From any given polyhedron another can be derived by reciprocating with respect to some sphere. The most rewarding cases arise when the sphere is either the inscribed or the circumscribed sphere of the first polyhedron. In this way every Archimedean polyhedron gives rise to a new polyhedron, all of whose faces are congruent. Taking the simplest Archimedean solid, the truncated regular tetrahedron, we may derive a solid which has been called* both the “triakis tetrahedron” and the “tristetrahedron”. This solid has twelve isosceles triangles as faces, which meet by threes and sixes in a total of twelve vertices. It enjoys a certain importance in crystallography; for example, the mineral eulytine occurs in crystals which are basically triakis tetrahedra. We shall here consider some of the solids formed by stellating the triakis tetrahedron.