Icosahedral star

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 . ${\displaystyle s={\frac {a}{2}}\left(1+{\sqrt {5}}\right)}$

• 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	${\displaystyle V={\frac {5}{4}}a^{3}\left(3+{\sqrt {5}}\right)}$
Surface area	${\displaystyle A_{O}=15a^{2}{\sqrt {5+2{\sqrt {5}}}}}$
Umkugelradius	${\displaystyle R={\frac {a}{4}}{\sqrt {3}}\left(3+{\sqrt {5}}\right)}$
Pyramid height	${\displaystyle k={\frac {a}{6}}{\sqrt {3}}\left(3+{\sqrt {5}}\right)}$
Ridge length	${\displaystyle s={\frac {a}{2}}\left(1+{\sqrt {5}}\right)}$
Pentagram diagonal	${\displaystyle c=a\left(2+{\sqrt {5}}\right)=a+2s}$
Face angle  ≈ 63 ° 26 ′ 6 ″	${\displaystyle \cos \,\alpha ={\frac {1}{5}}{\sqrt {5}}}$

Remarks

1. ↑ Let the sides of the triangle be denoted by a (base) and s ( side) .
2. ↑ ^{a b} The edge length of the inscribed icosahedron be with a designated.

Web links

• Eric W. Weisstein : Icosahedral star . In: MathWorld (English).