# What is the value of 81 factorials

## Red dodecahedron star for the University of Vienna

**An interesting cross-over project in the fields of science, mathematics and plastics technology will be realized on Thursday, November 28th, in front of the new building of the Faculty of Mathematics and the Faculty of Economics of the University of Vienna at Oskar-Morgenstern-Platz 1: The sculpture " Dodecahedral star ". With a diameter of four meters and an installation height of five meters, this figure visualizes all the solutions to a special mathematical equation. This and the mathematical theory of the star were developed by Alexandra Fritz as part of a diploma thesis in the Algebraic Geometry group by Herwig Hauser, professor at the Faculty of Mathematics at the University of Vienna. **

The star can serve as a symbol for both faculties - mathematics as well as economics - for scientific curiosity and ambition: to understand the structure of complex issues in depth and to reproduce their essential aspects in a coherent form and to apply them successfully.

**Mathematical background of the figure**

The symmetrical star shape of the figure results from the interplay of the mathematical areas of invariant theory, algebraic geometry and singularity theory. The surface of the sculpture is a real, compact algebraic surface with isolated singularities in the twenty points. Their symmetry group is that of the dodecahedron, that is, that Platonic solid that consists of twelve regular pentagons. The meeting points of the 20 individual parts of the sculpture - one part per point - lie at the center points of the surface of the dodecahedron, i.e. describe the corners of the icosahedron, which is dual to it. The curvature of the star between the tips is regulated by a parameter c, which can be freely selected in the equation. The value c = 0 corresponds to the sphere, positive values produce outwardly directed peaks, with negative values the peaks point inward (anti-star). The value c = 81 was chosen for the figure. The attraction of the sculpture, in addition to its immediately appealing aesthetic, is the fact that it can be defined by a single, albeit complicated, algebraic equation. It is:

5c (2φ - 3) (x

^{2}- φ

^{2}y

^{2}) (y

^{2}- φ

^{2}z

^{2}) (e.g.

^{2}- φ

^{2}x

^{2}) = (1− (x

^{2}+ y

^{2}+ z

^{2}))

^{3}- 5/27 c (x

^{2}+ y

^{2}+ z

^{2})

^{3}

Each solution to this equation corresponds to a point in three-dimensional space. The totality of the solutions forms a surface, namely the surface of the star. The geometric structure is thus clearly defined by the algebraic equation.

**Manufacture and financing of the dodecahedral star**

The twenty individual parts of the star sculpture are made of fiberglass composite material (GRP). To produce the parts, a negative mold is first coated with colored gelcoat. Glass fabric and glass mats, as well as the core material, are then built up in layers and impregnated with resin and hardened. The segments are screwed to one another via flanges and sealed. The GRP structure is self-supporting. The star is suspended by pulling ropes that attach to an internal steel frame. Production: Steiner Kunststofftechnik, Ried im Traunkreis. The project was financed with the kind support of Raiffeisen-Holding NÖ-Wien and the University of Vienna.

**Oskar-Morgenstern-Platz: the new location of the University of Vienna**

At the beginning of the winter semester, the Faculty of Mathematics and the Faculty of Economics at the University of Vienna moved into their new joint building on the Danube Canal. The renaming of the location address "Oskar-Morgenstern-Platz 1" (formerly Rossauer Lände) commemorates the great Austrian scientist Oskar Morgenstern. His research area, game theory, is located at the interface between economics and mathematics. The renaming of the two faculties reflects the scientific proximity of these two faculties.

There are 800 workplaces on a net floor area of around 30,000 square meters. Numerous modern lecture halls, seminar, teaching and work rooms as well as a separate library area and a cafeteria are available in the newly adapted building for the 7,500 students of both subjects. A meeting and conference area for up to 160 people was created on the 12th floor. The new location is owned by Raiffeisen-Holding NÖ-Wien and is leased to the University of Vienna on a long-term basis. The building was planned by ARGE Maurer, Neumann + Partner under the specifications of the University of Vienna and implemented in the period from September 2011 to August 2013.

**More information at:**

**www.dodekaederstern.cc**

**Scientific contact**

Univ.-Prof. Mag. Dr. Herwig Hauser

Faculty of Mathematics

University of Vienna

1090 Vienna, Oskar-Morgenstern-Platz 1

T + 43-1-4277-506 90

herwig.hauser (at) univie.ac.at

http://homepage.univie.ac.at/herwig.hauser/

**Consultation notes**

Mag. Veronika Schallhart

Press office of the University of Vienna

Research and Teaching

1010 Vienna, Universitätsring 1

T + 43-1-4277-175 30

M + 43-664-602 77-175 30

veronika.schallhart (at) univie.ac.at

### Scientific contact

#### Univ.-Prof. Mag. Dr. Herwig Hauser

Faculty of MathematicsUniversity of Vienna

1090 - Vienna, Oskar-Morgenstern-Platz 1

+43-1-4277-506 90

[email protected]

### Consultation notice

#### Mag. Veronika Schallhart

Press office of the University of Vienna*Research and Teaching*

1010 - Vienna, Universitätsring 1

+43-1-4277-175 30

+43-664-602 77-175 30

[email protected]

### Downloads:

Roter_Dodekaederstern.jpg

Visualization_Dodekaeder_Steiner.jpg

Dodekaeder_OMP_provisorisch_aufnahm.jpg

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