In 2019, my Bridges paper and my submission to the Art Exhibit are inspired by a 18-inch diameter sand-cast sculpture by Charles O. Perry – which he called “Star Cinder.” This is an “Orderly Tangle” (thank you, Alan Holden!) in which ten triangular frames with rounded corners interlink with icosahedral symmetry. These ten loops then form the border curves of a soap-film-like, 2-manifold surface suspended in this configuration. This surface is confined to the outermost 30% of its circumsphere. My goal is to extend this type of intricate surface closer towards the center of the sphere. I am creating a suitable tangle of border curves by replacing the triangular loops with more complicated interlinked torus knots.
This is the most complex "Star Cinder" design that I have constructed so far. Like Perry’s sculpture, it is based on the icosahedral tangle of ten triangular loops. But for this sculpture, the original (3,1)-torus-knot loops have been replaced with (3,3)-torus-knots, which correspond to three mutually interlinked circles. Thus, the whole configuration consists of a tangle of 30 circles – one each for every edge of a dodecahedron. It was quite challenging to connect a single 2-manifold “soap-film” surface between those 30 border curves, and slight changes in the tilt of the circles may result in quite different soap films. This solution has five concentric, radially coupled shells, each with twelve pentagonal openings.
This is one of the simplest structures in this series of extended, multi-level "Star Cinders." It is derived from an interlinked tangle of only four triangular loops, arranged with octahedral symmetry. The triangle frames, which are topologically equivalent to (3,1)-torus-knots, are now replaced with four (3,2)-torus-knots – also known as Trefoil-knots. This provides additional border-curve segments near the center of the sphere. The resulting “soap-film” has the symmetry of an oriented cube. It exhibits two distinct levels of laterally connected lobes that roughly follow the edges of a cube.