Large icosahedron

Large icosahedron

Cardboard model from the University of Tübingen (around 1860)

The large icosahedron is a regular polyhedron and belongs to the Kepler star bodies ; it is bordered by 60 isosceles triangles and 120 irregular triangles .

construction

The basic body is the dodecahedral star . The great icosahedron is the result of 20 mutually intersecting equilateral triangles found in the dodecahedron star; the triangles intersect at an angle of ≈ 70.5 ° (or ≈ 109.5 ° supplementary ). Two triangles meet at one of their edges and form a "rib angle" of ≈ 41.8 °. This star body is quasi a reduced, "scraped off" dodecahedron star, whereby the 60 sections have the shape of irregular tetrahedra .

Formulas

Sizes in the cut out tetrahedron

Sizes of a large icosahedron with edge length a
volume	${\displaystyle V={\frac {a^{3}}{4}}\left(25+9{\sqrt {5}}\right)}$
Surface area	${\displaystyle A_{O}=3a^{2}{\sqrt {3}}\left(5+4{\sqrt {5}}\right)}$
Umkugelradius	${\displaystyle R={\frac {a}{4}}{\sqrt {50+22{\sqrt {5}}}}}$
Pyramid height	${\displaystyle k={\frac {a}{10}}{\sqrt {10+2{\sqrt {5}}}}}$
1. Burr length	${\displaystyle s={\frac {a}{2}}\left(1+{\sqrt {5}}\right)}$
2. Ridge length	${\displaystyle t={\frac {a}{10}}{\sqrt {2}}\left(5+3{\sqrt {5}}\right)}$
3. Ridge length	${\displaystyle u={\frac {a}{5}}{\sqrt {10}}}$
1. Face angle  ≈ 109 ° 28 ′ 16 ″	${\displaystyle \cos \,\alpha _{1}=-{\frac {1}{3}}}$
2. Face angle  ≈ 70 ° 31 ′ 44 ″	${\displaystyle \cos \,\alpha _{2}={\frac {1}{3}}}$
3. Face angle  ≈ 41 ° 48 ′ 37 ″ ('rib')	${\displaystyle \cos \,\alpha _{3}={\frac {1}{3}}{\sqrt {5}}}$

Remarks

1. ↑ Let the sides of the triangle be designated with a (base) and u (leg).
2. ↑ ^{a b} The edge length of the inscribed dodecahedron be with a designated.
3. ↑ Let the ridge lengths of the tetrahedron be denoted by t (long side) and u (short side).

Web links

• Eric W. Weisstein : Large icosahedron . In: MathWorld (English).