Kepler–Poinsot polyhedron
The small and great stellated dodecahedron have nonconvex regular pentagram faces.
Such lines of intersection are not part of the polyhedral structure and are sometimes called false edges.
For example, the small stellated dodecahedron has 12 pentagram faces with the central pentagonal part hidden inside the solid.
The visible parts of each face comprise five isosceles triangles which touch at five points around the pentagon.
We could treat these triangles as 60 separate faces to obtain a new, irregular polyhedron which looks outwardly identical.
Because of this, they are not necessarily topologically equivalent to the sphere as Platonic solids are, and in particular the Euler relation does not always hold.
Two relationships described in the article below are also easily seen in the images: That the violet edges are the same, and that the green faces lie in the same planes.
John Conway defines the Kepler–Poinsot polyhedra as greatenings and stellations of the convex solids.
Greatening maintains the type of faces, shifting and resizing them into parallel planes.
If the intersections are treated as new edges and vertices, the figures obtained will not be regular, but they can still be considered stellations.
(See Golden ratio) (The midradius is a common measure to compare the size of different polyhedra.)
Traditionally the two star polyhedra have been defined as augmentations (or cumulations), i.e. as dodecahedron and icosahedron with pyramids added to their faces.
Kepler calls the small stellation an augmented dodecahedron (then nicknaming it hedgehog).
E.g. MathWorld states that the two star polyhedra can be constructed by adding pyramids to the faces of the Platonic solids.
If they were, the two star polyhedra would be topologically equivalent to the pentakis dodecahedron and the triakis icosahedron.
In the great dodecahedron and its dual all faces and vertices are on 5-fold symmetry axes (so there are no yellow elements in these images).
The table below shows orthographic projections from the 5-fold (red), 3-fold (yellow) and 2-fold (blue) symmetry axes.
A small stellated dodecahedron appears in a marble tarsia (inlay panel) on the floor of St. Mark's Basilica, Venice, Italy.
[8] It is clear from the general arrangement of the book that he regarded only the five Platonic solids as regular.
[9] He obtained them by stellating the regular convex dodecahedron, for the first time treating it as a surface rather than a solid.
He noticed that by extending the edges or faces of the convex dodecahedron until they met again, he could obtain star pentagons.
Each has the central convex region of each face "hidden" within the interior, with only the triangular arms visible.
Kepler's final step was to recognize that these polyhedra fit the definition of regularity, even though they were not convex, as the traditional Platonic solids were.
In 1809, Louis Poinsot rediscovered Kepler's figures, by assembling star pentagons around each vertex.
Three years later, Augustin Cauchy proved the list complete by stellating the Platonic solids, and almost half a century after that, in 1858, Bertrand provided a more elegant proof by faceting them.
The following year, Arthur Cayley gave the Kepler–Poinsot polyhedra the names by which they are generally known today.
A hundred years later, John Conway developed a systematic terminology for stellations in up to four dimensions.
A small stellated dodecahedron is depicted in a marble tarsia on the floor of St. Mark's Basilica, Venice, Italy, dating from ca.
Norwegian artist Vebjørn Sand's sculpture The Kepler Star is displayed near Oslo Airport, Gardermoen.
