Carlo Sequin

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

I work on the boundary between Art and Mathematics. Sometimes I create artwork by using mathematical procedures; at other times I enhance a mathematical visualization model to the point where it becomes a piece of art. For the art exhibit at Bridges 2013 my submissions complement my oral presentation: "Cross-Caps -- Boy Caps -- Boy Cups" and provide visualization models made from three different materials: paper, wool, and ABS plastic. A compact model of a punctured projective plane is equivalent to a Moebius band. If this Moebius band is warped through itself so that it exhibits a circular rim in the form of an annulus, cone, or cylinder, that shape can inspire unusual designs of pouches, drinking vessels, hats, or caps.

3" x 6" x 6"
VisiJet® Proplast Plastic Material

Eight Boy caps join to form a non-orientable surface of genus 8 with octahedral symmetry. It is quite
challenging to realize this complex, single-sided, self-intersecting structure on today's rapid prototyping
machines. To realize the complete spherical shell in an effective and less expensive way, it should be
built as two half-shells. Here is a first half-shell realized on a ProJet HD 3000 layered manufacturing
machine. The CAD file for this object is 23Mb! A second half-shell will be built, so that the whole
octahedral genus-8 surface can be displayed at the Bridges 2013 Mathematical Art Gallery.

Then the viewers may wonder, whether this shape represents some strange virus or the structure of the
expanding universe...

Cubist Boy Cap on Mirror
Cubist Boy Cap on Mirror
4" x 8" x 7"
Printed Paper

When a Boy surface (a compact, smooth model of the projective plane) is punctured, it becomes a Moebius band. The single rim of this band can be un-warped and stretched into the rim of an annulus, forcing the surface to self-intersect on the inside. It is possible to impose a 6-fold D3 symmetry onto this Boy cap configuration. To make the construction of the paper model as simple as possible, its geometry has been approximated with a "cubist" polyhedron. The result has been place on a mirror so that the viewer can compare upper and lower sides of this surface.

Knitted Cross Cap
Knitted Cross Cap
7" x 5" x 5"

Margareta SĂ©quin created this prototype of a knitted version of a punctured cross surface (a compact model of the projective plane), called a cross cap, which also serves as a warm, attention-grabbing skull cap for outdoor adventures.