In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass of prismatoids, and are a (degenerate) type of snub polyhedron. Antiprisms are similar to prisms, except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are triangles, rather than quadrilaterals. The dual polyhedron of an -gonal antiprism is an -gonal trapezohedron. History At the intersection of modern-day graph theory and coding theory, the triangulation of a set of points have interested mathematicians since Isaac Newton, who fruitlessly sought a mathematical proof of the kissing number problem in 1694. The existence of antiprisms was discussed, and their name was coined by Johannes Kepler, though it is possible that they were previously known to Archimedes, as they satisfy the same conditions on faces and on vertices as the Archimedean solids. According to Ericson and Zinoviev, Harold Scott MacDonald Coxeter wrote at length on the topic, and was among the first to apply the mathematics of Victor Schlegel to this field. Knowledge in this field is "quite incomplete" and "was obtained fairly recently", i.e. in the 20th century. For example, as of 2001 it had been proven for only a limited number of non-trivial cases that the -gonal antiprism is the mathematically optimal arrangement of points in the sense of maximizing the minimum Euclidean distance between any two points on the set: in 1943 by László Fejes Tóth for 4 and 6 points (digonal and trigonal antiprisms, which are Platonic solids); in 1951 by Kurt Schütte and Bartel Leendert van der Waerden for 8 points (tetragonal antiprism, which is not a cube). The chemical structure of binary compounds has been remarked to be in the family of antiprisms; especially those of the family of boron hydrides (in 1975) and carboranes because they are isoelectronic. This is a mathematically real conclusion reached by studies of X-ray diffraction patterns, and stems from the 1971 work of Kenneth Wade, the nominative source for Wade's rules of polyhedral skeletal electron pair theory. Rare-earth metals such as the lanthanides form antiprismatic compounds with some of the halides or some of the iodides. The study of crystallography is useful here. Some lanthanides, when arranged in peculiar antiprismatic structures with chlorine and water,