An augmented polyhedron is a polyhedron in which additional polyhedra are connected to one or more of the faces of the original polyhedron, such that the contacting faces are identical polygons. In this example, to each of the 20 triangular faces of the original icosahedron, is attached a tetrahedron. The resulting polyhedra, while looking like a threedimensional star, is similar to but not identical to what is called a stellated ("star") icosahedron, of which there are several types. In a stellated icosahedron, the additional edges (magnetic rods in Roger's Connection) must conform to the intersections of the planes defined by extending the faces of the icosahedron beyond the boundaries of the original icosahedron. See the additional notes below in the section Stellated Polyhedra and Roger's Connection. 

Shown here is a dodecahedron. The dodecahedron is one of the five platonic solids. It has 12 pentagonal faces, 20 vertices, and 30 edges. By adding a pentagonal pyramid to each face, as shown in the movie frame on the left in blue, an augmented dodecahedron is formed. Without the triangular bracing provided by the pentagonal pyramids, a Roger's Connection version of the dodecahedron would not be a stable design. The movie shows the structure rotating, and will give you a much clearer idea of its three dimensional details. The red rods highlight the original dodecahedron. Requires three sets to build. 
An augmented polyhedron is a polyhedron in which additional polyhedra are connected to one or more of the faces of the original polyhedron, such that the contacting faces are identical polygons. In this example, to each of the 12 pentagonal faces of the original dodecahedron, is attached a pentagonal pyramid. The resulting polyhedron is similar to but not identical to what is called a stellated ("star") dodecahedron, of which there are several types. In a stellated dodecahedron, the additional edges (magnetic rods in Roger's Connection) must conform to the intersections of the planes defined by extending the faces of the dodecahedron beyond the boundaries of the original dodecahedron. See the additional notes below in the section Stellated Polyhedra and Roger's Connection. 
In Roger's Connection, all edge lengths (magnetic rod lengths) are the same, but in order to form a stellated icosahedron or dodecahedron, different lengths from those available would be required to add the stellations. Roger's Connection can be used to make only one of the stellated platonic solids, a stellated octahedron, also called a stella octangula, which has the appearance of two interlocked tetrahedrons. For a detailed discussion of stellated polyhedra, see http://mathworld.wolfram.com/Stellation.html. Once there, click on the links for threedimensional examples of stellated polyhedra. If your browser supports Java, then many of the figures can be rotated by clicking and dragging on the figures with the mouse. The stella octangula may be viewed and rotated at: http://mathworld.wolfram.com/StellaOctangula.html. Here is a stellated octangula made with Roger's Connection.