My 2018 presentation pays homage to Charles O. Perry. I study his
"Topological Ribbon Sculptures" and analyze the mathematical
classification of these 2-manifold surfaces. I also analyze the graphs
that result, when all the ribbons are skeletonized into simple, thin
wires. Perry’s ribbon sculptures represent 1-sided as well as 2-sided
surfaces, with a wide variation in their genus and their numbers of
borders. But, surprisingly, the skeleton graphs of all the sculptures
that I have studied turn out to be planar, i.e., they can be drawn
without any self-intersections on a flat sheet of paper.
Using custom CAD software, I have reproduced some of his sculptures and have also created new models, similar in style, but with new topologies.
This is the result of my attempt to make a Perry-style sculpture based on the simplest non-planar graph: the "Utility Graph," also known as the bipartite graph K_3,3. I found a nice symmetrical way to embed this graph in 3D Euclidean space, by starting with a simple trefoil knot and adding three cross-connecting struts at the points where the typical depiction of this knot would show three crossings. A curled cross section, which is found in many of Perry’s ribbon sculptures, is swept along a trefoil path to form a knotted ribbon. The highly curved, cross-connecting struts form the type of loopy T-intersections with this ribbon that are also prominent features in Perry's “Continuum” in Washington, DC.
In this sculpture, I follow Perry’s approach that he used in most
of his topological sculptures: Nice smooth 3D space curves are
first defined (e.g., with the help of steel cables) and these then
form the borders of the final 2-manifold. With the borders defined
in space, a nice surface is fitted in between; it often assumes
the shape of a minimal surface, as one finds in soap films
suspended between wires.
Here, I start with four identical trefoil loops and place them onto the faces of a tetrahedron, letting them interlink across the edges of the tetrahedron. Then I fill in a soap-film like surface defined by the given trefoil borders. The result is a single-sided surface of genus 10 with (of course) four borders.