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 ∆ of excess floppy modes by comparing generic and symmetric geometries with identical topologies, and show that ∆ 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.Floppy modes (FMs) play a fundamental role in the mechanics of a wide variety of disordered physical systems, from elastic networks [1-7] to jammed particle packings [8][9][10]. Floppy modes also play a role in many engineering problems, ranging from robotics to deployable structures, where the goal is to design structures that feature one or more mechanisms [11]. Mechanisms are collections of rigid elements linked by flexible hinges, designed to allow for a collective, floppy motion of the elements. More recently, floppy modes and mechanisms have received renewed attention in the context of mechanical metamaterials, which are architected materials designed to exhibit anomalous mechanical properties, including negative response parameters, shape morphing, and self-folding [6,7,[12][13][14][15][16][17][18][19][20][21][22]]. An important design strategy for mechanical metamaterials borrows the geometric design of mechanisms, and replaces their hinges by flexible parts which connect stiffer elements [22]. In all these examples, understanding how the geometric design controls the number and character of the floppy modes plays a central role.For systems consisting of objects with a total of n d degrees of freedom, connected by hinges that provide n c constraints, the number of non-trivial floppy modes n f and states of self-stress n ss are related by Maxwell-Calladine counting as n f − n ss = n d − n c − n rb where n rb counts the trivial rigid body modes (n rb = 3 in two dimensions) [23]. For generic, disordered systems n f and n ss can be determined separately from the connection topology [1, 2], but when symmetries are present such approaches break down and counting only yields the difference n f − n ss . For example, spring lattices which feature perfectly aligned bonds can generate excess floppy modes (and associated states of self stress) that disappear under generic perturbations and thus escape topology-based counting methods [1,2,[24][25][26][27][28][29]. Mechanical metamaterials often feature symmetric architectures where excess (a) (b) θ × FIG. 1. (color online) (a) N × N systems of...