $d=2$ in [*Geometric Folding Algorithms: Linkages, Origami, Polyhedra*](http://www.gfalop.org/), Chapter 3. There we asked if there is a *planar* (non-crossing) linkage that "signs your name" (traces any semi-algebraic region), a question posed by Don Shimamoto in 2004. <hr /> [![JohnHancock][1]][1] <hr /> This was recently settled positively by Zachary Abel in his Ph.D. thesis: any polynomial curve $f(x,y) = 0$ can be traced by a non-crossing linkage. > Abel, Zachary Ryan. "On folding and unfolding with linkages and origami." PhD diss., Massachusetts Institute of Technology, 2016. [MIT link](https://dspace.mit.edu/handle/1721.1/107547). [1]: https://i.sstatic.net/Ge9Pr.jpg