The advent of accessible automated tools — 3d printers, laser cutters, CNC-controlled mills, vinyl cutters, et.c. — that through the Maker movement reaches commodity prices opens up a number of new approaches to art: especially algorithmic and mathematical art works. The computational control allows us to write algorithms to generate concrete physical art; and their precision allows a higher resolution than what the eye can discern.
In my mathematical art I seek to reify the abstract, to make mathematical concepts and shapes available to touch, to trace, to twist and turn. To create physical artifacts meant for interaction to bring the complex closer and make the abstract concrete.
Most will know the Klein bottle as a bottle-like shape, where the top swings over, goes through a side and connects to the bottom - the iconic shape widely available. That shape emphasizes the Klein bottle as a glued cylinder.
The most immediate alternative construction starts from a Möbius strip, and glues two of them together along the boundary. An emphasis on this construction highlights the gluing circle as the center from which the entire structure emanates.
My Circular Klein Bottle demonstrates the embedding in the circular Möbius construction, built to pick up and inspect, to trace the geometry and topology.
The Roman Surface is an immersion of the real projective plane into 3-dimensional space. Using the Veronese embedding, often seen early when studying algebraic geometry, this immersion combines topology with algebraic geometry into a single tangible representation.
Topologically, the projective plane can be produced by gluing sides of a square together. This operation can produce both the torus and the klein bottle depending on how many of the gluing lines are reflected before gluing -- the projective plane is the case when everything that can be reflected is.
The self-intersections that the projective plane forces in any immersion can be seen from the way the grid self-intersects allowing us to trace the intersections closely.