Carlo Sequin

Professor of Computer Science
University of California, Berkeley
Berkeley, CA, USA

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.

Tetra-Cluster of Seven Klein-Bottles
15 x 15 x 15 cm
3D-print, PLA plastic

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.

Dodecahedral-Cluster of 25 Klein-Bottles
24 x 24 x 24 cm
3D-print, PLA plastic

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.