Dodecahedral star

Dodecahedral star (12 points)

The dodecahedron star (also called the small star dodecahedron ) is a regular polyhedron and belongs to 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 116.57 °.

The dodecahedron star has 12 points, the corresponding " antister " 20 points. Each point of this star surrounds (equidistant) 5 other points, with Antister there are only 3.

Origin & properties

If all the edges of a pentagon dodecahedron are extended beyond its corners until five of them intersect at a point, a dodecahedron star is created, which can be imagined as a dodecahedron with twelve pyramids on top . The points of the dodecahedron star form the corner points of a regular icosahedron with the side length . ${\displaystyle v={\frac {a}{2}}\left(3+{\sqrt {5}}\right)=a+s}$

The dodecahedron star is the circumscribed body of twelve mutually intersecting pentagrams which are coincident with the boundary surfaces of a dodecahedron. Inside the dodecahedron star there are 20 equilateral triangles, cut surfaces, the side length of which is the same as the pentagram diagonal, and which span a large icosahedron .

• The surfaces of the dodecahedral and icosahedral stars are the same, with the latter including the smaller volume.
• Pentakis dodecahedron and dodecahedron star are topologically equivalent.

Formulas

Sizes of a dodecahedral star with edge length a
volume	${\displaystyle V={\frac {5}{4}}a^{3}\left(7+3{\sqrt {5}}\right)}$
Surface area	${\displaystyle A_{O}=15a^{2}{\sqrt {5+2{\sqrt {5}}}}}$
Umkugelradius	${\displaystyle R={\frac {a}{4}}{\sqrt {50+22{\sqrt {5}}}}}$
Pyramid height	${\displaystyle k={\frac {a}{5}}{\sqrt {25+10{\sqrt {5}}}}}$
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  ≈ 116 ° 33 ′ 54 ″	${\displaystyle \cos \,\alpha =-{\frac {1}{5}}{\sqrt {5}}}$

Sculpture in the form of a dodecahedral star

Sculpture dodecahedron star , in Vienna IX., Alluding to the place name and the Morgenstern weapon

On the occasion of the adaptation of the building at Oskar-Morgenstern-Platz 1 in Vienna-Alsergrund, which was previously used by the pension insurance company, for the faculties of mathematics and economics at the University of Vienna , the sculpture dodecahedron star was installed. The surface represents the solution of an equation in 3 dimensions and, with its 20 curved tips, resembles a dodecahedral antister.

Based on an idea and a design by Herwig Hauser from the Faculty of Mathematics, the twenty individual parts of the star sculpture were made of fiberglass composite material (GRP) . The individual segments of the self-supporting GRP structure are screwed together using flanges. The pull cables for the suspension are connected to an internal steel frame.

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 dodecahedron be with a designated.
3. ↑ dodekaederstern.cc

Web links

• Eric W. Weisstein : Dodecahedral star . In: MathWorld (English).
• Kepler's dodecahedron star - the hedgehog
• Alexandra Fritz: Platonic stars - Construction of algebraic curves and surfaces with prescribed symmetries and singularities Diploma thesis, University of Innsbruck, December 11, 2009. (English)