Taneli Luotoniemi
The geometric concept of the fourth spatial dimension has had a lasting effect on Western culture and science. Kinetic models based on the 4-dimensional regular polychora would be beneficial for understanding these structures, but are difficult to implement because of the distortions resulting from the ‘flattening’ required. Bitruncated versions of the polychora are, however, possible to realize as kinetic models by employing hyperbolic patchwork surfaces composed of cloth hexagons. The 4-dimensional rotations of the polychoron can be visualized as a partial ‘inside-out turning’ of the surface – pulling neighboring cells out through the openings. As a result the object changes its color.
The bitruncation of the pentachoron (5-cell) results in a polychoron composed of 10 truncated tetrahedra. When the triangular faces are removed, the remaining 20 hexagons form a closed surface. This surface is hyperbolic with 4 hexagons meeting at a vertex. Topologically it is a torus with 6 handles. The 4-dimensional rotations takes us to 5 different cells, each of its own color.