Smooth 3D manifolds based on thin plates

This paper demonstrates a fractal system of thin plates connected with hinges. The system can be studied using the methods of the mechanics of solids with internal degrees of freedom. The structure is deployable, and initially, it is similar to a small-diameter one-dimensional manifold, which occupies significant volume after deployment. The geometry of solids is studied using the method of the moving hedron. The relationships enabling the definition of the geometry of the introduced manifolds are derived based on the Cartan structure equations. The proof substantially makes use of the fact that the fractal consists of thin plates that are not long compared to the sizes of the system. The mechanics are described for solids with rigid plastic hinges between the plates, and the hinges are made of shape memory material. Based on the ultimate load theorems, estimates are performed to specify the internal pressure that is required to deploy the package into a three-dimensional (3D) structure and the heat input needed to return the system into its initial state. Some possible applications of the smooth 3D manifolds are demonstrated.