Editing Regular polyhedron (section)

=== Regular star polyhedra ===

Regular star polygons such as the [[pentagram]] (star pentagon) were also known to the ancient Greeks – the [[pentagram]] was used by the [[Pythagoreans]] as their secret sign, but they did not use them to construct polyhedra. It was not until the early 17th century that [[Johannes Kepler]] realised that pentagrams could be used as the faces of regular [[star polyhedron|star polyhedra]]. Some of these star polyhedra may have been discovered by others before Kepler's time, but Kepler was the first to recognise that they could be considered "regular" if one removed the restriction that regular polyhedra be convex. Two hundred years later [[Louis Poinsot]] also allowed star [[vertex figure]]s (circuits around each corner), enabling him to discover two new regular star polyhedra along with rediscovering Kepler's. These four are the only regular star polyhedra, and have come to be known as the [[Kepler–Poinsot polyhedra]]. It was not until the mid-19th century, several decades after Poinsot published, that Cayley gave them their modern English names: (Kepler's) [[small stellated dodecahedron]] and [[great stellated dodecahedron]], and (Poinsot's) [[great icosahedron]] and [[great dodecahedron]].

The Kepler–Poinsot polyhedra may be constructed from the Platonic solids by a process called [[stellation]]. The reciprocal process to stellation is called [[facetting]] (or faceting). Every stellation of one polyhedron is [[Dual polyhedron|dual]], or reciprocal, to some facetting of the dual polyhedron. The regular star polyhedra can also be obtained by facetting the Platonic solids. This was first done by Bertrand around the same time that Cayley named them.

By the end of the 19th century there were therefore nine regular polyhedra – five convex and four star.