Icosahedral star
The icosahedron star (also called the large star dodecahedron ) is a regular polyhedron and is one of the Kepler star bodies ; it is bounded by 60 isosceles triangles . The star body is characterized by the equality of all surface angles - both inside and outside - of 63.44 °. A fold of the star made of paper is the Bascetta star .
Origin & properties
If all edges of an icosahedron are extended beyond its corners until three of them intersect at a point, an icosahedron star is created, which can be imagined as an icosahedron with 20 pyramids on top . The points of the icosahedron star form the corner points of a regular dodecahedron with the length of the side .
- The great stellated dodecahedron is the circumscribed body of twelve mutually intersecting pentagrams , which coincide with the pentagonal cut surfaces of an icosahedron are
- Triakis icosahedron and icosahedron star are topologically equivalent.
- The surfaces of the dodecahedral star and icosahedral star are the same, with the former enclosing the larger volume.
Formulas
Sizes of an icosahedral star with edge length a | |
---|---|
volume | |
Surface area | |
Umkugelradius | |
Pyramid height | |
Ridge length | |
Pentagram diagonal | |
Face angle ≈ 63 ° 26 ′ 6 ″ |
Remarks
- ↑ Let the sides of the triangle be denoted by a (base) and s ( side) .
- ↑ a b The edge length of the inscribed icosahedron be with a designated.
Web links
- Eric W. Weisstein : Icosahedral star . In: MathWorld (English).