A closed linkage mechanism having the shape of a discrete Möbius strip

A closed linkage mechanism in three-dimensional space is an object comprising rigid bodies connected with hinges in a circular form like a rosary. Such linkages include Bricard6R and Bennett4R. To design such a closed linkage, it is necessary to solve a high-degree algebraic equation, which is generally difficult. In this lecture, the author proposes a new family of closed linkage mechanisms with an arbitrary number of hinges as an extension of a certain Bricard6R. They have singular properties, such as one-dimensional degree of freedom (1-DOF), and certain energies taking a constant value regardless of the state. These linkage mechanisms can be regarded as discrete M\"obius strips and may be of interest in the context of pure mathematics as well. However, many of the properties described here have been confirmed only numerically, with no rigorous mathematical proof, and should be interpreted with caution.