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.
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
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.
Margareta Séquin 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.