Natural realizations of closed orientable surfaces A beautiful fact is that

The space of configurations of a 5-vertex polygon with unit length sides, two of whose  vertices are fixed, is a closed orientable surface of genus $3$.

Similarly, but much more simply, the torus is the space of configurations of a double pendulum.
Are there other natural realizations of high genus closed orientable surfaces?
 A: Actually, there are several relatively simple planar linkages whose configuration spaces are surfaces of higher genera. For instance, if $S_n$ is a linkage which is a "spider" with $n$ legs each of which has one joint and such that tips of the legs are glued to the plane, then the configuration space of $S_n$ is a surface of certain genus $g$. Such linkages (with $n=3$) first were analyzed (to the best of my knowledge) in a paper by Thurston and Weeks:
W. Thurston and J. Weeks, The Mathematics of three dimensional manifolds,  Scientific American, 251 (1984) 94-106.  

The following is a theorem proven in 
P.Mounoud, Sur l’espace des configurations d’une araignee. Osaka J. Math., 48(1):149–178, 2011.
Theorem. Let $g$ be an natural number and $r$ the biggest integer
such that $2r$ divides $g-1$. A compact orientable surface of genus $g$ is diffeomorphic to the configuration space of some spider if and only if 
$$
2^{-r}(g − 1) \le 6r + 12.$$
On the other hand, if one allows linkages which are "centepides" then the oriented surface of any given genus is realized:
D. Jordan and M. Steiner
Compact surfaces as configuration spaces of mechanical linkages, Israel Journal of Mathematics, 122 (2001) 175–187, 
