"I was a somnambule guided by Plato’s world of ideas, dreaming of Plato’s grandmother, drawing circles in the sand and now I am drawing circles into silicon."
The pictures presented here were worked out in 3D animated geometry during the Quantum Cinema project (funded by FWF) with a group of scientists and digital artists. We were examining the geometrical properties of 5-dimensional space and its significance in quantum physics.
Currently I am pursuing my doctoral studies „On the Hermeneutics of the Penrose Tilings".
My finding, the 3D representation of the Penrose kites and darts (P2), the unit cell of 5-dimensional space, named "epitahedron" is exhibited here in different configurations and media as prints and sculpture.
The Epita-dodecahedron is a composition of 12 polyhedra. Each of the pentagonal faces is filled with one heptahedron, named epitahedron.
It exhibits space filling properties in 4D space which can be iterated ad infinitum - cf.Penrose Patterns - uniting dodecahedral and icosahedral symmetries.
This picture is inspired by Henri Poincaré's description of the homology sphere: instead of the pentagons, the epitahedra are twisted against each other in counter-movements. It is the first visual access to the resulting different evolving symmetries, crystallizing in steps of 36 degrees, as foretold by Poincaré in 1904. Here the central lines of the edges are exposing the Petersen graph which is used for the description of quantum structures.
The dodecahedron can be created as a boundary shape of the faces of two intersecting heptahedra, named epitahedra, in the framework of an icosahedron. It is not mentioned by Plato, because it does not fit into his concept of triangles and its discovery is ascribed only to his student Theatetos, who first proved that there are only five regular convex polyhedra, the platonic solids, the shapes of the "four elements".
Plato added instead an obscure remark about the fifth element: ‘[…] since there is another fifth configuration, the god used it for the ornamentation of the whole universe’. In fact epitahedra are revealing one of his most beautiful triangles in projection and can therefor be regarded as Plato's 5th element,
The Complex Center of the Epita-dodecahedron
24" x 24"
Print on paper (3D animated geometry, still frame)
In the above epita-dodecahedron the 12 apices are intersecting each other in a way that a four-dimensional analogue of the dodecahedron arises, which contains naturally another dodecahedron within a complex intersecting space configuration in the center.
This picture shows the center with a complex intersecting space configuration consisting of four different solids: another dodecahedron and its dual the icosahedron, the icosidodecahedron and a stellated icosidodecahedron.
This 4-dimensional dodecahedron unites the dodecahedral and icosahedral symmetries and can be related to the 120-cell and the 600-cell (cf. drawing by Alicia Boole Scott around1900. This print exposes the central forms in overlapping transparent layers.
