It has remained a great challenge to develop coupled mechanisms in kinematics. Polyhedrons with various appearance and symmetric configuration attract more and more attentions and provide possible basic modes for the design of coupled mechanisms. Based on the symmetry and duality of the regular dodecahedron, a movable icosahedron mechanism with single mobility was designed, where the coupling nodes were composed of triangles, quadrilaterals and pentagons. Firstly, in order to design the coupling nodes with triangles, quadrilaterals and pentagons, each vertex was added to the ten faces of the icosahedron and the node variety was increased by adding types of vertexes. The common edge of the original regular pentagon was replaced by four edges. Four edges made up a quadrilateral. The common edge of the original regular pentagon became a diagonal of the quadrilateral. Then four kinds of replacement components were designed according to the connection of each component, and the replacement components sequentially were connected by rotational joints to construct a basic model of the coupled mechanism. Secondly, mobility of the mechanism model was calculated by the shunting method based on screw theory. The coupled mechanism was divided into five identical branches according to the shunting tags. As long as the constraints of one branch were determined based on screw theory, the constraints of all branches to the moving platform was obtained. Then mobility of the coupled mechanism was obtained, which showed the coupled mechanism had a translational freedom. Finally, active input selection was analyzed and the three-dimensional model of the coupled mechanism was established. Two opposite driving conditions were set up in the simulation, and position curves of the moving platform and the node components were got by Matlab. The simulation results were consistent with theoretical results. It further showed that the proposed mechanism had the characteristics of stretching motion.