Stellations

    Recall that when you fly inside certain polyhedra (e.g., the five cubes) and see the innermost region within its planes, you find a smaller inner object (e.g., the rhombic triacontahedron) whose face planes are parts of the outer polyhedron. Stellation is the reverse relationship, or process, of starting with a polyhedron and extending its face planes until they meet other face planes, to create a new larger polyhedron.

    Depending on how the face planes of a given polyhedron divide space, there may be several ways to extend them and choose regions which they bound. Starting with an octahedron, stellation produces only one possibility: the compound of two tetrahedra (stella octangula). This model shows two tetrahedra in outline; their face planes are generated by extending the faces of the inner octahedron.

    Observe that two planes which meet at an edge of the stellation will not share an edge of the inner polyhedron. So there are no stellations of the cube --- nonadjacent faces are parallel, and so never meet to form a finite solid.





    Starting with the dodecahedron, the three stellations possible: the small stellated dodecahedron, the great dodecahedron, and the great stellated dodecahedron are shown above.  In that order, each is a continuation of the face planes of the previous one. These three stellations fall within the class of Kepler-Poinsot polyhedra. Try flying inside them to find the inner dodecahedron in each.

    The remaining Kepler-Poinsot solid, the great icosahedron, is a stellation of the icosahedron. Travel inside this great icosahedron to find its inner icosahedron. Starting with the icosahedron, it turns out that there are 59 possible stellations, of which the great icosahedron is only one. To learn about the others, read about the 59 Stellations of the Icosahedron.



    The first stellation of a convex quasi-regular solid is a compound of two dual regular solids. For example, the first stellation of the cuboctahedron is the compound of the cube and octahedron. The first stellation of the icosidodecahedron is the compound of the icosahedron and the dodecahedron. 



    There are three stellations of the rhombic dodecahedron:

    • The first stellation of the rhombic dodecahedron, illustrated at right, shows up in popular take-apart puzzles and is an interesting space-filling solid. Notice how similar it is to the compound of three octahedra; just change the shape of each triangle in the compound slightly from equilateral (i.e., reduce the altitudes of the square dipyramids) to get the stellation. Both show up in the work of M.C. Escher.
    • The second stellation can be seen as a compound of four flat rhombic parallelepipeds. (Each with its 3-fold axis aligned with a 3-fold axis of the original rhombic dodecahedron.)
    • The third stellation can be seen as a compound of six tetragonal disphenoids, very similar to the rigid compound of 6 tetrahedra; again only a slight change in triangle shape from equilateral is necessary to obtain the stellated rhombic dodecahedron.
    Interestingly, the original rhombic dodecahedron and the three stellations can all be assembled from copies of a single simple rhombic pyramid --- one twelfth of the original rhombic dodecahedron --- as discussed in Cundy and Rollett's Mathematical Models listed in the references.

    You can also read about the many stellations of the cuboctahedron, stellations of the rhombic triacontahedron, stellations of the triakis tetrahedron, and tetrahedral stellations of the dodecahedron.

    Virtual Polyhedra, (c) 1996, George W. Hart