Nonconvex Embeddings of the Exceptional Simplicial 3-Spheres with 8 Vertices

As shown by D. Barnette (1973, J. Combin. Theory Ser. A14, 37–53) there are precisely 39 simplicial 3-spheres with 8 vertices. Thirty-seven of these are boundary complexes for convex 4-polytopes. In this paper we supply nonconvex embeddings in Euclidean 4-space for the remaining two 3-spheres. We discuss the properties of the embeddings as well as the techniques used to demonstrate their validity.