Thurston's "tinker toy" problem In the article "On Being Thurstonized" by Benson Farb (located here), a particular result of Thurston is mentioned. 
Namely, suppose a "tinker toy" $T$ is a contraption consisting of a multitude of rods. They can either be bolted to a table, or attached to each other along hinges. The configuration space $C(T)$ consists of all possible configurations of $T$. For example, if $T$ is a single segment, then $C(T)$ is a circle. 
The claim is then that any compact, smooth manifold can be obtained as a component of some $C(T)$. 
I've thought about it a bit (though perhaps not enough), and this theorem escapes me completely. Are there any notes or explanations out there that describe the proof of this theorem?
 A: I was Thurston's (undergraduate) student in the mid 1980s, when he was thinking about linkages.  Here's how Thurston explained the proof to me.
By a theorem of Nash (see https://en.wikipedia.org/wiki/Nash_functions), any smooth manifold is diffeomorphic to a solution space of a set of real polynomial equations.  So now all one needs to do is devise planar linkages which implement addition and multiplication of real numbers, and hook them together in a way that mirrors the algebraic equations from Nash's theorem.  
I never worked through the details of the above sketch myself.

[Added: Igor Rivin's answer overlaps with mine, and has additional details.]
A: It was published here:
M. Kapovich, J. Millson, Universality theorems for configuration spaces of planar linkages,
Topology 41 (2002), no. 6, 1051–1107. 
A: The result comes by way of Nash's theorem which states that every smooth manifold is a component of a real algebraic variety.
Nash, John, Real algebraic manifolds, Ann. Math. (2) 56, 405-421 (1952). ZBL0048.38501.
The game is now to represent a real algebraic variety as a configuration space of a linkage. This is a very old game, going back to Kempe 1876 - the so called Kempe Universality Theorem which states that any real plane algebraic curve can be thus represented. The result of Kapovich and Millson in Andy Putman's answer is a direct descendant of Kempe's result. As far as I know, Thurston did not prove this result, but did give it to Bill Goldman as a Senior Thesis problem. I was under the impression that Goldman solved the problem, but maybe it was a restricted version.
A: Sometimes the Kempe Universality Theorem 
(see Igor Rivin's answer)
is expressed as: There is a linkage
that signs your name.

          


As an indication of how difficult it is to achieve this in practice, here is a little
linkage that when driven by $b$ rotating on the pinned circle center $a$,
draws a rough approximation to a handwritten letter J (purple), drawn out
by joint $z$.

          


          

Fig. 2.16 in How To Fold It.
Linkage designed by Don Shimamoto.
Computations in Cinderella.


