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

more »

... that are multibranched. 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 nontrivial 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 topologybased counting methods [1, 2, [24] [25] [26] [27] [28] [29] . Mechanical metamaterials often feature symmetric architectures where excess floppy modes (EFMs) may arise, but their geometries are more complex than spring lattices [22, [30] [31] [32] [33] [34] [35] . We focus on understanding the EFMs of a geometry which underlies a range of metamaterials [14, 16, 17, [20] [21] [22] 36, 37] : rigid quadrilaterals connected by flexible hinges. We define n s and n g as the number of nontrivial floppy modes for symmetric systems consisting of squares, and stress-free generic systems obtained by randomly displacing the corners of linked squares with magnitude = 0.1 [38] . Each quadrilateral has three degrees of freedom (DOF), and in a fully connected lattice each quadrilateral has four connections in the bulk, and less near the boundary. By counting the degrees of freedom and constraints one finds that M × N lattices of generic quadrilaterals are rigid (n g = 0) when M 3 and N 3; however, replacing the generic quadrilaterals by equally sized squares, such lattices always exhibit an EFM where the squares can counter-rotate [14], i.e., n s = 1 [ Fig. 1(a) ]. Here we address two key issues. First, what is the multiplicity and statistics of EFMs in diluted lattices as considered recently [21, 36, 37] [ Fig. 1(b) ]? Second, do EFMs in diluted lattices possess anomalous properties, and if so, how can we leverage these to embed new functionalities into metamaterials? System and methods. We consider diluted N × N lattices of quadrilaterals connected by springs of unit stiffness and zero (a) (b) θ × FIG. 1. (a) N × N systems of generic quadrilaterals are rigid for N 3, but have a floppy, "hinging" mode characterized by the opening angle θ ∈ [0, π] for perfectly symmetric squares. (b) Diluted square tiling (N = 10, ρ = 0.8).