Meenakshi: A stellated octahedron can be thought of as a compound of two tetrahedra, where the spikes are smaller tetrahedra themselves ( https://en.wikipedia.org/wiki/Stellated_octahedron). The Sonobe constructions have spikes that are pyramids bound by 3 right isosceles triangles and a triangular base. In the case of the Stellated Octahedron, the spikes are bound by 4 equilateral triangles. The Sonobe type constructions in no way satisfy the criteria for a stellated solid ( https://en.wikipedia.org/wiki/Stellation).
But then, I haven't found a good name for those Sonobe solids. How about pyramidized octahedron or icosahedron? I've seen the terms 'elevated' or 'augmented' used to refer to these types of structures. The pyramid sticking out from the base polyhedron does not need to be anything special, unless you are looking for the stellation. George Hart has some pictures from Leonardo Da Vinci's (1509) http://www.georgehart.com/virtual-polyhedra/leonardo.html Faye Goldman
