Carlo Sequin
As in previous years, my art submissions again complement my Bridges
conference presentations. The first entry is a non-orientable surface
of genus 8; it relates to the paper "From Klein Bottles to Modular
Super-Bottles" which describes a building-block approach for the
construction of 2-manifolds of higher genus by forming the connected
sum of multiple Klein bottles.
The second entry shows two tightly intertwined tetrahedral frames; it
relate to the paper “Interlinking Polyhedral Wire-Frames” which
investigates in how many topologically different ways two identical
wire-frames, representing cubes or tetrahedra, can be interlinked.
I designed a set of parts that can be combined in different ways to form single-sided, as well as double-sided 2-manifolds of higher genus that have some aesthetic appeal regardless of their mathematical message. The parts are tubular components that combine the geometry of the classical Klein-bottle mouth with a 3-way or 4-way junction of pipe segments. These modules are connected with a small set of curved toroidal pipe segments. Depending on how the modular components are combined to form these “Super-Bottles”, the resulting surfaces may be single-sided or double-sided. The sculpture depicted here uses three 4-way junction modules and six curved connectors. The resulting surface is single-sided and of genus 8 with six borders.
There are 11 known topologically different interlinkings of two tetrahedral frames. For all of them we have determined the maximum thickness of the edge tubes that do not yet cause intersections. In this simple configuration with 6-fold (D3d) symmetry, only three face-loops in each tetrahedron are linked with a loop in the other frame. A tight fit results when the pipe diameter is 14.96% of the tetrahedron edge length. The PVC pipes are joined together at the corners with custom-made connector pieces fabricated on an inexpensive 3D printer.