Kepler-Poinsot polyhedron

related topics
{math, energy, light}
{@card@, make, design}
{work, book, publish}
{line, north, south}
{water, park, boat}

In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic faces or vertex figures.



Kepler-Poinsot solids.svg

The four Kepler–Poinsot polyhedra are illustrated above. Each is identified by its Schläfli symbol, of the form {p, q}, and by its name. One face of each figure is shown yellow and outlined in red.



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 golden balls at the true vertices, and silver 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 the Euler relation 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.

Full article ▸

related documents
Wave plate
Electric charge
Elongation (astronomy)
Mössbauer effect
Ionization potential
Fresnel equations
Infrared astronomy
Sidereal time
Pioneer 11
Eclipse cycle
Large Magellanic Cloud
Celestial pole
Exotic atom
Johnson solid
Orbital period
Strong interaction
Yarkovsky effect
Kristian Birkeland
Titius–Bode law
Atomic nucleus
Radio astronomy
Standing wave ratio