Stimulated by the LEGO-Knot project, I aimed to design a set of modular parts that permits to compose not only various handle-bodies, but also single-sided surfaces of higher genus. The modular parts employed in my sculptures are tubular 3-way junctions, where one of the tubular stubs exposes the opposite side of the surface shown by the other two stubs. Depending on how the parts are connected, the resulting compositions remains orientable or becomes single-sided; in the latter case they correspond to sums of multiple Klein bottles; which I call “Super-Bottles.” For a few special assemblies, the resulting surface remains 2-sided (σ = 2); the genus then drops to half the value that it would be for the single-sided Super-Bottle.

Artworks

The eight parts demonstrate four different ways in which one leg of a tubular 3-way junction can be made to switch surface orientation. The eight parts can be put together in hundreds of different ways. In most cases the resulting surface is single-sided (σ = 1), but in a few cases it is still double-sided (σ = 2). The genus of the resulting surface is 10/σ. The configuration shown is a non-orientable surface of genus 10, corresponding to the connected sum of five Klein Bottles, with a total of 12 punctures.

The two identical parts of which the sculpture is composed can be put together in three different ways. In two cases the resulting surface is single-sided (σ = 1) and in the third case it is double-sided (σ = 2). The genus of the resulting surface is 4/σ. The configuration shown is a non-orientable surface of genus 4, corresponding to the connected sum of two Klein Bottles, with two punctures.
The insets show the two individual parts, and an assembly of them resulting in a 2-hole torus of genus 2 (with two punctures).