Topological antiqued mechanical toy

Abstract

{\it Jacob's ladder} -- a classic children's toy -- is a simple mechanical frame comprising rigid blocks connected by strings that shows curious unidirectional flipping waves. Nonetheless, its physical origin remains elusive. By combining experiment, numeral simulation, and theory, we show that understanding the underlying design principle of this toy requires diverse physical ideas. First, we conduct a water-tank experiment that excludes the domino-like mechanism, thus defying widespread expectations. Subsequently, we analytically demonstrate that the toy is bistable under gravity, thus implying its kink wave as a class of topological solitons. The waves are surprisingly reminiscent -- both experimentally and theoretically -- to those in the Kane--Lubensky topological chain, owing to the stiffening of zero modes by the pretension under gravity. However, a close examination based on the index theorem reveals that the similarity remains superficial and that the floppiness of the toy underlies the kink and antikink coexistence -- a forbidden mode in the topological chain. By analyzing a generalized asymmetric toy, we reveal that its symmetric connection renders it topologically singular, thus resulting in amusing motions. We demonstrate these ideas by experimentally observing a dramatic pair annihilation of kink and antikink waves.