Symmetric Configuration spaces of linkages

A $configuration$ of a linkage $\mathrm{\Gamma }$ is a possible positioning of $\mathrm{\Gamma }$ in ${\mathbb{R}}^d$ and the collection of all such forms the configuration space $\mathcal{C}(\mathrm{\Gamma })$ of $\mathrm{\Gamma }$. We here introduce the notion of the $symmetricconfigurationspace$ of a linkage, in which we identify configurations which are geometrically indistinguishable. We show that the symmetric configuration space of a planar polygon has a regular cell structure, provide some principles for calculating this structure, and give a complete description of the symmetric configuration space of all quadrilaterals and of the equilateral pentagon.