I'm a mathematician who is interested in visualising different topological shapes. I was originally attracted to mathematical art by the possibilities offered by 3D-printing. However, I quickly realised that many printable shapes can also be crocheted. Nowadays I create shapes with several different materials, but I still find crocheting the easiest way of making complex shapes.
Three different embeddings of the Möbius band into R^3.
The shape on the left resembles the classical paper strip model but it is mathematically rather different. It has a simple parametrisation, its centre forms a circle, and it is a non-developable and ruled surface.
The one sided shell in the middle is a Möbius band with perfectly circular boundary. It is known as a Sudanese Möbius band or a Möbius snail.
The rightmost model is an embedding that has a perfectly circular boundary in the middle. The surface should extend to infinity, but to save some time and yarn I stopped when a nice planet shape was reached.
The fist two embeddings are homeomorphic with the classical paper strip model.