A regular star prism consists of two regular N-sided star polygons joined
together by N squares. A regular star antiprism consists of two regular
N-sided star polygons joined together by 2N equilateral triangles.
They are considered uniform polyhedra because they have regular polygons as
faces and are vertex-transitive. Vertex transitivity means that for any
two vertices of the polyhedron, there exists a translation, rotation, and/or
reflection that leaves the outward appearance of the polyhedron unchanged yet
moves one vertex to the other. Like regular prisms and antiprisms, they can
be distinguished from other uniform polyhedra by the fact that they do not
have polyhedral group (tetrahedral, octahedral, or icosahedral) rotational
Jean Paul Albert Badoureau, Mémoire sur les Figures Isocèles, Journal de l'École polytechnique49 (1881), 47-172.
H. S. M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller, Uniform Polyhedra, Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences246 (1954), 401-450.