# Carlo Sequin

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

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.

Emulation of Charles Perry’s “Tetra” sculpture
20 x 15 x 15 cm
ABS plastic, shaped by Fused Deposition Modeling
2015

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.

Tetra_4-half-twists
20 x 15 x 15 cm
ABS plastic, shaped by Fused Deposition Modeling, hand-painted
2015

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.