  Floppy modes --- deformations that cost zero energy --- are central to the
mechanics of a wide class of systems. For disordered systems, such as random
networks and particle packings, it is well-understood how the number of floppy
modes is controlled by the topology of the connections. Here we uncover that
symmetric geometries, present in e.g. mechanical metamaterials, can feature an
unlimited number of excess floppy modes that are absent in generic geometries,
and in addition can support floppy modes that are multi-branched. We study the
number $\Delta$ of excess floppy modes by comparing generic and symmetric
geometries with identical topologies, and show that $\Delta$ is extensive,
peaks at intermediate connection densities, and exhibits mean field scaling. We
then develop an approximate yet accurate cluster counting algorithm that
captures these findings. Finally, we leverage our insights to design
metamaterials with multiple folding mechanisms.
