Kepler–Poinsot polyhedron
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.[1]
They may be obtained by stellating the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic faces or vertex figures. They can all be seen as three-dimensional analogues of the pentagram in one way or another.
Characteristics[edit]
Sizes[edit]
The great icosahedron edge length is times the original icosahedron edge length. The small stellated dodecahedron, great dodecahedron, and great stellated dodecahedron edge lengths are respectively and times the original dodecahedron edge length.
Non-convexity[edit]
These figures have pentagrams (star pentagons) as faces or vertex figures. The small and great stellated dodecahedron have nonconvex regular pentagram faces. The great dodecahedron and great icosahedron have convex polygonal faces, but pentagrammic vertex figures.
In all cases, two faces can intersect along a line that is not an edge of either face, so that part of each face passes through the interior of the figure. Such lines of intersection are not part of the polyhedral structure and are sometimes called false edges. Likewise where three such lines intersect at a point that is not a corner of any face, these points are false vertices. The images below show spheres at the true vertices, and blue rods along the true 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. Each edge would now be divided into three shorter edges (of two different kinds), and the 20 false vertices would become true ones, so that we have a total of 32 vertices (again of two kinds). The hidden inner pentagons are no longer part of the polyhedral surface, and can disappear. Now Euler's formula holds: 60 − 90 + 32 = 2. However, this polyhedron is no longer the one described by the Schläfli symbol {5/2, 5}, and so can not be a Kepler–Poinsot solid even though it still looks like one from outside.
Euler characteristic χ[edit]
A Kepler–Poinsot polyhedron covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the figures which have pentagrammic faces, and the vertices in the others. Because of this, they are not necessarily topologically equivalent to the sphere as Platonic solids are, and in particular the Euler relation
The stellated dodecahedra[edit]
Hull and core[edit]
The small and great stellated dodecahedron
can be seen as a regular and a great dodecahedron with their edges and faces extended until they intersect.
The pentagon faces of these cores are the invisible parts of the star polyhedra's pentagram faces.
For the small stellated dodecahedron the hull is times bigger than the core, and for the great it is times bigger.
(See Golden ratio)
(The midradius is a common measure to compare the size of different polyhedra.)
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