Geometrical elements of DNA nanostructure are faces, edges, and vertices. Research focusing on the faces and edges has been reported, such as DNA boxes and DNA tweezers. Since no mechanism has been established to change the number and position of vertices, here we propose a new mechanism that controls the shape by swapping the positions of recessed and protruding vertices using the theory of dual polyhedron. The structure consists of a rhombic dodecahedron created using DNA origami wireframe and ssDNA branched from the edges. When Another ssDNA is added as a signal, the corresponding ssDNA reacts and transform the structure. There are three states: a rhombic dodecahedron, a hexahedron, and an octahedron. Therefore, by arranging the structures regularly, it is possible to create crystals whose density can be changed in response to signals, and there is a possibility that this structure is used as a new building material.