# Carlo Sequin

My 2015 submissions complement my Bridges conference presentations.
The first two entries relate to: "2-Manifold Sculptures," describing a
topological analysis of different sculptures by various artists that
are thin, complex, single-sided or double-sided surfaces. In
particular, they focus on Charles Perry’s “Tetra” sculpture, which can
be seen as a tetrahedral frame with gracefully curved ribbons forming
the tetrahedron edges. A model of Perry’s original geometry is
contrasted with a variation that results from changing the amount of
twist on some of the ribbons.

The third entry relates to “Introducing the Möbius-Twisted Turk’s Head
Knot,” discussing how this classical cylinder-shaped braid can be
turned into a Möbius band.

One of the basic patterns that shows up in many different forms in the rawhide weavings found in the gear used around a horse is a flat braid that is formed into a wreath or a cylindrical ring by means of a single “thong.” Our challenge (jointly with Lorenzo Larrucea) was to find a way to weave such a braid in the shape of a Möbius band. Our entry shows the result woven from paper strips that join to form a single contiguous thong that is rainbow-colored in the longitudinal direction, passing once around the complete hue circle.

In this variation, the twist of the four twisted tetrahedron edges has been reduced from 360 degrees to 180 degrees. Surprisingly, this does not produce a single-sided surface, but it changes the genus of the original sculpture from g = 0 to g = 1, because it reduces the number of borders from four to only two (which are different from each other). Two make the two sides of the sculpture and its two different edges clearly visible, the FDM model has been enhanced by painting these features with different colors.

Charles Perry’s “Tetra” sculpture is modeled as a subdivision surface connecting four interlocking circular borders. It can also be understood as four Y-shaped junctions at the corners of a tetrahedron, connected with six ribbons, four of which exhibit a twist of 360 degrees, while the other two ribbons are untwisted. The result is an orientable, double-sided surface of Euler characteristic χ = -2 with four separate border loops.