For several years, I have admired the ceramic creations and metal
sculptures by Eva Hild. They not only enchant me with their
free-flowing organic beauty; they also intrigue the mathematical part
of my mind. How many tunnels and border curves are there? What might
be their genus. For a topologist, many of Hild’s surfaces represent
challenging exercises in surface classification.
Analyzing many images of her sculptures, I was surprised that I could not find any single-sided ones; they are all orientable! Would a non-orientable surface look quite different and stand out from her portfolio?
Here I present two models of single-sided 2-manifold sculptures that use some of the geometrical elements found in Eva Hild’s creations.
The key building block is a “4-stub Dyck funnel”, a disk with a
pair of tubular extrusions emerging from both sides. Six of these
elements have been placed at right angles to the 6 edges of a
tetrahedron; they are mutually interconnected with 12 tunnels.
This yields a surface of genus 14 – the equivalent of the
connected sum of 7 Klein bottles with 6 punctures, exhibiting the
12-fold symmetry of the oriented tetrahedron.
This model maximizes symmetry – in stark contrast to Eva Hild’s sculptures, which exhibit more organic forms and avoid perfect geometrical symmetry.
In this expanded second sculpture, 24 of the “4-stub Dyck funnels”
have been aligned with the 24 edges of a rhombic dodecahedron, and
their stubs have been connected with 48 tunnels. This yields a
surface of genus 50 – the equivalent of the connected sum of 25
Klein bottles with 24 punctures, exhibiting the 24-fold symmetry
of the oriented cube.
It took 132 hours to build this model on a LULZBOT 3D-printer. Support removal took several more hours.