Andreas Groß, Peter Hilgers
Karl Andreas Groß, (born 1977) is a digital artist from Germany. His work can be described as surreal, with an abstract touch of dark emotions. Coming from a professional 3D modelling background, he uses both fotographic, as well as digital 3D content to create his artworks. He has worked with several national and international cutomers on both personal and commercial projects.
Peter Hilgers is a retired engineer, amateur in mathematics and computer hobbyist. He has always been fascinated by geometry. Current computers and sophisticated programs make it possible to create, study and visualise complex structures. For him this is a constant source of challenges and delight.
William Hamilton invented a game requiring to find a closed route along the edges of a dodecahedron, visiting all vertices exactly once. The problem has been generalized to other graphs. Such a route is called a Hamiltonian path.
André Sainte-Laguë showed that there are 17 topologically different Hamiltonian paths on an icosahedron, each passing 12 edges .
Felix Klein studied the rotation group of the icosahedron.
Here we combine these three topics: The Hamiltonian paths are generated by elements of the rotation group. We match two of the 17 paths in such a way that no edges are used twice.
1: André Sainte-Laguë: Avec des nombres et des lignes (Recréations mathématiques), Paris 1937
The symmetries of the dodecahedron are a classical example for the application of group theory. Rotations of three different types preserve the orientation of the object: These are the rotations around the axes defined by the centers of opposing faces, vertices or edges. The symmetry group has 60 elements, 15 of them are edge rotations.
The dodecahedron contains 5 inscribed cubes, that is there are 5 possibilities to select 8 vertices of the dodecahedron to define a cube. Imagine the rotating dodecahedron in a dark room with 8 small lights at those vertices.
The collage plays with the simulated effect caused by long time exposure in photography. All selected "shots" of edge rotations are taken from the same "camera position".